HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS · Reza Salehi and Ali Shadrokh The bias reduction...

HACETTEPE UNIVERSITY

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HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

A Bimonthly Publication

Volume 47 Issue 4

2018

ISSN 1303 5010

HACETTEPE JOURNAL OF

MATHEMATICS AND

STATISTICS

Volume 47 Issue 4

August 2018

A Peer Reviewed Journal

Published Bimonthly by the

Faculty of Science of Hacettepe University

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ISSN 1303 5010

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Hacettepe Journal of Mathematics and Statistics

Cilt 47 Say� 4 (2018)

ISSN 1303 � 5010

KÜNYE

YAYININ ADI:

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS

YIL : 2018 SAYI : 47 - 4 AY : A§ustos

YAYIN SAHBNN ADI : H.Ü. Fen Fakültesi Dekanl�§� ad�na

Prof. Dr. F. Güler Ekmekçi

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Hacettepe Journal of Mathematics and Statistics

A Bimonthly Publication � Volume 47 Issue 4 (2018)

ISSN 1303 � 5010

EDITORIAL BOARD

Honorary Editor

Lawrence Micheal Brown

Editors in Chief

MathematicsAy³e Çi§dem Özcan (Hacettepe University - Mathematics - [email protected])

StatisticsÇa§da³ Hakan Alada§ (Hacettepe University - Statistics - [email protected])

Associate Editors

Bülent Saraç (Hacettepe University- Mathematics - [email protected])Asl� Y�ld�z (Hacettepe University- Mathematics - [email protected])Ceren Eda Can (Hacettepe University-Statistics - [email protected])

Production Editors

Kerime Kall� (Hacettepe University - Mathematics - [email protected])O. O§ulcan Tuncer (Hacettepe University - Mathematics - [email protected])Nurbanu Bursa (Hacettepe University - Statistics - [email protected])

Area Editors

Mathematics

Evrim Akalan (Algebra-Associative rings and modules,[email protected])Gary F. Birkenmeier (Algebra-Associative rings and algebras, [email protected])Okay Çelebi (Partial Di�erential Equations, [email protected])Angel del Rio (Algebra-Group Theory, [email protected])Gülin Ercan (Algebra-Group Theory, [email protected])Varga Kalantarov (Appl. Math., [email protected])Cihan Orhan (Analysis, [email protected])Murat Özayd�n (Di�erential Geometry, Global Analysis,[email protected])Abdullah Özbekler (App. Math., [email protected])Ekin Özman (Number Theory,Algebraic Geometry, [email protected])Mehmetçik Pamuk (Topology-Manifolds and Cell Complexes, [email protected])Ivan Reilly (Topology, [email protected])Alexander P. ostak (Analysis, [email protected])

Statistics

Ali Allahverdi (Operational research statistics, [email protected])Olcay Arslan (Robust statistics, [email protected])Narayanaswany Balakrishnan (Applied Statistics, Theory of Statistics, [email protected])

Sat Gupta (Sampling, Time Series, [email protected])Birdal eno§lu (Experimental Design, Statistical Distributions, [email protected])Tahir Hanalio§lu (Stochastic Processes Theory, Probability Theory, [email protected])Zeynep I³�l Kalayl�o§lu (Bayesian Inference, Model Selection, [email protected])

Published by Hacettepe University, Faculty of Science

CONTENTS

Mathematics

Murat Bodur, Fatma Ta³delen and Gülen Ba³canbaz-Tunca

On multivariate Lupa³ operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783

Gülen Ba³canbaz-Tunca, Ay³egül Erençin and Hatice Gül nce-larslan

Bivariate Cheney-Sharma operators on simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793

Muhammad Ahsan Binyamin, Ha�z Muhammad Afzal Siddiqui andAmir Shehzad

A combinatorial approach to the classi�cation of resolution graphs of weighted

homogeneous plane curve singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

Asena Çetinkaya and Ya³ar Polato§lu

q− Harmonic mappings for which analytic part isq− convex functions of complex order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .813

Muhey U Din, Hari Mohan Srivastava and Mohsan Raza

Univalence of Certain Integral Operators Involving

Generalized Struve functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .821

Manal Ghanem and Khalida Nazzal

Some properties of the total graph and regular graph of

a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

M. Habil Gürsoy and lhan c.en

Coverings of structured Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Khizar Hayat, Muhammad Irfan Ali, Bing-Yuan Cao andFaruk Karaaslan

New results on type-2 soft sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

Faqir Muhammad and M. Kazim Khan

Basic sequences and unbiased estimation in quasi

power series distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877

Sevda Sa§�ro§lu and Mehmet Ünver

Statistical convergence of sequences of sets in hyperspaces . . . . . . . . . . . . . . . . . . . 889

Co³kun Yakar and Mehmet Arslan

Terminal value problems with causal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897

Statistics

Raghunath Arnab and Antonio Arcos

Jackknife variance estimation from complex survey designs . . . . . . . . . . . . . . . . . .909

Mehmet Ali Balc�

Hierarchies in Communities of Borsa Istanbul Stock Exchange . . . . . . . . . . . . . . 921

Gauss M. Cordeiro, Morad Alizadeh, Rodrigo B. Silva and Thiago G.Ramires

A new wider family of continuous models:

The Extended Cordeiro and de Castro Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

Ekaette Inyang Enang and Emmanuel John Ekpenyong

On the optimal search for e�cient estimators of population mean in simple random

sampling in the presence of an auxiliary variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963

Juanfang Liu, Liugen Xue and Ruiqin Tian

Generalized empirical likelihood inference in partially linear errors-in-variables

models with longitudinal data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983

Reza Salehi and Ali Shadrokh

The bias reduction in density estimation using a geometric

extrapolated kernel estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003

Yuzhu Tian, Aijun Yang, Erqian Li and Maozai Tian

Parameters estimation for mixed generalized inverted exponential distributions

with type-II progressive hybrid censoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023

Mingqiu Wang, Xiuli Wang and Muhammad Amin

Identi�cation and estimation for generalized varying

coe�cient partially linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041

Vali Zardasht

A bootstrap test for symmetry based on quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . .1061

MATHEMATICS

Hacettepe Journal of Mathematics and StatisticsVolume 47 (4) (2018), 783 � 792

On multivariate Lupa³ operators

Murat Bodur∗†, Fatma Ta³delen‡ and Gülen Ba³canbaz-Tunca

Abstract

This paper is primarily concerned with multivariate Lupa³ operator.We demonstrate that multivariate Lupa³ operator preserves the prop-erties of the general function of modulus of continuity, Lipschitz's con-stant and order of a Lipschitz continuous function. Moreover, we obtainmonotonicity of the multivariate Lupa³ operator under the conditionthat the original function is convex. Lastly, two modi�ed extensionsare constructed.

Keywords: Lupa³ operator, convexity, Lipschitz continuity, function of modulus

of continuity, monotonic convergence.

Mathematics Subject Classi�cation (2010): 41A25, 41A36

Received : 09.05.2017 Accepted : 17.07.2017 Doi : 10.15672/HJMS.2017.500

1. Introduction

Recall the following identity

(1− a)−α =∞∑

k=0

(α)kk!

ak, |α| < 1,

where (.)k is the Pochammer's symbol de�ned as

(α)0 = 1 (α 6= 0) ,(α)k = (α) (α+ 1) ... (α+ k − 1) , k ∈ N.(1.1)

Taking α = nx (x ≥ 0) in this identity, Lupa³ constructed the linear positive operatorsgiven by

∗Ankara University, Faculty of Science, Department of Mathematics, 06100, Tando-gan, Ankara, Turkey, Email: [email protected]†Corresponding Author.‡Ankara University, Faculty of Science, Department of Mathematics, 06100, Tandogan,

Ankara, Turkey Email: [email protected] University, Faculty of Science, Department of Mathematics, 06100, Tandogan,

Ankara, Turkey Email: [email protected]

784

Ln(f ;x) = (1− a)nx∞∑

k=0

(nx)kk!

akf(k

n), x ≥ 0,

for suitable real valued functions f de�ned on [0,∞), n ∈ N [9]. In [1], Agratini consideredthe special case of the operators Ln by taking a =

12in its formula to get the property

of preservation of linear functions. Namely, he considered the following linear positiveoperators

(1.2) Ln,1(f ;x) = 2−nx

∞∑

k=0

(nx)k2kk!

f(k

n), n ∈ N,

for real valued, continuous and bounded functions on [0,∞) so that Ln,1 (1;x) = 1, andLn,1 (t;x) = x. So, it is clear that

2nx =

∞∑

k=0

(nx)k2kk!

.

We simply call the operatos Ln,1 as (univariate) Lupa³ operators. For some classicalapproximation results related to Lupa³ operators (1.2) we refer to [1], [6], [7], [11]. Inthis work, among others, we especially are interested in the multivariate extension of theLupa³ operators.

Let D ⊂ Rm, m ∈ N, denote the setD = {x =(x1, x2, ..., xm) ∈ Rm : 0 ≤ xi

785

1.2. De�nition. A continuous, real valued, nonnegative function ω de�ned in D is saidto be a modulus of continuity, if the ensuing are satis�ed.

i) ω (0) = ω (0, 0, ..., 0) = 0,ii) ω (u) is nondecreasing, namely ω (u) ≤ ω (v) whenever u ≤ v,iii) ω (u) is sub-additive, namely ω (u+ v) ≤ ω (u) + ω (v) ,for all u = (u1, u2, ..., um) , and v = (v1, v2, ..., vm) ∈ D.

1.3. De�nition. A continuous function from D ⊂ Rm into R is said to be Lipschitzcontinuous of order α, α ∈ (0, 1], if there exists a constant M > 0 such that f satis�es

|f (x)− f (y)| ≤Mm∑

i=1

|xi − yi|α

for every x = (x1, x2, ..., xm) , y = (y1, y2, ..., ym) ∈ D. The set of Lipschitz continuousfunctions is denoted by LipM (α,D) .

Let f be a real valued continuous function on D. Then n−th multivariate Lupa³operator Ln,mf (with m−dimension) can be de�ned as

(1.3) Ln,m (f ;x) = 2−n|x|

∞∑

k=0

(nx)k2|k|k!

f

(k

n

)

for x ∈D and n ∈ N. It is not to di�cult to see that Ln,m (1;x) = 1 and for f (t) =ti, t ∈D, Ln,m (ti;x) = xi, i = 1, 2, ...,m.

Motivated from the excellent work of Cao, Ding and Xu as for multivariate Baskakovoperator [5], in this work, apart from the approximation properties of Ln,m, we deal withsome classical shape properties that the multivariate Lupa³ operator Ln,m satis�es. Weshow that the operator Ln,m preserves the properties of a general function of modulusof continuity, Lipschitz's constant and order of a Lipschitz continuous function, and alsothat the monotonic convergence of the sequence of the multivariate Lupa³ operators Ln,m,when the attached function is convex. It is clear that the case m = 1 gives the results of[6]. On the other hand, similar results for multivariate Szász-Mirakyan operator, whichhas a close similarity with the Lupa³ operators, were obtained in [10]. Note that a briefhistory for this kind of approach for some univariate operators can also be reached in [5].

2. Shape Preserving Properties

Firstly, we study the monotonicity of the sequence of multivariate Lupa³ operatorsLn,m (f ;x) de�ned by (1.3) from inspiring [6].

2.1. Theorem. Let f be a convex function de�ned on D. Then Ln,mf is monotonicallynonincreasing in n for all n.

Proof. For simplicity, we take m = 2. The proof for higher dimensions will be similar.From the de�nition of Ln (f ;x) , with taking into account of multivariate notation, we

786

have

Ln,m (f ;x)− Ln+1,m (f ;x)

= 2−n|x|−|x|{2|x|

∞∑

k=0

(nx)k2|k| k!

f

(k

n

)−∞∑

k=0

((n+ 1)x)k2|k| k!

f

(k

n+ 1

)}

= 2−n|x|−|x|

2|x|

∞∑

k1=0

∞∑

k2=0

(nx1)k12k1 k1!

(nx2)k22k2 k2!

f

(k1n,k2n

)(2.1)

−∞∑

k1=0

∞∑

k2=0

((n+ 1)x1)k12k1 k1!

((n+ 1)x2)k22k2 k2!

f

(k1

n+ 1,k2

n+ 1

) .

Using the facts 2xi =∞∑li=0

(xi)li2li li!

, i = 1, 2, then the summations in the bracket in (2.1)

reduce to

∞∑

l1=0

∞∑

k1=0

∞∑

l2=0

∞∑

k2=0

(x1)l12l1 l1!

(nx1)k12k1 k1!

(x2)l22l2 l2!

(nx2)k22k2 k2!

f

(k1n,k2n

)(2.2)

−∞∑

k1=0

∞∑

k2=0

((n+ 1)x1)k12k1 k1!

((n+ 1)x2)k22k2 k2!

f

(k1

n+ 1,k2

n+ 1

).

Replacing ki with ki − li, i = 1, 2, then (2.2) gives rise to

∞∑

l1=0

∞∑

l2=0

∞∑

k1=l1

∞∑

k2=l2

(x1)l1 (x2)l2l1! l2!2l1+l2

(nx1)k1−l1 (nx2)k2−l22k1+k2−l1−l2 (k1 − l1)! (k − l2)!

×f(k1 − l1n

,k2 − l2n

)

−∞∑

k1=0

∞∑

k2=0

((n+ 1)x1)k1 ((n+ 1)x2)k22k1+k2 k1! k2!

f

(k1

n+ 1,k2

n+ 1

).

Changing the order of the summations and replacing ki − li, with li, i = 1, 2, then thelast formula reduces to

∞∑

k1=1

∞∑

k2=1

1

2k1+k2

k1∑

l1=0

k2∑

l2=0

(nx1)l1 (nx2)l2 (x1)k1−l1 (x2)k2−l2l1! l2! (k1 − l1)! (k2 − l2)!

f

(l1n,l2n

)

−((n+ 1)x1)k1 ((n+ 1)x2)k2

k1! k2!f

(k1

n+ 1,k2

n+ 1

)}

+

∞∑

k1=1

1

2k1

k1∑

l1=0

(nx1)l1 (x1)k1−l1l1! (k1 − l1)!

f

(l1n, 0

)−

((n+ 1)x1)k1k1!

f

(k1

n+ 1, 0

)

+

∞∑

k2=1

1

2k2

k2∑

l2=0

(nx2)l2 (x2)k2−l2l2! (k2 − l2)!

f

(0,l2n

)−

((n+ 1)x2)k2k2!

f

(0,

k2n+ 1

)

+f ((0, 0))− f ((0, 0)) .

787

Denoting

I : =

k1∑

l1=0

k2∑

l2=0

(nx1)l1 (nx2)l2 (x1)k1−l1 (x2)k2−l2l1! l2! (k1 − l1)! (k2 − l2)!

f

(l1n,l2n

)

−((n+ 1)x1)k1 ((n+ 1)x2)k2

k1! k2!f

(k1

n+ 1,k2

n+ 1

)},

I1 : =

k1∑

l1=0

(nx1)l1 (x1)k1−l1l1! (k1 − l1)!

f

(l1n, 0

)−

((n+ 1)x1)k1k1!

f

(k1

n+ 1, 0

),

I2 : =

k2∑

l2=0

(nx2)l2 (x2)k2−l2l2! (k2 − l2)!

f

(0,l2n

)−

((n+ 1)x2)k2k2!

f

(0,

k2n+ 1

),

it su�cies to show that I, I1, and I2 are nonnegative.For I1, let

αl1 :=

(k1l1

)(nx1)l1 (x1)k1−l1((n+ 1)x1)k1

≥ 0

and

xl1 =

(l1n, 0

), l1 = 0, 1, ..., k1.

Then, as in [6], using the formula ((n+ 1)xi)ki =ki∑li=0

(kili

)(nxi)li (xi)ki−li , i = 1, 2, It

readily follows that

k1∑

l1=0

αl1 = 1

and

k1∑

l1=0

αl1xl1 =

(k1

n+ 1, 0

).

Hence, convexity of f gives that

f

(k1

n+ 1, 0

)≤ 1

((n+ 1)x1)k1

k1∑

l1=0

(nx1)l1 (x1)k1−l1

(k1l1

)f

(l1n, 0

),

which implies that I1 ≥ 0. The case I2 ≥ 0 is obtained in a similar way by taking

βl2 :=

(k2l2

)(nx2)l2 (x2)k2−l2((n+ 1)x2)k2

≥ 0.

and

xl2 =

(0,l2n

), l2 = 0, 1, ..., k2.

Similarly, it holds

k2∑

l2=0

βl2 = 1,

788

and

k2∑

l2=0

βl2xl2 =

(0,

k2n+ 1

),

so, the result I2 ≥ 0 follows from convexity of f. Finally, for I, we may take

α′l1 := αl1

k2∑

l2=0

βl2 ≥ 0

as nonnegative constants satisfyingk1∑l1=0

(αl1

k2∑l2=0

βl2

)= 1, and x′l1 =

(l1n, l2n

), where

αl1 and βl2 are the same as given above. So, from the convexity of f we reach to thefollowing.

k1∑

l1=0

k2∑

l2=0

(k1l1

)(k2l2

)(nx1)l1 (nx2)l2 (x1)k1−l1 (x2)k2−l2

((n+ 1)x1)k1 ((n+ 1)x2)k2f

(l1n,l2n

)

≥ f

k1∑

l1=0

k2∑

l2=0

(k1l1

)(k2l2

)(nx1)l1 (nx2)l2 (x1)k1−l1 (x2)k2−l2

((n+ 1)x1)k1 ((n+ 1)x2)k2

(l1n,l2n

)

= f

(k1

n+ 1,k2

n+ 1

),

which shows that I ≥ 0. This completes the proof. �

2.2. Theorem. Let ω be a function of modulus of continuity. Then, for each n ∈N, Ln,mω is also a function of modulus of continuity.

Proof. Let x =(x1, x2, ..., xm) , y =(y1, y2, ..., ym) ∈ D, and x ≤ y. Then, taking multi-variate notation into consideration, we have

Ln,m (ω ;y) = 2−n|y|

∞∑

k=0

(ny)k2|k|k!

ω

(k

n

)

= 2−n|y|∞∑

k=0

(n (x+ (y − x)))k2|k|k!

ω

(k

n

)

= 2−n|y|∞∑

k1=0

∞∑

k2=0

...

∞∑

km=0

k1∑

i1=0

k2∑

i2=0

...

km∑

im=0

1

i!2|k| (k− i)!

×(nx)i (n(y − x))k−i ω(k1n,k2n, ...,

kmn

)

for i =(i1, i2, ..., im) ∈ Nm0 . Changing the order of the summations and taking kr − ir =jr, r = 1, 2, ...,m, we can easily obtain

(2.3) Ln,m (ω ;y) = 2−n|y|

∞∑

i=0

∞∑

j=0

1

i!j!2|i+j|(nx)i (n(y − x))j ω

(i+ j

n

),

789

where j =(j1, j2, ..., jm) . On the other hand, for x ∈ D we have

Ln,m (ω ;x) = 2−n|x|

∞∑

i=0

(nx)i2|i|i!

ω

(i

n

)

= 2−n|(y−(y−x))|∞∑

i=0

(nx)i2|i|i!

ω

(i

n

)

= 2−n|y|2n|y−x|∞∑

i=0

(nx)i2|i| i!

ω

(i

n

)

= 2−n|y|∞∑

i=0

(nx)i2|i| i!

∞∑

j=0

(n(y−x))j2|j|j!

ω

(i

n

)

= 2−n|y|∞∑

i=0

(nx)i2|i| i!

∞∑

j=0

(n (y − x))j2|j|j!

ω

(i

n

).(2.4)

If we subtract (2.3) from (2.4), we arrive at

(2.5)

Ln,m (ω ;y)−Ln,m (ω ;x) = 2−n|y|∞∑

i=0

∞∑

j=0

(nx)i2|i| i!

(n (y − x))j2|j|j!

[ω

(i+ j

n

)− ω

(i

n

)].

Using the sub-additivity property of ω, we have

Ln,m (ω;y)− Ln,m (ω;x) ≤ 2−n|y|∞∑

i=0

∞∑

j=0

(nx)i2|i| i!

(n (y − x))j2|j|j!

ω

(j

n

)

= 2−n|x|∞∑

i=0

(nx)i2|i|i!

2−n|y−x|∞∑

j=0

(n (y − x))j2|j|j!

ω

(j

n

)

= 2−n|y−x|∞∑

j=0

(n (y − x))j2|j|j!

ω

(j

n

)

= Ln,m (ω;y − x) .

The last inequality shows the sub-additivity of Ln,m. Moreover, from (2.5) we easilyobtain that Ln,m (ω;x) ≤ Ln,m (ω;y) when x ≤ y. Obviously, this gives that Ln,m isnondecreasing. Finally, from the de�nition of the operators Ln,m, Ln,m (ω;0) = ω (0) =0 is obvious. Therefore Ln,mω is also a function of modulus of continuity, this completesthe proof. �

Here, we present the preservation of the Lipschitz constant and order of a Lipschitzcontinuous function by the multivariate Lupa³ operators Ln,m.

2.3. Theorem. Let f ∈ LipM (α,D) . Then Ln,mf ∈ LipM (α,D).

790

Proof. Assume that x ≤ y. Then, writing (2.5) for f, we get|Ln,m (f ;y)− Ln,m (f ;x)|

≤ 2−n|y|∞∑

i=0

∞∑

j=0

(nx)i2|i|i!

(n (y − x))j2|j|j!

∣∣∣∣f(i+ j

n

)− f

(i

n

)∣∣∣∣

≤ M2−n|x|∞∑

i=0

(nx)i2|i|i!

∞∑

j=0

(n (y − x))j2|j|j!

2−n(|y|−|x|)m∑

k=1

(jkn

)α

= M2−n|y−x|∞∑

j=0

(n (y − x))j2|j|j!

m∑

k=1

(jkn

)α

= M {Ln,1 (tα1 ; y1 − x1) + Ln,1 (tα2 ; y2 − x2) + ...+ Ln,1 (tαm; ym − xm)}by the hypothesis that f ∈ LipM (α,D) , where i =(i1, i2, ..., im) and j =(j1, j2, ..., jm) belongto Nm0 . Following the same steps of the proof of Theorem 2 in [6] for each sum in the lastformula, we reach to

|Ln,m (f ;y)− Ln,m (f ;x)|

≤ Mm∑

k=1

|yk − xk|α ,

which shows that Ln,mf ∈ LipM (α,D) . In a similar way, we can show the case whenx ≥ y. Finally, we can consider the case that x1 ≥ y1, x2 ≥ y2, ..., xi−1 ≥ yi−1, xi+1 ≥yi+1, ..., xm ≥ ym, and xi ≤ yi. Since (y1, y2, ..., yi−1, xi, yi+1,..., ym) ∈ D, then we obtainfrom the above arguments that

|Ln,m (f ;y)− Ln,m (f ;x)|≤ |Ln,m (f ; (y1, y2, ..., ym))− Ln,m (f ; (y1, y2, ..., yi−1, xi, yi+1,..., ym))|

+ |Ln,m (f ; (y1, y2, ..., yi−1, xi, yi+1,..., ym))− Ln,m (f ; (x1, x2, ..., xm))|

≤ Mm∑

k=1

|yk − xk|α .

Clearly, if the last case holds for more than one components, then the result followssimilarly. �

3. Extensions

So as to gain approximation process in space of integrable functions, we propose newintegral modi�cation, i.e. the Kantorovich type [1]:

L∗n,m(f,x) :=∞∑

k=0

Pn,k(x)φn,k(f), (n ∈ N) .

where

Pn,k(x) := 2−n|x| (nx)k

2|k|k!

and

φn,k(f) := nm

k1+1n∫

k1n

k2+1n∫

k2n

...

km+1n∫

kmn

f(t1, t2, ...tm)dtm...dt2dt1.

791

A number of solution can be found for multivariate Lupa³-Kantorovich operator, whichhas been identi�ed above, about shape preserving and approximation properties. How-ever, the objective in this study is to give de�nition only.

Now, we establish n−th multivariate generalized Lupa³ operator to investigate andunderstand their properties reckon weighted approximation in [8]. A lot of work has beendone in this regard, some are [3], [4], [2].

Let ρ be a function de�ned on R+ := [0,∞) and have the following properties:(ρ1) ρ is a continuously di�erentiable function on R+,(ρ2) ρ(0) = 0, inf

x∈R+ρ′(x) ≥ 1.

These conditions ensure that ρ is strictly increasing and the inverse ρ−1 (x) of ρ existson R+. For example, ρ(x) = x + x2 is a function which is given from [2] satis�es theconditions (ρ1) and (ρ2). Let f be a real valued continuous function de�ned on D, whichis explained above for multivariate Lupa³ operator and ρ (x) , x =(x1, ..., xm) ∈ D denotea function acting from the set D onto D such that each component of which is given byρ(xi), 1 ≤ i ≤ m, namely

ρ (x) := (ρ(x1), ρ(x2), ..., ρ(xm)) .

Denoting the inverse of ρ by ρ−1, which means that

ρ−1 (x) :=(ρ−1(x1), ρ

−1(x2), ..., ρ−1(xm)

),

Then n−th multivariate generalized Lupa³ operator Lρn,mf is de�ned as

(3.1) Lρn,m (f ;x) = 2−n|ρ(x)|

∞∑

k=0

(f ◦ ρ−1

)(kn

)(nρ (x))k2|k|k!

,

where(f ◦ ρ−1

) (kn

):= f

(ρ−1

(k1n

), ρ−1

(k2n

), ..., ρ−1

(kmn

)), n ∈ N.

Here, we recall the extended version of the notion of the ρ−convexity, due to Aralet.al [2], to the multivariate case as in [3].

3.1. De�nition. A continuous, real valued function f is called as ρ−convex in D, iff ◦ ρ−1 is convex in the sense of De�nition 1.1.

We need the following de�nition for the generalized Lipschitz class used, for bivariatecase, in [3].

3.2. De�nition. A continuous function from D ⊂ Rm into R is said to be ρ−Lipschitzcontinuous of order α, α ∈ (0, 1], if there exists a constant M > 0 such that f satis�es

|f (x)− f (y)| ≤Mm∑

i=1

|ρ (xi)− ρ (yi)|α

for every x = (x1, x2, ..., xm) , y = (y1, y2, ..., ym) ∈ D.The set of ρ−Lipschitz continuous functions is denoted by LipρM (α,D) .Note that clearly when ρ (x) = x one obtaines the multivariate Lupa³ operators given

by (1.3).It can be shown that the multivariate generalized Lupa³ operators Lρn,mf preserve

some properties.The following two theorems can be given in the light of all these;

3.3. Theorem. Let f be a ρ−convex function de�ned on D. Then Lρn,m is monotonicallynondecreasing in n.

Proof. The theorem can be proved similar with multivariate Lupa³ operator so it can beomitted. �

792

3.4. Theorem. Let f ∈ LipρM (α,D) , 0 < α ≤ 1. Then Lρn,m (f ;x) ∈ LipρM (α,D).Proof. Also, omitted. �

References

[1] Agratini, O. On a sequence of linear and positive operators, Facta Univ. Ser. Math. Inform.14 41-48, 1999.

[2] Aral, A., Inoan, D. and Ra³a, I. On the generalized Szász-Mirakyan operators, Results Math.65 no. 3-4, 441-452, 2014 .

[3] Ba³canbaz-Tunca,G., nce-larslan, H. G. and Erençin, A. Bivariate Bernstein type opera-tors, Appl. Math. Comput., 273 543-552, 2016.

[4] Cárdenas-Morales, D., Garrancho, P. and Rasa, I. Bernstein-type operators which preservepolynomials, Compt. Math. Appl. 62 158-163, 2011.

[5] Cao,F., Ding, C. and Xu, Z. On Multivariate Baskakov operator, J. Math. Anal. Appl. 307no. 1, 274-291, 2005.

[6] Erençin, A., Ba³canbaz-Tunca, G. and Ta³delen Ye³ildal, F. Some properties of the operatorsde�ned by Lupa³, Revue D'Analyse Numérique et de Théorie de L'Approximation, 43 no.2, 158-164, 2014.

[7] Finta,Z. Quantitative estimates for some linear and positive operator, Studia Univ. Babe³-Bolyai Math. 47 no. 3, 71-84, 2002.

[8] Gadjiev, A.D. and Aral, A. The estimates of approximation by using a new type of weightedmodulus of continuity, Computers and Mathematics with Applications 54 127-135, 2007 .

[9] Lupa³, A. The approximation by some positive linear operators In: Proceedings of the Inter-national Dortmund Meeting on Approximation Theory (M.W. Müller et al.,eds.),AkademieVerlag, Berlin, 201-229, 1995.

[10] Olgun, A. Some properties of the Multivariate Szász Operators, Comptes rendus de l'Academie bulgare des Sciences, 65, no. 2, 139-146, 2012.

[11] Sofonea, D. F. On a sequence of linear and positive operators, Results Math. 53 no. 3-4,435-444, 2009.


Bivariate Cheney-Sharma operators on simplex

Gülen Ba³canbaz-Tunca∗, Ay³egül Erençin†‡ and Hatice Gül nce-larslan

Abstract

In this paper, we consider bivariate Cheney-Sharma operators whichare not the tensor product construction. Precisely, we show that theseoperators preserve Lipschitz condition of a given Lipschitz continuousfunction f and also the properties of the modulus of continuity functionwhen f is a modulus of continuity function.

Keywords: Lipschitz continuous function, modulus of continuity function, bi-variate Cheney-Sharma operators.

Mathematics Subject Classi�cation (2010): 41A36


1. Introduction

The most celebrated linear positive operators for the uniform approximation of con-tinuous real valued functions on [0, 1] are Bernstein polynomials. As it is well known,besides approximation results, Bernstein polynomials have some nice retaining proper-ties. The most referred study in this direction was due to Brown, Elliott and Paget [7]where they gave an elementary proof for the preservation of the Lipschitz constant andorder of a Lipschitz continuous function by the Bernstein polynomials. Whereas, Lindvallpreviously obtained this result in terms of probabilistic methods in [20]. Moreover, in[19] Li proved that Bernstein polynomials also preserve the properties of the function ofmodulus of continuity. The same problems for some other type univariate or multivariatelinear positive operators were solved by either using an elementary or probabilistic way(see, e.g. [3]-[6], [8], [9], [14], [15], [17], [18], [28]).

∗Ankara University, Faculty of Science, Department of Mathematics, 06100, Tando§an,Ankara, Turkey,Email : [email protected]†Abant zzet Baysal University, Faculty of Arts and Science, Department of Mathematics,

14280, Bolu, Turkey,Email : [email protected]‡Corresponding Author.Gazi University, Faculty of Arts and Science, Department of Mathematics, 06500,

Teknikokullar, Ankara, TurkeyEmail : [email protected]

794

In Abel-Jensen identity (see [2], p.326)

(1.1) (u+ v) (u+ v +mβ)m−1 =m∑

k=0

(m

k

)u (u+ kβ)k−1 v [v + (m− k)β]m−k−1

where u, v and β ∈ R, by taking u = x, v = 1 − x and m = n, Cheney-Sharma [11]introduced the following Bernstein type operators for f ∈ C[0, 1], x ∈ [0, 1] and n ∈ N,

Qβn (f ;x) = (1 + nβ)1−n

n∑

k=0

f

(k

n

)(n

k

)x (x+ kβ)k−1

× (1− x)[1− x+ (n− k)β

]n−k−1,

where β is a nonnegative real parameter. For these operators, tensor product of themand their some generalizations we can cite the papers [1], [10], [12], [21]-[27] and themonograph [2]. Remark that from [11] and [21] we know that Qβn operators reproduceconstant functions and linear functions. Very recently, in [6] the authors showed that uni-variate Cheney-Sharma operators preserve the Lipschitz constant and order of a Lipschitzcontinuous function as well as the properties of the function of modulus of continuity.

We now introduce the notations, some needful de�nitions and the construction of thebivariate operators.Throughout the paper, we shall use the standard notations given below.Let x = (x1, x2) ∈ R2, k = (k1, k2) ∈ N20 , e = (1, 1), 0 = (0, 0), 0 ≤ β ∈ R and n ∈ N.We denote as usual

|x| := x1 + x2, xk := xk11 xk22 , |k| := k1 + k2, k! := k1!k2!, βx = (βx1, βx2)

and(n

k

):=

n!

k!(n− |k|)! ,∑

|k|≤n:=

n∑

k1=0

n−k1∑

k2=0

.

We also denote the two dimensional simplex by

S :={x = (x1, x2) ∈ R2 : x1, x2 ≥ 0, |x| ≤ 1

}.

Moreover, x ≤ y stands for xi ≤ yi, i = 1, 2.We now construct the non-tensor product Cheney-Sharma operators. From (1.1) it is

clear that

(1 + nβ)n−1 =n∑

k1=0

(n

k1

)x1 (x1 + k1β)

k1−1 (1− x1)

× [1− x1 + (n− k1)β]n−k1−1 .

In (1.1), taking u = x2, v = 1− x1 − x2 and m = n− k1 we have

(1− x1) [1− x1 + (n− k1)β]n−k1−1

=

n−k1∑

k2=0

(n− k1k2

)x2 (x2 + k2β)

k2−1 (1− x1 − x2)

× [1− x1 − x2 + (n− k1 − k2)β]n−k1−k2−1 .

795

Using this result in the above equality we �nd that

(1 + nβ)n−1

=

n∑

k1=0

n−k1∑

k2=0

(n

k1

)(n− k1k2

)x1x2 (x1 + k1β)

k1−1 (x2 + k2β)k2−1

× (1− x1 − x2) [1− x1 − x2 + (n− k1 − k2)β]n−k1−k2−1

=∑

|k|≤n

(n

k

)xe(x+ kβ)k−e (1− |x|) [1− |x|+ (n− |k|)β]n−|k|−1

or

1 = (1 + nβ)1−n∑

|k|≤n

(n

k

)xe(x+ kβ)k−e (1− |x|)

× [1− |x|+ (n− |k|)β]n−|k|−1 .In this paper, for a continuous real valued function f de�ned on S we consider the

non-tensor product bivariate extension of the operators Qβn(f ;x) de�ned by

(1.2)Gβn(f ;x) = (1 + nβ)

1−n ∑

|k|≤nf

(k

n

)(n

k

)xe(x+ kβ)k−e

× (1− |x|) [1− |x|+ (n− |k|)β]n−|k|−1

where β is a nonnegative real parameter, x ∈ S and n ∈ N. We observe that forβ = 0 these operators reduce to non-tensor product bivariate Bernstein polynomials (see[13],[16]).

1.1. De�nition. (see, e.g.[9]) A continuous real valued function f de�ned on A ⊆ R2is said to be Lipschitz continuous function of order µ , 0 < µ ≤ 1 on A, if there existsM > 0 such that

|f(x)− f(y)| ≤M2∑

i=1

|xi − yi|µ

for all x,y ∈ A. The set of Lipschitz continuous functions of order µ with Lipschitzconstant M on A is denoted by LipM (µ,A).

1.2. De�nition. (see, e.g.[8]) If a bivariate non-negative and continuous function ω(t)satis�es the following conditions, then it is called a function of modulus of continuity.(a) ω(0) = 0,(b) ω(t) is a non-decreasing function in t, i.e., ω(t) ≥ ω(v) for t ≥ v,(c) ω(t) is semi-additive, i.e., ω(t+ v) ≤ ω(t) + ω(v).

2. Main results

In this section, inspired by the paper of Cao, Ding and Xu [9], including preservationproperties of multivariate Baskakov operators, we show that non-tensor product bivari-ate Cheney-Sharma operators de�ned by Gβn(f ;x) := G

βn(f)(x) preserve the Lipschitz

condition of a given Lipschitz continuous function f and properties of the function ofmodulus of continuity when the attached function f is a modulus of continuity function.

2.1. Theorem. If f ∈ LipM (µ, S), then Gβn(f) ∈ LipM (µ, S) for all n ∈ N.

796

Proof. Let x,y ∈ S such that y ≥ x. From (1.2) we have

Gβn(f ;y) = (1 + nβ)1−n

n∑

i1=0

n−i1∑

i2=0

f

(i

n

)(n

i

)ye (y + iβ)i−e

× (1− |y|) [1− |y|+ (n− |i|)β]n−|i|−1

=(1 + nβ)1−nn∑

i1=0

n−i1∑

i2=0

f

(i

n

)(n

i

)y1 (y1 + i1β)

i1−1

× y2 (y2 + i2β)i2−1 (1− |y|) [1− |y|+ (n− |i|)β]n−|i|−1 .

Setting u = x1, v = y1 − x1, m = i1 and u = x2, v = y2 − x2, m = i2, respectively, in(1.1), we �nd

y1 (y1 + i1β)i1−1 =

i1∑

k1=0

(i1k1

)x1 (x1 + k1β)

k1−1 (y1 − x1)

× [y1 − x1 + (i1 − k1)β]i1−k1−1

and

y2 (y2 + i2β)i2−1 =

i2∑

k2=0

(i2k2

)x2 (x2 + k2β)

k2−1 (y2 − x2)

× [y2 − x2 + (i2 − k2)β]i2−k2−1 .

Therefore,

Gβn(f ;y)

= (1 + nβ)1−nn∑

i1=0

n−i1∑

i2=0

f

(i

n

)(n

i

)i1∑

k1=0

i2∑

k2=0

(i1k1

)(i2k2

)x1x2

× (x1 + k1β)k1−1 (x2 + k2β)k2−1 (y1 − x1) (y2 − x2)× [y1 − x1 + (i1 − k1)β]i1−k1−1 [y2 − x2 + (i2 − k2)β]i2−k2−1

× (1− |y|) [1− |y|+ (n− |i|)β]n−|i|−1

=(1 + nβ)1−nn∑

i1=0

i1∑

k1=0

n−i1∑

i2=0

i2∑

k2=0

f

(i

n

)n!

k!(n− |i|)!(i1 − k1)!(i2 − k2)!

× xe(x+ kβ)k−e(y − x)e [y − x+ (i− k)β]i−k−e

× (1− |y|) [1− |y|+ (n− |i|)β]n−|i|−1 .

797

Changing the order of the above summations and then letting i− k = l we obtain

(2.1)

Gβn(f ;y)

= (1 + nβ)1−nn∑

k1=0

n∑

i1=k1

n−i1∑

k2=0

n−i1∑

i2=k2

f

(i

n

)n!

k!(n− |i|)!(i1 − k1)!(i2 − k2)!

× xe(x+ kβ)k−e(y − x)e [y − x+ (i− k)β]i−k−e

× (1− |y|) [1− |y|+ (n− |i|)β]n−|i|−1

=(1 + nβ)1−nn∑

k1=0

n−k1∑

l1=0

n−k1−l1∑

k2=0

n−|k|−l1∑

l2=0

f

(k+ l

n

)n!

k!l!(n− |k| − |l|)!

× xe(x+ kβ)k−e(y − x)e(y − x+ lβ)l−e

× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Now we consider

Gβn(f ;x) = (1 + nβ)1−n

n∑

k1=0

n−k1∑

k2=0

f

(k

n

)(n

k

)xe(x+ kβ)k−e

× (1− |x|) [1− |x|+ (n− |k|)β]n−|k|−1 .

In (1.1), if we put y1 − x1, 1− y1 − x2 and n− |k| in place of u, v and m, respectively,one has

(1− |x|) [1− |x|+ (n− |k|)β]n−|k|−1

=

n−|k|∑

l1=0

(n− |k|l1

)(y1 − x1)(y1 − x1 + l1β)l1−1(1− y1 − x2)

× [1− y1 − x2 + (n− |k| − l1)β]n−|k|−l1−1 .

Again in the equality (1.1), we replace u, v and m by y2−x2, 1−|y| and m = n−|k|− l1,respectively, we �nd

(1− y1 − x2) [1− y1 − x2 + (n− |k| − l1)β]n−|k|−l1−1

=

n−|k|−l1∑

l2=0

(n− |k| − l1

l2

)(y2 − x2)(y2 − x2 + l2β)l2−1

× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Making use of this in the above equality leads to

(1− |x|) [1− |x|+ (n− |k|)β]n−|k|−1

=

n−|k|∑

l1=0

n−|k|−l1∑

l2=0

(n− |k|

l

)(y − x)e(y − x+ lβ)l−e

× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

798

Thus we conclude that

Gβn(f ;x)

= (1 + nβ)1−nn∑

k1=0

n−k1∑

k2=0

n−|k|∑

l1=0

n−|k|−l1∑

l2=0

f

(k

n

)n!

k!l!(n− |k| − |l|)!


× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Now changing the order of the two summations in the middle, we obtain

(2.2)

Gβn(f ;x)

= (1 + nβ)1−nn∑

k1=0

n−k1∑

l1=0

n−k1−l1∑

k2=0

n−|k|−l1∑

l2=0

f

(k

n

)n!

k!l!(n− |k| − |l|)!


× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

So, from (2.1) and (2.2) it follows that

Gβn(f ;y)−Gβn(f ;x)

= (1 + nβ)1−nn∑

k1=0

n−k1∑

l1=0

n−k1−l1∑

k2=0

n−|k|−l1∑

l2=0

[f

(k+ l

n

)− f

(k

n

)]

× n!k!l!(n− |k| − |l|)!x

e(x+ kβ)k−e(y − x)e(y − x+ lβ)l−e

× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Again interchanging the order of the summations two times successively, we �nd

(2.3)

Gβn(f ;y)−Gβn(f ;x)

= (1 + nβ)1−nn∑

l1=0

n−l1∑

k1=0

n−k1−l1∑

l2=0

n−k1−|l|∑

k2=0

[f

(k+ l

n

)− f

(k

n

)]

× n!k!l!(n− |k| − |l|)!x


× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1

=(1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

n−|l|∑

k1=0

n−k1−|l|∑

k2=0

[f

(k+ l

n

)− f

(k

n

)]

× n!k!l!(n− |k| − |l|)!x


× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Using the fact f ∈ LipM (µ, S) and the following equality

n!

k!l!(n− |k| − |l|)! =(n

l

)(n− |l|k1

)(n− k1 − |l|

k2

)

799

one can write∣∣∣Gβn(f ;y)−Gβn(f ;x)

∣∣∣

≤M (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

n−|l|∑

k1=0

n−k1−|l|∑

k2=0

[(l1n

)µ+

(l2n

)µ]

×(n

l

)(n− |l|k1

)(n− k1 − |l|

k2

)xe(x+ kβ)k−e(y − x)e

× (y − x+ lβ)l−e (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1

=M (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

[(l1n

)µ+

(l2n

)µ](n

l

)(y − x)e

× (y − x+ lβ)l−en−|l|∑

k1=0

(n− |l|k1

)x1(x1 + k1β)

k1−1

×n−k1−|l|∑

k2=0

(n− k1 − |l|

k2

)x2(x2 + k2β)

k2−1

× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Taking u = x2, v = 1− |y| and m = n− k1 − |l| in (1.1), it is easily seen that

(2.4)

(x2 + 1− |y|) [x2 + 1− |y|+ (n− k1 − |l|)β]n−k1−|l|−1

=

n−k1−|l|∑

k2=0

(n− k1 − |l|

k2

)x2(x2 + k2β)

k2−1 (1− |y|)

× [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Hence we can write∣∣∣Gβn(f ;y)−Gβn(f ;x)

∣∣∣

≤M (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

[(l1n

)µ+

(l2n

)µ](n

l

)(y − x)e

× (y − x+ lβ)l−en−|l|∑

k1=0

(n− |l|k1

)x1(x1 + k1β)

k1−1(x2 + 1− |y|)

× [x2 + 1− |y|+ (n− k1 − |l|)β]n−k1−|l|−1 .

Again in (1.1), we replace u, v and m by x1, x2 + 1− |y| and m = n− |l|, respectively,to obtain

(2.5)

(x1 + x2 + 1− |y|) [x1 + x2 + 1− |y|+ (n− |l|)β]n−|l|−1

=(1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1

=

n−|l|∑

k1=0

(n− |l|k1

)x1(x1 + k1β)

k1−1(x2 + 1− |y|)

× [x2 + 1− |y|+ (n− k1 − |l|)β]n−k1−|l|−1 .

800

This leads to

(2.6)

∣∣∣Gβn(f ;y)−Gβn(f ;x)∣∣∣

≤M (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

[(l1n

)µ+

(l2n

)µ](n

l

)(y − x)e

× (y − x+ lβ)l−e(1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1

=M[Gβn (t

µ1 ;y − x) +Gβn (tµ2 ;y − x)

].

Now consider the term Gβn (tµ1 ;y − x). With the help of the equality(

n

l

)=

(n

l1

)(n− l1l2

)

we can write

Gβn (tµ1 ;y − x)

= (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

(l1n

)µ(n

l

)(y − x)e(y − x+ lβ)l−e

× (1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1

=(1 + nβ)1−nn∑

l1=0

(l1n

)µ(n

l1

)(y1 − x1)(y1 − x1 + l1β)l1−1

×n−l1∑

l2=0

(n− l1l2

)(y2 − x2)(y2 − x2 + l2β)l2−1

× (1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1 .In the equality (1.1), if we take y2 − x2, 1 − |y − x| and n − l1 in place of u, v and m,respectively, then we get

[1− (y1 − x1)] [1− (y1 − x1) + (n− l1)β]n−l1−1

=

n−l1∑

l2=0

(n− l1l2

)(y2 − x2)(y2 − x2 + l2β)l2−1(1− |y − x|)

× [1− |y − x|+ (n− |l|)β]n−|l|−1 .Therefore,

Gβn (tµ1 ;y − x)

= (1 + nβ)1−nn∑

l1=0

(l1n

)µ(n

l1

)(y1 − x1)(y1 − x1 + l1β)l1−1

× [1− (y1 − x1)] [1− (y1 − x1) + (n− l1)β]n−l1−1

=Qβn (tµ1 ; y1 − x1) .

Applying the Hölder inequality with conjugate pairs p = 1µand q = 1

1−µ , we �nd

Gβn (tµ1 ;y − x) =Qβn (tµ1 ; y1 − x1)

≤[Qβn(t1; y1 − x1)

]µ [Qβn(1; y1 − x1)

]1−µ.

801

As mentioned before, since the univariate Cheney-Sharma operators given by Qβn repro-duce constant and linear functions we reach to

Gβn (tµ1 ;y − x) ≤ (y1 − x1)µ.

Now in the following equality

Gβn (tµ2 ;y − x) = (1 + nβ)1−n

n∑

l1=0

n−l1∑

l2=0

(l2n

)µ(n

l

)(y − x)e(y − x+ lβ)l−e

× (1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1 ,

if we change the order of summations and use the equality

(n

l

)=

(n− l2l1

)(n

l2

), then

we can write

Gβn (tµ2 ;y − x) = (1 + nβ)1−n

n∑

l2=0

n−l2∑

l1=0

(l2n

)µ(n

l

)(y − x)e(y − x+ lβ)l−e

× (1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1

=(1 + nβ)1−nn∑

l2=0

(l2n

)µ(n

l2

)(y2 − x2)(y2 − x2 + l2β)l2−1

×n−l2∑

l1=0

(n− l2l1

)(y1 − x1)(y1 − x1 + l1β)l1−1

× (1− |y − x|) [1− |y − x|+ (n− |l|)β]n−|l|−1 .

Taking u = y1 − x1, v = 1− |y − x| and m = n− l2 in (1.1), it is clear that

[1− (y2 − x2)] [1− (y2 − x2) + (n− l2)β]n−l2−1

=

n−l2∑

l1=0

(n− l2l1

)(y1 − x1)(y1 − x1 + l1β)l1−1(1− |y − x|)

× [1− |y − x|+ (n− |l|)β]n−|l|−1 .

Hence, one gets

Gβn (tµ2 ;y − x) = (1 + nβ)1−n

n∑

l2=0

(l2n

)µ(n

l2

)(y2 − x2)(y2 − x2 + l2β)l2−1

× [1− (y2 − x2)] [1− (y2 − x2) + (n− l2)β]n−l2−1

=Qβn (tµ2 ; y2 − x2) .

Application of Hölder's inequality with p = 1µand q = 1

1−µ gives

Gβn (tµ2 ;y − x) ≤ (y2 − x2)µ.

Thus from (2.6) it follows that

∣∣∣Gβn(f ;y)−Gβn(f ;x)∣∣∣ ≤M [(y1 − x1)µ + (y2 − x2)µ]

802

which implies that Gβn(f) ∈ LipM (µ, S). In a similar way the same result can be foundfor x ≥ y. If x1 ≥ y1, x2 ≤ y2, then we obtain from the above result for (y1, x2) ∈ S that

∣∣∣Gβn(f ;y)−Gβn(f ;x)∣∣∣ ≤∣∣∣Gβn (f ; (x1, x2))−Gβn (f ; (y1, x2))

∣∣∣

+∣∣∣Gβn (f ; (y1, y2))−Gβn (f ; (y1, x2))

∣∣∣≤M [(x1 − y1)µ + (y2 − x2)µ]

Finally, for the case x1 ≤ y1, x2 ≥ y2 we have the same result. This completes theproof. �

2.2. Theorem. If ω is a modulus of continuity function, then Gβn(ω) is also a modulusof continuity function for all n ∈ N.

Proof. Let x,y ∈ S such that y ≥ x. Regarding f as a modulus of continuity functionω in (2.3) we have

Gβn(ω;y)−Gβn(ω;x)

= (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

n−|l|∑

k1=0

n−k1−|l|∑

k2=0

[ω

(k+ l

n

)− ω

(k

n

)]

× n!k!l!(n− |k| − |l|)!x


× (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

Using the property (b) of ω we have Gβn(ω;y)−Gβn(ω;x) ≥ 0 when y ≥ x.Moreover, from the property (c) of modulus of continuity function ω and the equality

n!

k!l!(n− |k| − |l|)! =(n

l

)(n− |l|k1

)(n− k1 − |l|

k2

)

we can write

Gβn(ω;y)−Gβn(ω;x)

≤ (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

n−|l|∑

k1=0

n−k1−|l|∑

k2=0

ω

(l

n

)(n

l

)(n− |l|k1

)

×(n− k1 − |l|

k2

)xe(x+ kβ)k−e(y − x)e(y − x+ lβ)l−e (1− |y|)

× [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1

=(1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

ω

(l

n

)(n

l

)(y − x)e(y − x+ lβ)l−e

×n−|l|∑

k1=0

(n− |l|k1

)x1(x1 + k1β)

k1−1n−k1−|l|∑

k2=0

(n− k1 − |l|

k2

)x2

× (x2 + k2β)k2−1 (1− |y|) [1− |y|+ (n− |k| − |l|)β]n−|k|−|l|−1 .

803

Using the equalities (2.4) and (2.5) respectively, one has

Gβn(ω;y)−Gβn(ω;x) ≤ (1 + nβ)1−nn∑

l1=0

n−l1∑

l2=0

ω

(l

n

)(n

l

)

× (y − x)e(y − x+ lβ)l−e(1− |y − x|)× [1− |y − x|+ (n− |l|)β]n−|l|−1

=Gβn(ω;y − x).

This shows that Gβn(ω) is semi-additive. Finally, from the de�nition of Gβn it is obvious

that Gβn(ω;0) = ω(0) = 0. Therefore Gβn(ω) itself is a function of modulus of continuity

when ω is so. �

References

[1] Agratini, O. and Rus, I. A. Iterates of a class of dicsrete linear operators via contractionprinciple, Comment. Math. Univ. Carolin. 44(3), 555-563, 2003.

[2] Altomare, F. and Campiti, M. Korovkin-type approximaton theory and its applications, (Wal-ter de Gruyter, Berlin-New York, 1994).

[3] Ba³canbaz-Tunca, G. and Tuncer, Y.On a Chlodovsky variant of a multivariate beta operator,J. Comput. Appl. Math. 235(16), 4816-4824, 2011.

[4] Ba³canbaz-Tunca, G. and Ta³delen, F. On Chlodovsky form of the Meyer-König and Zelleroperators, An. Univ. Vest Timi³. Ser. Mat.- Inform. 49(2), 137-144, 2011.

[5] Ba³canbaz-Tunca, G., nce-larslan, H. G. and Erençin, A. Bivariate Bernstein type operators,Appl. Math. Comput. 273, 543-552, 2016.

[6] Ba³canbaz-Tunca, G., Erençin, A. and Ta³delen, F. Some properties of Bernstein type Cheneyand Sharma operators(accepted).

[7] Brown, B.M., Elliott, D. and Paget, D.F. Lipschitz constants for the Bernstein polynomialsof a Lipschitz continuous function, J. Approx. Theory 49(2), 196-199, 1987.

[8] Cao, F. Modulus of continuity, K-functional and Stancu operator on a simplex, Indian J.Pure Appl. Math. 35(12), 1343-1364, 2004.

[9] Cao, F., Ding, C. and Xu, Z. On multivariate Baskakov operator, J. Math. Anal. Appl.307(1), 274-291, 2005.

[10] C�atina³, T. and Otrocol, D. Iterates of multivariate Cheney-Sharma operators, J. Comput.Anal. Appl. 15(7), 1240-1246, 2013.

[11] Cheney, E. W. and Sharma, A. On a generalization of Bernstein polynomials, Riv. Mat.Univ. Parma 2(5), 77-84, 1964.

[12] Cr�aciun, M. Approximation operators constructed by means of She�er sequences, Rev. Anal.Numér. Théor. Approx. 30(2), 135-150, 2001.

[13] Ding, C. and Cao, F. K-functionals and multivariate Bernstein polynomials, J. Approx.Theory, 155(2), 125-135, 2008.

[14] Erençin, A., Ba³canbaz-Tunca, G. and Ta³delen, F. Some preservation properties of MKZ-Stancu type operators, Sarajevo J. Math., 10(22)(1), 93-102, 2014.

[15] Erençin, A., Ba³canbaz-Tunca, G. and Ta³delen, F. Some properties of the operators de�nedby Lupa³, Rev. Anal. Numér. Théor. Approx. 43(2), 168-174, 2014.

[16] Farca³, M. D. About Bernstein polynomials, An. Univ. Craiova Ser. Mat. Inform. 35, 117-121, 2008.

[17] Khan, M. K. Approximation properties of Beta operators, Progress in approximation theory,(Academic Press, Boston, MA, 1991), 483-495.

[18] Khan, M. K. and Peters, M. A. Lipschitz constants for some approximation operators of aLipschitz continuous function, J. Approx. Theory 59(3), 307-315, 1989.

[19] Li, Z. Bernstein polynomials and modulus of continuity, J. Approx. Theory 102(1), 171-174,2000.

[20] Lindvall, T. Bernstein polynomials and the law of large numbers, Math. Sci. 7(2), 127-139,1982.

804

[21] Stancu, D. D. and Cisma³iu, C. On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numér. Théor. Approx. 26(1-2), 221-227, 1997.

[22] Stancu, D. D., C�abulea, L. A. and Pop, D. Approximation of bivariate functions by meansof the operators Sα,β;a,bm,n , Stud. Univ. Babe³-Bolyai Math. 47(4), 105-113, 2002.

[23] Stancu, D. D. Use of an identity of A. Hurwitz for construction of a linear positive operatorof approximation, Rev. Anal. Numér. Théor. Approx. 31(1), 115-118, 2002.

[24] Stancu, D. D. and Stoica, E. I. On the use Abel-Jensen type combinatorial formulas for con-struction and investigation of some algebraic polynomial operators of approximation, Stud.Univ. Babe³-Bolyai Math. 54(4), 167-182, 2009.

[25] Stoica, E. I. On the combinatorial identities of Abel-Hurwitz type and their use in con-structive theory of functions , Stud. Univ. Babe³-Bolyai Math. 55(4), 249-257, 2010.

[26] Ta³cu, I. Approximation of bivariate functions by operators of Stancu-Hurwitz type, FactaUniv. Ser. Math. Inform. 20, 33-39, 2005.

[27] Ta³cu, I. and Horvat-Marc, A. Construction of Stancu-Hurwitz operator for two variables,Acta Univ. Apulensis Math. Inform. 11, 97-101, 2006.

[28] Trif, T. An elementary proof of the preservation of Lipschitz constants by the Meyer-Königand Zeller operators, J. Inequal. Pure Appl. Math. 4(5), Article90, 3pp., 2003.


A combinatorial approach to the classi�cation ofresolution graphs of weighted homogeneous plane

curve singularities

Muhammad Ahsan Binyamin∗, Ha�z Muhammad Afzal Siddiqui† and AmirShehzad‡

Abstract

In this article we describe the classi�cation of the resolution graphsof weighted homogeneous plane curve singularities in terms of theirweights by using the concepts of graph theory and combinatorics. Theclassi�cation shows that the resolution graph of a weighted homoge-neous plane curve singularity is always a caterpillar.

Keywords: Plane Curves, Weights, Caterpillar.

Mathematics Subject Classi�cation (2010): 14Q05,14H20


1. Introduction

The history of resolution of plane curve singularities is very old. It started withNewton in 1676, who showed the existence of Puiseux series. The resolution of planecurve singularities is an easy consequence. There is a large group of mathematicians whointroduced new methods to resolve a plane curve singularity and they found deep andimportant applications of resolution of plane curve singularities. János Kollár lists about20 ways of resolution of plane curve singularities (cf. [5]). Moreover an algebraic andcombinatorial information about plane curve singularities can be found in [7], [9].A graph is an ordered pair G = (V,E), where V is called vertex set and E is called edgeset. |V | and |E| denote the order and the size of a graph, respectively. A tree is an acyclicconnected graph with n vertices and n − 1 edges and a caterpillar is a special type oftree with the property that a path remains if all leaves are deleted. A vertex labeling is

∗Department of Mathematics, Government College University, Faisalabad, Pakistan,Email : [email protected]†Department of Mathematics, COMSATS Institute of Information Technology Lahore, Pak-

istan,Email : [email protected]‡Department of Mathematics, Government College University, Faisalabad, Pakistan,

Email : [email protected]

806

a bijection from the set of vertices V to the set of labels {1, 2, . . . , |V |}.In [2], S. Dale Cutkosky and H. Srinivasan compute the resolution graph combinatoricallyby using the characteristic pairs of an irreducible plane curve singularity. Also Y. Jingenin [4], associates each singularity of a curve on a surface to a tree called S-tree which issome kind of "structured graph" and obtained by using the procedure of blow-ups.In this article we introduce a combinatorial approach to compute the resolution graph ofa weighted homogeneous plane curve singularity. As a �rst step we compute the orderand the size of the resolution graph of a weighted homogeneous plane curve singularity.Then we de�ne a vertex labeling on the set of vertices obtained from the weights of aweighted homogeneous plane curve singularity and �nally construct the resolution graph.

2. Basic De�nitions

In this section, we give some basic de�nitions related to the resolution of plane curvesingularities. De�nitions can be found in [3].

2.1. De�nition. Let Bl(0)C2 denote the blowing − up of 0 ∈ C2, which is the subset ofC2 × P , where P is the projective line and it is de�ned as

Bl(0)C2 = {(a, la) : a ∈ la},where a is a point and la is a line in C2 on which point a lies.Then there is a projection map

π : Bl(0)C2 → C2which is called the blowing-up map.

We denote E := π−1(0) the exceptional divisor of π.Bl(0)C2 can also be de�ned in coordinates as follows

Bl(0)C2 = {(x, y, u : v) : xv = yu} ⊂ C2 × P.here (x, y) ∈ C2 and (u : v) are the homogeneous coordinates of P .Let

V1 = {(x, xv, 1 : v) : x, v ∈ C2} ∼= C2

V2 = {(yu, u, u : 1) : y, u ∈ C2} ∼= C2then V1 ∪ V2 = Bl(0)C2 is an a�ne covering.2.2. De�nition. Let (V (f), 0) be a curve singularity. Then the closure of π−1(f \ 0) iscalled the strict transform of f , and the inverse image π−1(f) is called the total transformof f .

Let f =r∪i=1

fi ⊂ C2 be a small representative of a reducible plane curve singu-larity with branches f1, ..., fr r ≥ 1. Assume that Xi

πi→ ... π2→X1π1→C2 is a sequence

of blowing up points. Denote by E(i) = (π1 ◦ ... ◦ πi)−1(0) the exceptional divisor,f (i) = (π1 ◦ ... ◦ πi)−1(f \ {0}) the strict transform and (π1 ◦ ...◦πi)−1(f) the total trans-form of f . Let Xi+1

πi+1→ Xi be the blowing up of Xi in all points of f (i)∩E(i) which arestill singular on f (i) or non-transversal intersection of f (i) with E(i) that is the pointswith intersection multiplicity of f (i) and E(i) greater than one or where two exceptionaldivisors and f (i) meet.

2.3. De�nition. (i) Xkπk→ ... π1→X1 → C2 is called a standard resolution of (V (f), 0) if all

branches of f (k) are smooth, do not intersect each other, do intersect just one componentof E(k) and do intersect this component transversally.

807

We consider the following weighted graph, the resolution graph of f .(i) To each component of E(k) a point • is associated.(ii) To each component of f (k) a point ∗ is associated.(iii) Two points are connected by an edge if the corresponding components intersect.

(iv) The points of type(i) are weighted. Let E be a component of E(k).We give to thecorresponding point the weight i if E is created in the i− th level of the blowing ups thatis i is minimal such that πi+1 ◦ ... ◦ πn(E) is not a point.

2.4. De�nition. Let n, m be positive integers then by the Euclidean algorithm, we canexpand n

min a continued fraction in nonnegative integers ci,

(2.1)n

m= c1 +

1

c2 +1

c3 +1

. . .+1

cn

We denote it by nm

= [c1, c2, . . . , cn].

3. Classi�cation of Resolution Graphs of Weighted Homogeneous

plane Curve Singularities

The type of nondegenerate quasihomogeneous polynomials in two variables accordingto V.I. Arnold, S. M. Gusein-Zade and A. N. Varchenko [1] is given in the following table.

Table 1

Type Quasihomogeneous polynomial Weighted Vector

I xa + yb a, b ∈ Z>0 (b, a, ab)II xay + yb a ∈ Z>0, b ∈ Z>1 (b− 1, a, ab)III xay + ybx a, b ∈ Z>1 (b− 1, a− 1, ab− 1)

3.1. Proposition. Any weighted homogeneous polynomial f ∈ C[x, y] de�ning an iso-lated singularity is of the type f = f0+h, where f0 is one of the form given in the table.1and h consist on terms having the same degree as f0.

Proof. See [1]. �

3.2. Remark. The resolution of weighted homogeneous polynomial f de�ning an iso-lated singularity does not depend on h and depends only on weights and degree (seeTheorem-3.5 in [6]).

3.3. De�nition. (V (f), 0) ⊆ (C2, 0) is called a quasihomogeneous plane curve singular-ity, if there exist an automorphism

φ : C[[x, y]]→ C[[x, y]]such that φ(f) is a weighted homogeneous plane curve singularity.

The following proposition gives us a combinatorial formula to compute the order andsize of the resolution graph of a weighted homogeneous plane curve singularity.

808

3.4. Proposition. Let Gf denote the resolution graph of a plane curve singularity(V (f), 0), where f is weighted homogeneous polynomial de�ning an isolated singularity isof the type f = f0 + h, where f0 is one of the form given in the table.1 and h consist onterms having the same degree as f0. Then if

i: f is of type-I then the order of Gf isn∑i=1

ci+ gcd(a, b), whereba= [c1, c2, . . . , cn].

ii: f is of type-II then the order of Gf isn∑i=1

ci + gcd(a, b − 1) + 1, where b−1a =[c1, c2, . . . , cn].

iii: f is of type-III then the order of Gf isn∑i=1

ci + gcd(a − 1, b − 1) + 2, whereb−1a−1 = [c1, c2, . . . , cn].

Moreover Gf is always a caterpillar.

Proof. i: If (V (f), 0) be a weighted homogeneous plane curve singularity of type-I. Then it is noted that number of branches of the plane curve singularity isgcd(a, b) and resolution graphs Gf and Gf0 are same (see remark 3.2). Weconsider a ≤ b. The case for a > b can be treated in a similar way.We start the resolution of singularity by the following blow up

x→ xy, y → y(This chart is only considered since the exceptional divisor does not intersectthe curve in the other chart.)Then we have the strict transformation xa+yb−a = 0 and exceptional divisor E1 :y = 0. After [ b

a] = c1 blow ups we have the strict transformation x

a+yb−c1a = 0such that b − c1a < a. Then multiplicity of strict transformation dropped andis equal to b− c1a and exceptional divisor Ec1 : y = 0. Then make the blow up

x→ x, y → xywe get the strict transformation xa−(b−c1a) + yb−c1a = 0 and exceptional divi-sor Ec1+1 : x = 0. After [

ac] = c2 blow ups we have the strict transformation

xa−c2(b−c1a) + yb−c1a = 0 such that a − c2(b − c1a) < b − c1a. Then multiplic-ity of strict transformation dropped and becomes equal to a − c2(b − c1a) andexceptional divisor Ec1+c2 : y = 0. Continue in this way, after c1 + c2 + · · ·+ cnblow ups we get the standard resolution. So the number of vertices of Gf isn∑i=1

ci + gcd(a, b) and if we construct the dual graph of this resolution as de-

scribed in section-2 then we �nd Gf is a caterpillar.ii and iii can be proved similarly to i. �

3.5. Remark. In the above propositionn∑i=1

ci is the number of dot vertices which rep-

resents the number of blow-ups required to make the standard resolution and gcd(a, b) isthe number of star vertices of the resolution graph which denote the number of branchesof weighted homogeneous plane curve singularity of type-I.

3.6. Remark. If nm

= [c1, c2, . . . , cn] then in the following proposition, the integers ei,

at and bs denote the sumi∑

k=1

ck, the weight of the vertex vt and the weight of the vertex

vs respectively.

In the following proposition we describe a combinatorial construction to compute theresolution graph of a weighted homogeneous plane curve singularity of type-I.

809

3.7. Proposition. Let (V (f), 0) be a weighted homogeneous plane curve singularity oftype-I then its resolution grpah Gf can be obtained by using the following construction.

Proof. Since (V (f), 0) be a weighted homogeneous plane curve singularity of type-I, thenfrom Proposition-3.4 we have the set of vertices

V = {vi : 1 ≤ i ≤n∑

i=1

ci + gcd(a, b)},

where ba= [c1, c2, . . . , cn]. Then we can de�ne the integers such that e1 < e2 < · · · < en,

where ei =i∑

k=1

ck and a partion on the set of vertices V such that V = V(1)• ∪ V (2)• ∪

V(3)• ∪ V∗, where

V (1)• = {vi : 1 ≤ i ≤ e1},

V (2)• = {vi : e1 + 1 ≤ i ≤ e1 + l},

V (3)• = {vi : e1 + l + 1 ≤ i ≤ en},where

l =

n−22∑j=1

(e2j+1 − e2j), if n is even;n−12∑j=1

(e2j+1 − e2j), if n is odd;

V∗ = {v∗i : 1 ≤ i ≤ gcd(a, b)}.Now de�ne

A := {e2 + 1, . . . , e3, e4 + 1, . . . , e5, . . . , e2[n−12

]+ 1, . . . , e

2[n−12

]+1} = {ae1+1, . . . , ae1+l}

and

B := {e1+1, . . . , e2, e3+1, . . . , e4, . . . , e2[n−12

]+1+1, . . . , e2[n

2]} = {ben , ben−1, . . . , be1+l+1}.

Note that |V (2)• | = |A| and |V (3)• | = |B|.Let

s =

{1, if n is even;0, if n is odd;

and q = e1 + l + 1 + s then we obtain the following resolution graph

21 e1 ae +11 a - 2q en bq be

n

e +1n

e +2n

e +gcd(a,b)n

* **

�

810

3.8. Example. Consider (V (f), 0) be a weighted homogeneous plane curve singularitywith weights (230, 1055).Then V = {v1, v2, . . . , v13, v∗1 , v∗2 , v∗3 , v∗4 , v∗5} (see Proposition-3.4). Now by using theProposition-3.7 we can construct the following data:Step− 1 : Construct

e1 = 4, e2 = 5, e3 = 6, e4 = 8, e5 = 10, e6 = 11, e7 = 13.

Step− 2 : Since n = 7 which is odd therefore V = V (1)• ∪ V (2)• ∪ V (3)• ∪ V∗, whereV (1)• = {v1, v2, v3, v4},

V (2)• = {v5, v6, v7, v8, v9},V (3)• = {v10, v11, v12, v13},

and

V∗ = {v∗1 , v∗2 , v∗3 , v∗4 , v∗5}.Step− 3 :

A = {6, 9, 10, 12, 13} = {a5, a6, a7, a8, a9}and

B = {5, 7, 8, 11} = {b13, b12, b11, b10}.Step− 4 : Attach all ∗ vertices with v9 then we get the resolution graph as given inFigure 1.

1 32 4 6 79 10 12 13 11 8 5

14

1516 17

18

* ****

Figure 1. Resolution graph of a weighted homogeneous plane curvesingularity with weights (230, 1055).

In the following two propositions we describe a combinatorial construction to computethe resolution graph of weighted homogeneous plane curve singularities of type II andIII.

3.9. Proposition. Let (V (f), 0) be a weighted homogeneous plane curve singularity oftype-II then its resolution graph Gf is one of the graphs given in Table 2.

Proof. Since (V (f), 0) be a weighted homogeneous plane curve singularity of type-II,then we have

V = {vi : 1 ≤ i ≤n∑

i=1

ci + (gcd(a, b− 1) + 1)},

Make a partition of V such that V = V′ ∪ V (1)∗ , where

V′= {vi : 1 ≤ i ≤

n∑

i=1

ci + gcd(a, b− 1)}

and

V (1)∗ = {v∗i : i = gcd(a, b− 1) + 1}.For V

′follow the construction as explained in Proposition-3.7. Assign the weight

en + gcd(a, b− 1) + 1 to the ∗ vertex v∗i for i = gcd(a, b− 1) + 1. If a ≤ b− 1 then attach

811

Table 2

Type Condition Resolution Graph

II a ≤ b− 121 e1 ae +11 a - 2q

en bq ben

e +1n

e +2n

e +gcd(a,b)n

* ***

e + gcd(a,b)+1n

II a > b− 121 e1 ae +11 a - 2q

en bq ben

e +1n

e +2n

e +gcd(a,b)n

* ** *

e + gcd(a,b)+1n

v∗i with the • vertex of weight 1. If a > b − 1 then attach v∗i with the vertex of V′(3)•

with smallest weight. �

3.10. Proposition. Let (V (f), 0) be a weighted homogeneous plane curve singularity oftype-III then its resolution grpah Gf is the following:

21 e1 ae +11 a - 2q en bq be

n

e +1n

e +2n

e +gcd(a,b)n

* ** *

e + gcd(a,b)+2n

*e + gcd(a,b)+1n

Figure 2. Resolution graph of (V (f), 0)

Proof. Since (V (f), 0) be a weighted homogeneous plane curve singularity of type-III,then we have

V = {vi : 1 ≤ i ≤n∑

i=1

ci + (gcd(a− 1, b− 1) + 2)},

Make a partition of V such that V = V′ ∪ V (1)∗ , where

V′= {vi : 1 ≤ i ≤

n∑

i=1

ci + gcd(a− 1, b− 1)}

andV (1)∗ = {v∗i : i = gcd(a− 1, b− 1) + 1, gcd(a− 1, b− 1) + 2},

For V′follow the construction as explained in Proposition-3.7. Assign the weights

en+gcd(a−1, b−1)+1, en+gcd(a−1, b−1)+2, to the ∗ vertices v∗i for i = gcd(a−1, b−1)+1and i = gcd(a− 1, b− 1)+ 2 respectively. Attach v∗i for i = gcd(a− 1, b− 1)+ 1 with the• vertex of weight 1 and attach v∗i for i = gcd(a− 1, b− 1) + 2 with the • vertex of V

′(3)•

with smallest weight. �

812

4. Acknowledgements

The authors would like to thank the referee for his/her corrections, comments anduseful criticism which improved the �rst version of this paper.

References

[1] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko,A. N. Singularities of Di�erentiable Maps,Volume I, Birkhäuser, Boston Basel Berlin (1985).

[2] Cutkosky, S. D. and Srinivasan, H.; The algebraic fundamental group of a curve singularity,Journal of Algebra 230, 101-126, (2000).

[3] De Jong, T. and P�ster, G.; Local Analytic Geometry, Vieweg (2000).[4] Jingen, Y.; Curve Singularities and Graphs, Acta Mathematica Sinica, 6 (1), 87-96, (1990).[5] Kollár, J.; Lectures on Resolution of Singularities, Princeton University Press (2007).[6] Kang, C.; Analytic Types of Plane Curve Singularities de�ned by Weighted Homogeneous

Polynomials, Trans. A.M.S. 352 (9), 3995-4006, (2000).[7] Muhly, H.T. and Zariski, O.; The Resolution of Singularities of an Algebraic curve,

Amer.J.Math., 61 (1), 107-114, (1939).[8] Saito, K.; Quasihomogene isolierte singularitäten von hyper�ächen, Invent. Math. 14, 123-

142, (1971).[9] Wall, C.T.C.; Singular Points of Plane Curves, Cambridge University Press (2004).


q− Harmonic mappings for which analytic part isq− convex functions of complex order

Asena Çetinkaya∗† and Ya³ar Polato�glu‡

Abstract

We introduce a new class of harmonic function f , that is subclass ofplanar harmonic mapping associated with q− di�erence operator. Leth and g are analytic functions in the open unit disc D = {z : |z| < 1}. Iff = h+ g is the solution of the non-linear partial di�erential equation

wq(z) =Dqg(z)

Dqh(z)= fz

fzwith |wq(z)| < 1, wq(z) ≺ b1 1+z1−qz and h is

q− convex function of complex order, then the class of such functionsare called q− harmonic functions for which analytic part is q− convexfunctions of complex order denoted by SHCq(b). Obviously that the classSHCq(b) is the subclass of SH. In this paper, we investigate propertiesof the class SHCq(b) by using subordination techniques.

Keywords: q− di�erence operator, q− harmonic mapping, q− convex functionof complex order.

Mathematics Subject Classi�cation (2010): 30C45


∗Department of Mathematics and Computer Sciences,stanbul Kültür University, stanbul,TURKEY ,Email : [email protected]†Corresponding Author.‡Department of Mathematics and Computer Sciences,stanbul Kültür University, stanbul,

TURKEY,Email : [email protected]

814

1. Introduction

A planar harmonic mapping in the open unit disc D is a complex valued harmonicfunction f , which maps D onto the some planar domain f(D). Since D is a simplyconnected domain, the mapping f has a canonical decomposition f = h+g, where h andg are analytic in D and have the following power series expansions

h(z) = z +

∞∑

n=2

anzn and g(z) =

∞∑

n=1

bnzn

where an, bn ∈ C, n = 0, 1, 2, 3, · · · . As usual, we call h the analytic part of f and gthe co-analytic part of f , respectively. An elegant and complete treatment theory ofthe harmonic mapping is given in Duren's monograph [3]. Lewy [11] proved in 1936that the harmonic mapping f is locally univalent in D if and only if its Jacobian Jf =|h′(z)|2 − |g′(z)|2 is di�erent from zero in D. In view of this result, locally univalentharmonic mappings in the open unit disc are either sense-preserving if |g′(z)| < |h′(z)| orsense-reversing if |g′(z)| > |h′(z)| in D. Throughout this paper, we will restrict ourselvesto the study of sense-preserving harmonic mappings. We also note that f = h + g issense-preserving in D if and only if h′ does not vanish in D and the second dilatationw(z) = g

′(z)h′(z) has the property |w(z)| < 1 for all z ∈ D. Therefore the class of all sense-

preserving harmonic mappings in D with a0 = b0 = 0 and a1 = 1 will be denoted bySH. Thus SH contains standard class S of analytic univalent functions. The family of allmappings f ∈ SH with the additional property that g′(0) = 0, i.e., b1 = 0 are denotedby S0H. Hence it is clear that S ⊂ S0H ⊂ SH.

In 1908 and 1910 Jackson [8, 9] initiated a study of q− di�erence operator by

(1.1) Dqf(z) =f(z)− f(qz)

(1− q)z for z ∈ B\{0}

where B is a subset of complex plane C, called q− geometric set if qz ∈ B, wheneverz ∈ B. Note that if a subset B of C is q− geometric, then it contains all geometricsequences {zqn}∞0 , zq ∈ B. Obviously, Dqf(z) → f ′(z) as q → 1−. The q− di�erenceoperator (1.1) is sometimes called Jackson q− di�erence operator. Note that such anoperator plays an important role in the theory of hypergeometric series and quantumphysics (see for instance [1, 4, 5, 10]).

Also, note that Dqf(0) → f ′(0) as q → 1− and D2qf(z) = Dq(Dqf(z)). In fact, q−calculus is ordinary classical calculus without the notion of limits. Recent interest in q−calculus is because of its applications in various branches of mathematics and physics.For de�nition and properties of q− di�erence operator and q− calculus, one may refer to[1, 4, 5, 10].

Under the hypothesis of the de�nition of q− di�erence operator, then we have thefollowing rules:

(1) For a function f(z) = zn, we observe that

Dqzn =

1− qn1− q z

n−1.

Therefore we have

Dqf(z) = 1 +

∞∑

n=2

an1− qn1− q z

n−1.

(2) If functions f and g are de�ned on a q− geometric set B ⊂ C such that q−derivatives of f and g exist for all z ∈ B, then

(i) Dq(af(z)±bg(z)) = aDqf(z)±bDqg(z) where a and b are real or complexconstants.

815

(ii) Dq(f(z).g(z)) = g(z)Dqf(z) + f(qz)Dqg(z).

(iii) Dq

(f(z)

g(z)

)=g(qz)Dqf(z)− f(qz)Dqg(z)

g(z)g(qz), g(z)g(qz) 6= 0.

(iv) As a right inverse, Jackson introduced q− integral∫ z

0

f(t)dqt = z(1− q)∞∑

n=0

qnf(zqn)

provided that the series converges.

The following theorem is an analogue of the fundamental theorem of calculus.

A. Theorem. ([10]) Let f be a q− regular at zero, de�ned on q− geometric set Bcontaining zero. De�ne

F (z) =

∫ z

c

f(ζ)dqζ, (ζ ∈ B)

where c is a �xed point in B, then F is q− regular at zero. Furthermore DqF (z) existsfor every z ∈ B and

DqF (z) = f(z)

for every z ∈ B.Conversely, if a and b are two points in B, then

∫ b

a

Dqf(ζ)dqζ = f(b)− f(a).

(3) The q− di�erential is de�ned asdqf(z) = f(z)− f(qz).

Therefore we can write

dqf(z) =f(z)− f(qz)

(1− q)z dqz.

(4) The partial q− derivative of a multivariable real continous functionsf(x1, x2, ..., xi−1, xi, xi+1, ..., xn) to a variable xi is de�ned by

Dq,xif(~x) =f(~x)− εq,xif(~x)

(1− q)xi, xi 6= 0, q ∈ (0, 1)

[Dq,xif(~x)

]

xi=0

= limxi→0

Dq,xif(~x)

where εq,xif(~x) = f(x1, x2, ..., xi−1, qxi, xi+1, ..., xn) and we use Dkk,x instead of

operator∂kq∂qxk

for some simpli�cation.

Finally, let Ω be the family of functions φ analytic in D, and satisfy the conditionsφ(0) = 0, |φ(z)| < 1 for all z ∈ D. Denote by Pq the family of functions p of the formp(z) = 1 + p1z + p2z

2 + · · · , analytic in D and satisfy the condition

(1.2)

∣∣∣∣ p(z)−1

1− q

∣∣∣∣ ≤1

1− q , z ∈ D

where q ∈ (0, 1) is a �xed real number. Let A be the family of functions f , de�ned byf(z) = z + a2z

2 + a3z3 + ..., that are analytic in D and satisfy the conditions f(0) =

0, f ′(0) = 1. If f satis�es the condition

1 +1

b

(qzDq(Dqf(z))

Dqf(z)

)≺ 1 + z

1− qz ,

where b ∈ C, b 6= 0, then f is called q− convex function of complex order, and the class ofsuch functions are denoted by Cq(b). If f1 and f2 are analytic functions in D, then we say

816

that f1 is subordinate to f2, written as f1 ≺ f2 if there exists a Schwarz function φ ∈ Ωsuch that f1(z) = f2(φ(z)), z ∈ D. We also note that if f2 univalent in D, then f1 ≺ f2 ifand only if f1(0) = f2(0) and f1(D) ⊂ f2(D). This implies that f1(Dr) ⊂ f2(Dr), whereDr = {z : |z| < r , 0 < r < 1} (Subordination principle [6] ).

We also need the following lemmas:

1.1. Lemma. Let φ be analytic in D with φ(0) = 0 and |φ(z)| < 1, z ∈ D. If |φ(z)|attains its maximum value on the circle |z| = r at a point z0 , then we have

z0φ′(z0) = mφ(z0), m ≥ 1.

For more details of Jack's lemma, one may refer to [7].

1.2. Lemma. ([12]) If h is an element of Cq(b), then

F2(|b|, Reb, q, r) ≤ |Dqh(z)| ≤ F1(|b|, Reb, q, r)where

F1(|b|, Reb, q, r) =[(1− qr)Reb+|b|.(1 + qr)Reb−|b|

]− 1−q22q2logq−1

,

F2(|b|, Reb, q, r) =[(1− qr)Reb−|b|.(1 + qr)Reb+|b|

]− 1−q22q2logq−1

.

The aim of this paper is to investigate properties of the class of q− harmonic functionsfor which analytic part is q− convex functions of complex order de�ned by

SHCq(b) =

{f = h+g : wq(z) =

Dqg(z)

Dqh(z)=fzfz, wq(z) ≺ b1 1 + z

1− qz , |wq(z)| < 1, h ∈ Cq(b)},

where

Dqh(z) =h(z)− h(qz)

(1− q)z = fz and Dqg(z) =g(z)− g(qz)

(1− q)z = fz.

2. Main Results

In this section, we �rst assume that the function f is sense-preserving q− harmonicfunction if and only if wq(z) =

fzfz

is analytic. To show that

(⇒) Let f = h+ g be sense-preserving q− harmonic function, then we will show thatwq is analytic. Since h(z) = z +

∞∑n=2

anzn and g(z) =

∞∑n=1

bnzn are analytic functions,

then we can write q− derivatives of these functions as

Dqh(z) = 1 +

∞∑

n=2

1− qn1− q anz

n−1 and Dqg(z) = b1 +∞∑

n=2

1− qn1− q bnz

n−1.

We must note that when q → 1−, Dqh(z) reduces to h′(z) and Dqg(z) reduces to g′(z).The second q− dilatation and q− Jakobian are de�ned by

wq(z) =Dqg(z)

Dqh(z)=fzfz,

Jfq(z) = |Dqh(z)|2 − |Dqg(z)|2.Also, the total q− di�erential of f(~x) can be written in the following manner,

dqf(~x) = Dq,x1dqx1 +Dq,x2dqx2 +Dq,x3dqx3 + · · ·+Dq,xndqxn.Therefore the q− di�erential can be written as

dqf = Dq,zdqz +Dq,zdqz.

817

Consequently, f is locally univalent and sense-preserving if |Dqh(z)| > |Dqg(z)| andsense-reversing if |Dqg(z)| > |Dqh(z)|. Note that fz 6= 0 whenever Jfq(z) > 0. Forsense-preserving f , one sees that

(|Dqh(z)| − |Dqg(z)|)|dqz| ≤ |dqf | ≤ (|Dqh(z)|+ |Dqg(z)|)|dqz|.With aid of these de�nitions, let f = h+g be the solution of the non-linear elliptic partialdi�erential equation

wq(z)fz = fzunder the condition |wq(z)| < 1 for all z ∈ D. A non-constant complex -valued functionf is q− harmonic and orientation sense-preserving mapping on D if and only if f is thesolution of the non-linear elliptic partial di�erential equation

(2.1) wq(z)fz = fz

where

fz = Dqh(z) =h(z)− h(qz)

(1− q)z and fz = Dqg(z) =g(z)− g(qz)

(1− q)z .

If we take the q− derivative of equation (2.1) with respect to z, we get

(2.2) fzz = fzzwq(z) + fz∂wq∂z

.

On the other hand, since f is q− harmonic, then we have 4f = 4 ∂2f∂z∂z

= 4fzz = 0 and

fzz = 0. Therefore the equality (2.2) reduces to

(2.3) fz∂wq∂z

= 0

and this shows that∂wq∂z

= 0, that is, wq is analytic.

(⇐) Conversely, if wq is analytic in D, then ∂wq∂z = 0. Therefore equality (2.2) reducesto

(2.4) fzz = fzzwq(z).

On the other hand, using the de�nition of wq, we have |wq(z)| < 1. Thus, we get(2.5) 1− |wq(z)| 6= 0.Considering (2.4) and (2.5), we obtain

(2.6) fzz = fzzwq(z)⇒ fzz = 0and the equality (2.6) shows that f is q− harmonic. This proves our assumption.

We now investigate properties of the class SHCq(b). For Theorem 2.4, we need thefollowing results. The �rst theorem is very important in order to obtain subordinationof the analytic functions involving q− di�erence operator.

2.1. Theorem. ([2]) p is an element of Pq if and only if p(z) ≺ 1 + z1− qz . This result is

sharp for the functions p(z) =1 + φ(z)

1− qφ(z) , where φ is a Schwarz function.

Proof. If p is an element of Pq, then we have∣∣∣∣ p(z)−1

1− q

∣∣∣∣ ≤1

1− q ⇔ |p(z)−m| ≤ m,

where m =1

1− q > 1. Therefore we can write∣∣∣∣1

mp(z)− 1

∣∣∣∣ ≤ 1.

818

Thus the function ψ(z) = 1mp(z)− 1 is analytic and has modulus at most 1 in D, and so

φ(z) =ψ(z)− ψ(0)1− ψ(0)ψ(z)

=( 1mp(z)− 1)− ( 1

m− 1)

1− ( 1m− 1)( 1

mp(z)− 1)

satis�es the conditions of Schwarz lemma. This shows that we can write

p(z) =1 + φ(z)

1− (1− 1m

)φ(z)⇒ p(z) ≺ 1 + z

1− qz .

Conversely, suppose that the function p is analytic in D and satis�es the condition p(0) =1 and

p(z) ≺ 1 + z1− qz ⇒ p(z) =

1 + φ(z)

1− (1− 1m

)φ(z)

p(z)−m = m1−mm

+ φ(z)

1 + 1−mm

φ(z).

On the other hand the function1−mm

+φ(z)

1+ 1−mm

φ(z)maps the unit circle onto itself, then we have

|p(z)−m| ≤ m⇔∣∣∣∣ p(z)−

1

1− q

∣∣∣∣ ≤1

1− q .

This shows that p ∈ Pq. �

We must note that the linear tranformation 1+z1−qz maps |z| = r onto the disc with

centre C(r) = 1+qr2

1−q2r2 and radius ρ(r) =(1+q)r

1−q2r2 .

2.2. Lemma. If f is a function (real or complex valued) de�ned on q− geometric set Bwith |q| 6= 1, then

Dq(logf(z)) =Dqf(z)

f(z).

Proof. Using the de�nition of q− di�erence operator, then we have

Dq(logf(z)) =logf(qz)− logf(z)

qz − z = log(

1 + hDqf(z)

f(z)

) 1h

.

If we take limit for h→ 0, we obtain the desired result. �

2.3. Lemma. (q−Jack's Lemma) Let φ be analytic in D with φ(0) = 0. If |φ(z)| attainsits maximum value on the circle |z| = r at a point z0 ∈ D, then we have

z0Dqφ(z0) = mφ(z0)

where m ≥ 1 is a real number.

Proof. Using the de�nition of q− di�erence operator and Jack's lemma, then we canwrite

Dqφ(z) =φ(z)− φ(qz)

z − qz =φ(z)− φ(z0)

z − z0, qz = z0.

If we take limit for z → z0, we get

limz→z0

Dqφ(z) = Dqφ(z0) = limz→z0

φ(z)− φ(z0)z − z0

= φ′(z0).

Therefore we have

z0Dqφ(z0) = mφ(z0).

�

819

2.4. Theorem. If f = h+ g is an element of SHCq(b), then

(2.7)g(z)

h(z)≺ b1 1 + z

1− qz .

Proof. Since f = h+ g ∈ SHCq(b), then we haveDqg(z)

Dqh(z)≺ b1 1 + z

1− qz .

The linear transformation w = b11+z1−qz maps |z| = r onto the disc with centre C(r) =(α1(1+qr2)

1−q2r2 ,α2(1+qr

2)

1−q2r2)and radius ρ(r) = |b1|(1+q)r

1−q2r2 , where α1 = Reb1 and α2 = Reb2.Thus using the subordination principle and the de�nition of the class SHCq(b), we canwrite

(2.8) wq(Dr) ={Dqg(z)

Dqh(z):

∣∣∣∣Dqg(z)

Dqh(z)− b1(1 + qr

2)

1− q2r2∣∣∣∣ ≤|b1|(1 + q)r

1− q2r2 , q ∈ (0, 1)}.

In order to verify Schwarz function conditions, we de�ne the function φ by

(2.9)g(z)

h(z)= b1

1 + φ(z)

1− qφ(z) .

Note that φ is a well de�ned analytic function and

g(z)

h(z)

∣∣∣∣z=0

= b1 = b11 + φ(0)

1− qφ(0) .

This proves that φ(0) = 0. We now need to show that |φ(z)| < 1 for all z ∈ D. If we takeq− derivative of both sides of (2.9) and simplify, we get

Dqg(z)

h(z)− g(qz)Dqh(z)

h(z)h(qz)= b1

Dqφ(z)− qφ(qz)Dqφ(z) + qDqφ(z) + qφ(qz)Dqφ(z)(1− φ(z))(1− φ(qz)) .

Multiplying both sides of this equation by h(z)/Dqh(z) and simplifying, we obtain

(2.10)Dqg(z)

Dqh(z)= b1

(1 + φ(qz)

1− qφ(qz) +(1 + q)zDqφ(z)

(1− qφ(z))(1− qφ(qz)) .h(z)

zDqh(z)

).

Applying Lemma 2.2 in the equation (2.10), we can write the following form

(2.11)Dqg(z)

Dqh(z)= b1

(1 + φ(qz)

1− qφ(qz) +(1 + q)zDqφ(z)

(1− qφ(z))(1− qφ(qz))(1− qφ(z)

)b 1−q2q2logq−1

).

Assume to the contrary that there esists a point z0 ∈ Dr such that |φ(z0)| = 1. In viewof Lemma 2.3, equation (2.11) gives

Dqg(z0)

Dqh(z0)= b1

(1 + φ(qz0)

1− qφ(qz0)+

(1 + q)mφ(z0)

(1− qφ(z0))(1− qφ(qz0))(1−qφ(z0)

)b 1−q2q2logq−1

)/∈ wq(Dr).

This contradicts our assumption (2.8) and therefore |φ(z)| < 1 for all z ∈ D. Thiscompletes the proof of our theorem. �

2.5. Corollary. If f = h+ g ∈ SHCq(b), then we have(2.12) F2(|b|, Reb, |b1|, q, r) ≤ |Dqg(z)| ≤ F1(|b|, Reb, |b1|, q, r),where

F1(|b|, Reb, |b1|, q, r) =[(

1− qr)Reb+|b|(

1 + qr)Reb−|b|

]− 1−q22q2logq−1 |b1|(1 + r)

1− qr ,

F2(|b|, Reb, |b1|, q, r) =[(

1− qr)Reb−|b|(

1 + qr)Reb+|b|

]− 1−q22q2logq−1 |b1|(1− r)

1 + qr.

820

Proof. Since f = h + g is an element of SHCq(b), from Theorem 2.4 we writeDqg(z)

Dqh(z)≺

b11+z1−qz , where h ∈ Cq(b). Therefore we have∣∣∣∣

Dqg(z)

Dqh(z)− b1(1 + qr

2)

1− q2r2∣∣∣∣ ≤|b1|(1 + q)r

1− q2r2 .

This inequality yields

|Dqg(z)| ≤ |Dqh(z)| |b1|(1 + r)1− qr .

If we use Lemma 1.2, we get the right side of (2.12). Similarly, we can prove the otherside of the inequality (2.12). �

2.6. Corollary. If f = h+ g ∈ SHCq(b), then we have

(2.13) f = h(z) + h(z)b11 + φ(z)

1− qφ(z) ,

where φ is a Schwarz function.

Proof. Using Theorem 2.4, then we can write

g(z)

h(z)≺ b1 1 + z

1− qz ⇒g(z)

h(z)= b1

1 + φ(z)

1− qφ(z) .

Therefore we obtain

g(z) = h(z)b11 + φ(z)

1− qφ(z) ,

which gives (2.13). �

References

[1] Andrews, G.E. Applications of basic hypergeometric

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS · Reza Salehi and Ali Shadrokh The bias reduction...

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