Turkish Journal of - sciepubpubs.sciepub.com/tjant/TJANT-3-4.pdf · 2020-05-30 · Turkish Journal...
Transcript of Turkish Journal of - sciepubpubs.sciepub.com/tjant/TJANT-3-4.pdf · 2020-05-30 · Turkish Journal...
Science and Education Publishing
Turkish Journal ofAnalysis and Number Theory
Scan to view this journalon your mobile device
ISSN : 2333-1100(Print) ISSN : 2333-1232(Online)
Volume 3, Number 4, 2015
http://tjant.hku.edu.tr
Hasan Kalyoncu University
http://www.sciepub.com/journal/tjant
Turkish Journal of Analysis and Number Theory
Owner on behalf of Hasan Kalyoncu University: Professor Tamer Yilmaz, Rector
Correspondence address: Science and Education Publishing.
Department of Economics, Faculty of Economics,
Administrative and Social Sciences, TR-27410
Gaziantep, Turkey.
Web address: http://tjant.hku.edu.tr
http://www.sciepub.com/journal/TJANT
Publication type: Bimonthly
Turkish Journal of Analysis and Number Theory ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 http://www.sciepub.com/journal/TJANT
Editor-in-Chief
Mehmet Acikgoz University of Gaziantep, Turkey
Feng Qi Henan Polytechnic University, China
Cenap Özel Dokuz Eylül University, Turkey
Assistant Editor
Serkan Araci Hasan Kalyoncu University, Turkey
Erdoğan Şen Namik Kemal University, Turkey
Honorary Editors
R. P. Agarwal Kingsville, TX, United States
M. E. H. Ismail University of Central Florida, United States
Tamer Yilmaz Hasan Kalyoncu University, Turkey
H. M. Srivastava Victoria, BC, Canada
Editors
Henry W. Gould West Virginia University, United States
Toka Diagana Howard University, United States
Abdelmejid Bayad Université d'éry Val d'Essonne, France
Hassan Jolany Université de Lille 1, France
István Mező Nanjing University of Information Science and Technology, China
C. S. Ryoo Hannam University, South Korea
Junesang Choi Dongguk University, South Korea
Dae San Kim Sogang University, South Korea
Taekyun Kim Kwangwoon University, South Korea
Guotao Wang Shanxi Normal University, China
Yuan He Kunming University of Science and Technology, China
Aleksandar Ivıc Katedra Matematike RGF-A Universiteta U Beogradu, Serbia
Cristinel Mortici Valahia University of Targoviste, Romania
Naim Çağman University of Gaziosmanpasa, Turkey
Ünal Ufuktepe Izmir University of Economics, Turkey
Cemil Tunc Yuzuncu Yil University, Turkey
Abdullah Özbekler Atilim University, Turkey
Donal O'Regan National University of Ireland, Ireland
S. A. Mohiuddine King Abdulaziz University, Saudi Arabia
Dumitru Baleanu Çankaya University, Turkey
Ahmet Sinan CEVIK Selcuk University, Turkey
Erol Yılmaz Abant Izzet Baysal University, Turkey
Hünkar Kayhan Abant Izzet Baysal University, Turkey
Yasar Sozen Hacettepe University, Turkey
I. Naci Cangul Uludag University, Turkey
İlkay Arslan Güven University of Gaziantep, Turkey
Semra Kaya Nurkan University of Uşak, Turkey
Ayhan Esi Adiyaman University, Turkey
M. Tamer Kosan Gebze Institute of Technology, Turkey
Hanifa Zekraoui Oum-El-Bouaghi University, Algeria
Siraj Uddin University of Malaya, Malaysia
Rabha W. Ibrahim University of Malaya, Malaysia
Adem Kilicman University Putra Malaysia, Malaysia
Armen Bagdasaryan Russian Academy of Sciences, Moscow, Russia
Viorica Mariela Ungureanu University Constantin Brancusi, Romania
Valentina Emilia Balas “Aurel Vlaicu” University of Arad, Romania
R.K Raina M.P. Univ. of Agriculture and Technology, India
M. Mursaleen Aligarh Muslim University, India
Vijay Gupta Netaji Subhas Institute of Technology, India
Hemen Dutta Gauhati University, India
Akbar Azam COMSATS Institute of Information Technology, Pakistan
Moiz-ud-din Khan COMSATS Institute of Information Technology, Pakistan
Roberto B. Corcino Cebu Normal University, Philippines
Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 4, 94-96
Available online at http://pubs.sciepub.com/tjant/3/4/1
© Science and Education Publishing
DOI:10.12691/tjant-3-4-1
Some Curvature Properties on a Special Paracontact
Kenmotsu Manifold with Respect to Semi-Symmetric
Connection
K. L. Sai Prasad1,*
, T. Satyanarayana2
1Department of Mathematics, Gayatri Vidya Parishad College of Engineering for Women, Visakhapatnam, Andhra Pradesh, India 2Department of Mathematics, Pragathi Engineering College, Surampalem, Near Peddapuram, Andhra Pradesh, India
*Corresponding author: [email protected]
Received April 28, 2015; Accepted July 03, 2015
Abstract The object of the present paper is to study some properties of curvature tensor R of a semi-symmetric
non-metric connection in a type of special paracontact Kenmotsu (briefly SP-Kenmotsu) manifold. We have
deduced the expressions for curvature tensor R and the Ricci tensor S of Mn with respect to semi-symmetric non-
metric connection . It is proved that in an SP-Kenmotsu manifold if the curvature tensor of the semi-symmetric
non-metric connection vanishes then the manifold is projectively flat.
Keywords: curvature tensor, ricci tensor, projective curvature tensor, non-metric connection, sp-kenmotsu
manifold
Cite This Article: K. L. Sai Prasad, and T. Satyanarayana, “Some Curvature Properties on a Special
Paracontact Kenmotsu Manifold with Respect to Semi-Symmetric Connection.” Turkish Journal of Analysis and
Number Theory, vol. 3, no. 3 (2015): 94-96. doi: 10.12691/tjant-3-4-1.
1. Introduction
Friedmann and Schouten [1,2] introduced the idea of
semi-symmetric linear connection on a differentiable
manifold. Hayden [3] introduced semi-symmetric metric
connection on a Riemannian manifold and it was further
developed by Yano [4]. Semi-symmetric connections play
an important role in the study of Riemannian manifolds.
There are various physical problems involving the semi-
symmetric metric connection. For example, if a man is
moving on the surface of the earth always facing one
definite point, say Jaruselam or Mekka or the North pole,
then this displacement is semi-symmetric and metric [1].
In 1975, Prvanovi c [5] introduced the concept of semi-
symmetric non-metric connection with the name pseudo-
metric, which was further studied by Andonie [6,7]. The
study of semi-symmetric non-metric connection is much
older than the nomenclature ‟non-metric‟ was introduced.
In 1992, Agashe and Chafle [8] introduced a semi-
symmetric connection satisfying 0X g on a
Riemannian manifold, and called such a connection as
semi-symmetric non-metric connection. Later, the
curvature properties of the connection in an SP-Sasakian
manifold were studied by Bhagwat Prasad [9], and many
others.
On the other hand, in 1976, Sato [10] defined the
notions of an almost paracontact Riemannian manifold.
After that, T. Adati and K. Matsumoto [11] defined and
studied para-Sasakian and SP-Sasakian manifolds which
are regarded as a special kind of an almost contact
Riemannian manifolds. Before Sato, in 1972, Kenmotsu
[12] defined a class of almost contact Riemannian
manifolds satisfying some special conditions. In 1995,
Sinha and Sai Prasad [13] have defined a class of almost
paracontact metric manifolds namely para Kenmotsu
(briefly P-Kenmotsu) and special para Kenmotsu (briefly
SP-Kenmotsu) manifolds.
In 1970, Pokhariyal and Mishra [14] have introduced
new tensor fields, called W and E-tensor fields in a
Riemannian manifold and studied their properties. In the
present paper, we consider the W-curvature tensor of a
semi-symmetric non-metric connection and obtained a
relation connecting the curvature tensors of Mn with
respect to semi-symmetric non-metric connection and the
Riemannian connection. It is proved that in an SP-
Kenmotsu manifold if the curvature tensor of the semi-
symmetric non-metric connection vanishes then the
manifold is protectively flat.
Let Mn be an n-dimensional differentiable manifold
equipped with structure tensors ( , ξ, η) where is a
tensor of type (1, 1), ξ is a vector field, η is a 1-form such
that
( ) = 1 (1.1)
2( ) = ( ) ; =X X X X X (1.2)
Then Mn is called an almost paracontact manifold.
Let g be the Riemannian metric in an n-dimensional
almost paracontact manifold Mn such that
( , ) = ( )g X X (1.3)
95 Turkish Journal of Analysis and Number Theory
= 0, ( ) = 0, rank = 1X n (1.4)
( , ) = ( , ) ( ) ( )g X Y g X Y X Y (1.5)
for all vector fields X and Y on Mn. Then the manifold Mn
[10] is said to admit an almost paracontact Riemannian
structure ( , ξ, η, g) and the manifold is called an almost
paracontact Riemannian manifold.
A manifold Mn with Riemannian metric „g‟ admitting a
tensor field of type (1, 1), a vector field ξ and 1-form η
satisfying equations (1.1), (1.3) along with
( ) ( ) = 0X YY X (1.6)
( ) = [ ( , ) ( ) ( )] ( )
[ ( , ) ( ) ( )] ( )
X Y Z g X Z X Z Y
g X Y X Y Z
(1.7)
2= = ( )X X X X (1.8)
is called a para Kenmotsu manifold or briefly P-Kenmotsu
manifold [13], where 𝛻 is the covariant differentiation
with respect to g.
It is known that [13] on a P-Kenmotsu manifold the
following relations hold:
( , ) = ( 1) ( )Ric X n X (1.9)
[ ( , ) , ] = [ ( , , )]
= ( , ) ( ) ( , ) ( )
g R X Y Z R X Y Z
g X Z Y g Y Z X
(1.10)
where R is the Riemannian curvature.
Let (Mn, g) be an n-dimensional Riemannian manifold
admitting a tensor field of type (1, 1), a vector field ξ
and 1-form η satisfying
( ) = ( , ) ( ) ( )X Y g X Y X Y (1.11)
( , ) = ( ) and ( ) = ( , ),
where is an associate of
Xg X X Y X Y
(1.12)
is called a special para Kenmotsu manifold or briefly SP-
Kenmotsu manifold [13].
A linear connection in a Riemannian manifold Mn is
said to be semi-symmetric connection if its torsion tensor
T satisfies
( , ) = ( ) ( ) .T X Y Y X X Y (1.13)
A semi-symmetric non-metric connection in an
almost paracontact metric manifold with torsion tensor
(1.13) is given by
= ( )X XY Y Y X (1.14)
where 𝛻 is a Riemannian connection with respect to
metric g [8].
Apart from conformal curvature tensor, the projective
curvature tensor is an other important tensor from the
differential geometric point of view. The Weyl-projective
curvature tensor W of type (1, 3) of a Riemannian
manifold Mn with respect to the Riemannian connection is
defined by [14]
( , )1
( , ) ( , ) ( , )1
Ric Y Z XW X Y Z R X Y Z
Ric X Z Yn
(1.15)
for X, Y, Z ,T M where R is the curvature tensor and
Ric is the Ricci tensor. If there exists a one-to-one
correspondence between each coordinate neighbourhood
of a Riemannian manifold Mn and a domain in Eucledian
space such that any geodesic of the Riemannian manifold
corresponds to a straight line in the Eucledian space, then
Mn is said to be locally projectively flat. For 3,n Mn is
locally projectively flat if and only if the projective
curvature tensor W vanishes. For n = 2, the projective
curvature tensor identically vanishes.
2. Curvature Tensor
The manifold Mn is considered to be an SP-Kenmotsu
manifold. Let us denote the curvature tensor of the semi-
symmetric non-metric connection by R and the
curvatre tensor of 𝛻 by R. By straight forward calculation,
we get
( , , )
= ( , , ) ( )( ) ( )( ) .X Y
R X Y Z
R X Y Z Z Y Z X (2.1)
As a consequence of equations (1.11) and (1.14),
equation (2.1) reduces to
( , , ) = ( , , ) ( , ) ( , )R X Y Z R X Y Z g X Z Y g Y Z X (2.2)
which is the relation between the curvature tensors of Mn
with respect to the semi-symmetric non-metric connection
and the Riemannian connection .
It is well known that a Riemannian manifold is of
constant curvature if and only if it is projectively flat or
conformally flat [15] and in general, the necessary and
sufficient condition for a manifold with a symmetric linear
connection to be projectively flat is that the projective
curvature tensor with respect to it vanishes identically on a
manifold [16].
As an example, if Mn is a Riemannian manifold with
vanishing curvature tensor with respect to semi-symmetric
non-metric connection, then Mn is projectively flat [8].
Analogus to this, we prove the following for an
SP-Kenmotsu manifold which is Riemannian.
Theorem 2.1: If in an SP-Kenmotsu manifold Mn the
curvature tensor of a semi-symmetric non-metric
connection vanishes, then the manifold is projectively
flat.
Proof: Since R = 0, then equation (2.2) gives
( , , ) = ( , ) ( , ) .R X Y Z g Y Z X g X Z Y (2.3)
On contracting the above equation, we get
( , ) = ( 1) ( , ).Ric Y Z n g Y Z (2.4)
Then, by (2.3) and (2.4), we get
1
( , , ) [ ( , ) ( , ) ] = 01
R X Y Z Ric Y Z X Ric X Z Yn
(2.5)
or W = 0 from (1.15), proves that the manifold is
projectively flat.
Theorem 2.2: If in an SP-Kenmotsu manifold the Ric
tensor of a semi-symmetric non-metric connection
vanishes, then the curvature tensor of is equal to the
projective curvature tensor of the manifold Mn.
Proof: From equation (2.2), we have
Turkish Journal of Analysis and Number Theory 96
( , , , )
= ( , , , ) ( , ) ( , ) ( , ) ( , ).
R X Y Z U
R X Y Z U g X Z g Y U g Y Z g X U (2.6)
On contracting the above equation, we get
' ( , ) = ( , ) ( 1) ( , ).Ric Y Z Ric Y Z n g Y Z (2.7)
Since 'Ric = 0, we have
1
( , ) = [ ( , )].1
g Y Z Ric Y Zn
(2.8)
From equations (2.2) and (2.8), we have R = W.
Theorem 2.3: In an SP-Kenmotsu manifold the projective
curvature tensor of a semi-symmetric non-metric
connection is equal to the projective curvature tensor
of the manifold.
Proof: From equations (2.2) and (2.7), we get
' ( , )1( , , ) = ( , , )
( , )1
1 [' ( , ) ( , )] .
1
Ric Y ZR X Y Z R X Y Z X
Ric Y Zn
Ric X Z Ric X Z Yn
(2.9)
The terms of the equation (2.9) can be rearranged as
1( , , ) [ ' ( , ) ' ( , ) ]
1
1= ( , , ) [ ( , ) ( , ) ]
1
R X Y Z Ric Y Z X Ric X Z Yn
R X Y Z Ric Y Z X Ric X Z Yn
(2.10)
which is „W = W, where „W is the Weyl projective
curvature tensor with respect to the semi-symmetric non-
metric connection.
Theorem 2.4: In an SP-Kenmotsu manifold with semi-
symmetric non-metric connection we have
a) ( , , ) ( , , ) ( , , ) = 0R X Y Z R Y Z X R Z X Y
b) ' ( , , , ) ' ( , , , ) = 0R X Y Z U R X Y U Z
Proof: Using the Bianchi‟s first identity with respect to
the Riemannian connection equation (2.2) gives (a). From
equation (2.6) we get (b).
Acknowledgement
The authors acknowledge Prof. Kalpana, Banaras
Hindu University, Dr. B. Satyanarayana of Nagarjuna
University and Dr. A. Kameswara Rao, G.V.P. College of
Engineering for Women for their valuable suggestions in
preparation of the manuscript. They are also thankful to
the referee for his valuable comments in the improvement
of this paper.
Competing Interest
The authors declare that there is no conflict of interests
regarding the publication of this paper.
References
[1] Friedmann and Schouten, J. A., Uber die Geometrie der
halbsymmetrischen Ubertragungen, Math Zeitschrift, 21, 211-223,
1924.
[2] Schouten, J. A., Ricci-calculus, Springer-Verlag. Berlin, 1954.
[3] Hayden, H. A., Subspaces of a space with torsion, Proc. London
Math. Soc., 34, 27-50, 1932.
[4] Yano, K., On semi-symmetric metric connection, Revue
Roumanine de Mathematiques Pures et Appliques, 15, 1579-1581,
1970.
[5] Prvanovi c, M., On pseudo metric semi-symmetric connections,
Pub. De L’Institut Math., N.S., 18(32), 157-164, 1975.
[6] Andonie, P.O.C., On semi-symmetric non-metric connection on a
Riemannian manifold. Ann. Fac. Sci. De Kinshasa, Zaire sect.
Math. Phys., 2, 1976.
[7] Andonie, P.O.C., Smaranda, D., Certains connections semi-
symmetriques, Tensor (N.S.), 31, 8-12, 1977.
[8] Agashe, N. S. and Chafle, M. R., A semi-symmetric non-metric
connection on a Riemannian manifold, Ind. J. of Pure and Appl.
Math., 23(6), 399-409, 1992.
[9] Prasad, On a semi-symmetric non-metric connection in an sp-
Sasakian manifold, Istambul Univ.Fen Fak. Mat. Der., 53, 77-80,
1994.
[10] Sato, I., On a structure similar to the almost contact structure,
Tensor (N.S.), 30, 219-224, 1976.
[11] Adati, T. and Matsumoto, K., On conformally recurrent and
conformally symmetric P-Sasakian manifolds, TRU Math., 13, 25-
32, 1977.
[12] Kenmotsu, K., A class of almost contact Riemannian manifolds,
Tohoku Math. Journal, 24, 93-103, 1972.
[13] Sinha, B. B. and Sai Prasad, K. L., A class of almost para contact
metric Manifold, Bulletin of the Calcutta Mathematical Society,
87, 307-312, 1995.
[14] Pokhariyal, G.P., Mishra, R.S., The curvature tensors and their
relativistic significance, Yokohoma Math.J., 18, 105-108, 1970.
[15] Yano, K. and Bochner, S., Curvature and Betti numbers, Annals of
Math Studies 32, Princeton University Press, 1953.
[16] Sinha, B. B., An introduction to modern Differential Geometry,
Kalyani Publishers, New Delhi, 1982.
Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 4, 97-103
Available online at http://pubs.sciepub.com/tjant/3/4/2
© Science and Education Publishing
DOI:10.12691/tjant-3-4-2
Study of Some Sequences of Prime Numbers Defined by Iteration
Idir Sadani*
Department of Mathematics, University of Mouloud Mammeri, Tizi-Ouzou. Algeria
*Corresponding author: [email protected]
Received May 11, 2015; Accepted July 22, 2015
Abstract We study the properties of prime number sequences obtained using a well-defined equivalence relation
. It will be seen that the elements of each class of are all prime numbers which constitute the fundamental
object of our study. The number of prime numbers of each class less than or equal to a given quantity , the number
of the different equivalence classes and some other results will be deduced.
Keywords: distribution of prime numbers, equivalence relation, iteration, asymptotic formula
Cite This Article: Idir Sadani, “Study of Some Sequences of Prime Numbers Defined by Iteration.” Turkish
Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 97-103. doi: 10.12691/tjant-3-4-2.
1. Introduction
Let be the set of prime numbers, and for all ,
let denote the number of prime numbers less than or
equal to . The prime number theorem which was shown
independently by de la Vallée Poussin [1], and Hadamard
[2] in , states that:
~ ,asln
xx x
x (1)
or
( ) ( ), ,x Li x as x
where is the logarithmic integral of defined by:
1
0 10lim .
ln ln
xdy dyLi x
y y
We can give an equivalent statement for this theorem as,
for example, let denote the n'th prime number. Then
1 1 ~ ln .nn p Li n n n as n (2)
One of our objectives here is to use a restriction of the
function to to study intrinsic properties of some
sequences of primes defined by iterations. The point of
departure for this study is the construction of an
equivalence relation that we denote by . The purpose of
this equivalence relation is to show first, that there is a
recurrence relation between prime numbers which can be
arranged in an infinity of well-defined classes dependent
on the initial value. Second, the use of its properties with
the famous prime number theorem is one way to find and
prove other results for the possible applications in number
theory. Part of our motivation came from the prime
number theorem and the Riemann hypothesis. Also, it was
an attempt to establish the relationship between these
prime numbers classes and several as-yet unproved
conjectures, such as Goldbach's Conjecture and Twin
Prime Conjecture.
The structure of this paper is as follows. In section 2,
we begin with the definition of our equivalence relation
and its equivalence classes where is a prime number,
which constitute the fundamental object of study, and we
propose some preliminary results. In section 3, we exhibit
the main results of this work by introducing and studying
the functions and . The methods used
here are part of elementary number theory and we have
attempted to present the ideas in as elementary a way as
possible. Finally, in section 4, we give some open
questions related this subject.
Notation
1) We set
1,
p x
x
this function count the number of primes less than .
2) We define the following functions:
times
( ) ( ) ,
n
n x x
times
1 1 1( ) ( ),
n
n y y
where is the composition operator.
3) In many situations, we search to estimate ∑ ,
where . Then, we use the following formula:
2
( ) ( )( ) .
ln ln
x
p x n x
f n f tf p dt
n t
(3)
2. Preliminary results
2.1. The Classes and Its Elements
We start with the following lemma:
Turkish Journal of Analysis and Number Theory 98
Lemma 2.1. Let be the restriction of to . Then,
is a bijection and its inverse function is
.
Notation. Throughout this paper, we simply use the
notation to designate the restriction of to the set
instead of using .
The proof of the following theorem is obvious.
Theorem 2.1. We define the relation on the set of prime
numbers defined by: if and are two prime numbers,
if and only if:
1) for any prime number ,
2) there exists such that .
Then, is an equivalence relation.
Notation. The elements of the equivalence class are
defined by:
2 1 2, , , , , , .p p p p p p
The smallest element of , which we denote by , is
called the origin of the class . Then, in this case, we note
that the sets and have the same elements.
Example 1.
2 2,3,5,11,31,127, ,7 7,17,59, .
Notation. We denote by the set of all origins . The
set is given explicitly by:
0 {2,7,13,19,23,29,37,43 },
and we denote by the set of all origins less than or
equal to .
Theorem 2.2. Let be a prime number. Then, is not
a prime number implies that, for all , there is no an
integer such that .
Proof. Let . By the prime number theorem,
represents the -th prime number which we denote by
.
Next, , implies that
which is the -th prime number, so we obtain the
formula:
1 1 1 21p p .p p
p p
(4)
Now, we use the equation (4) which composed by
gives
2 1 1 21 1( ) ( )
p ( ) (p ) ( ( )),p p
p p
that is the -th prime number. We can write
1 1 2 31 2( ) ( )
(p ) ( ( )) p ( )p p
p p
Inductively, we obtain the following general formula:
( 1) ( )p ( ), 1.n
n pp n
(5)
To each number , we associate the set :
1 2{ ( ), 0} { , ( ), ( ), }npA p n p p p .
Now, suppose that is not a prime number, we
must prove that for all , there is no such that
. To show this, suppose that there exists a
positive integer such that , i.e.,
and we show that is a prime
number. Hence, we have, if and only if
, which implies that
But, for all is prime,
which proves what we wanted.
Remark. The cardinality of the set is infinite.
Theorem 2.3. There exists a partition of the set of
prime numbers defined by
1 2
{ , , , , },p p pnP A A A
such that
, where
and is not prime number.
Proof. On the one hand, according to the prime number
theorem, the number of prime numbers belonging to
is equal to . On the other hand, by
the Theorem 2.2, there are prime
numbers which do not belong to
. We denote these numbers
by , where . So, we
obtain of sets which are defined by:
1{ , ( ), , ( )},lpk
p k k kkA p p p
such that ∑ and
. Finally, it is not difficult to see that the sets
constitute a partition of the set of prime
numbers less than or equal to . So, since is arbitrary in
, letting tend to we obtain an infinity of sets
which then form a partition of the set .
Theorem 2.4. Let be a finite set of prime numbers
defined by
1 2{ , , , },nA p p p
such that are consecutive. Then,
there exists at least one set where , such that
. Before giving the proof of this result, we give an
illustrative example.
Example 2. Let be a set which defined by
{5,7,11,13,17}.A
• The class of the integer is and its
cardinality is greater than .
• We notice that , therefore the integer constitutes
the origin of a new class which is and its
cardinality is greater than . It only remains to see that the
prime number does not belong to the two classes and
. Thus the prime number constitutes the origin of a
new class and clearly its cardinality is .
Proof of theorem 2.4. We suppose that each class
containing at least two elements i.e., the
cardinality of is equal or greater than , and we suppose
that is prime. According to the theorem of prime
numbers, the interval contains prime numbers. This leads to the two
following cases:
• If , i.e., there exists only one prime number in
, namely . Therefore, we obtain
and is not a prime number since and are
99 Turkish Journal of Analysis and Number Theory
consecutive. Then contains only one element in which
is the prime number .
• If , i.e., there exist at least two prime numbers in
, namely . We suppose that where
, is a prime number. Then is not prime
and since , then, the unique element of
is .
Finally, in both cases, there exist at least one class
where , such that is a unique element of this class
in , i.e., . The proof now is completed.
We have the following definition:
Definition 1. Let be the set defined as in the Theorem
2.4 and is a prime number belonging to . We say that
is an outside class of , if
1 ( ) . .,| | 1 .p is not prime in A and p A i e p in A
if in , we say that this class is an inside class of
.
Remark. According to the Definition 1, the number is
the smallest element of the class in , therefore, it is the
origin of this class i.e., .
2.2. Study of
The function
, , is a contraction.
Therefore, the sequence defined by:
1 ( ) ,ln
nn n
n
xx p x
x
admits a fixed point. Since decreasing and is bounded
below by , then it converges to the single
fixed point .
Notation. ⏞
, where is the
function composition operator.
Theorem 2.5. Let be an initial value. Then
10 0
0 0
( ) ln ( ) ln ( ),ii
i i
p x e p x e p x
(6)
Proof. We set and we have
01 0 1 0 2 0
0
12 1 2 1 2 1
1
( ) ln ln lnln
( ) ln ln lnln
xx p x x x x
x
xx p x x x x
x
1 1 2( ) ln ln lnln
nn n n n n
n
xx p x x x x
x .
And combining all these, we obtain
1 0 2
0
ln ln ( ) ln ln .n
n n i
i
x p x x x
(7)
Which is equivalent to
0
1 0
l .nn
in i
xx
x
(8)
Then, passing to the limit, we obtain
0 0
1 0
1
0
0
lim lim lim ln
lim ln ln ln ( ).
n
in n nn n i
n
in
i
x xx
x p x
x p x
(9)
Consequently,
00
0 0
ln ln ln ( ),i i
i i
xx x p x
e
(10)
and since
lim ,nn
x e
the formula (10) is obviously equivalent to
0
0
( ) ln ( ).i
i
p x e p x
Which is the desired result.
Lemma 2.2. Let be the sequence which is
decreasing and bounded below by , and let be a
real number. Then, the number of iterations, which denote
by , is depend on and the initial value , and
given by :
0 0
ln ln( , ) ~ ,as ,
lnln lnlniter
c xn x x
(11)
where is a bounded value, and , i.e.,
depend on .
Proof. We have
2 1 1 1 0 1 1 0'( ) , [ , ],x x p x x x x
where is the derivative of the function
'3 2 2 2 1 2 1 1 0'( ) '( ) ,x x p x x p p x x
where, Inductively, we obtain
1 1 0 1
1
'( ) . , [ , ].n
n n i n n n
i
x x p x x x x
Next, there exists a real number , such that
1 0 1 0'( ) .n
n nx x p x x
Or in an equivalent way, since, and
,
1 0 0 1'( ) . .n
n nx x p x x
We can extract the value of from the above equality:
1 0 1
0 0
ln( ) ln( ),
ln '( ) ln '( )
n nx x x xn
p p
and replacing by its value, we get
1 0 1
0 0 0 0
ln( ) ln( )
ln(ln 1) 2lnln ln(ln 1) 2lnln
n niter
x x x xn
Now, according to the definition of , the sequences
and are bounded, then it holds for the difference
Turkish Journal of Analysis and Number Theory 100
which we denote it by . Then, we have, by
setting and :
0 0
1
0 0
ln,
ln ln 1 2lnln
ln( )
ln(ln 1) 2lnln
iterc
n x
x x
(12)
Finally, we have the following limits:
0 as ,x
0 0 0ln(ln 1) 2lnln lnln as ,x
1 as ,x x x x
then,
0 0
ln ln( , ) ~ , as .
lnln lnlniter
c xn x x
3. Principal Results
3.1. Definition and Estimate of the Functions
and
We set the following definition:
Definition 2. Let . We define
0
0 0
0 ( )0
( , ) 1 1, .iterp p x p p x
p p lp p
x p l
Then counts the number of prime numbers
less than or equal to belonging to the class .
Lemma 3.1. We have
0 0( , )~ ( , )~ ( , ), as .iter iter iterx p n x n x p x
Proof. According to the definition of , there exists
such that . Next, from the prime number
theorem, we have
( )~ ( ) , ( )~ ( ), .ln( )
n nxx p x x x p x x
x
Moreover, supposing that with not a
prime number, it follows that
0( ( ))~( ( ) ), .n np x p x x
And,
0
0 0
0 0( ) ~ ( )0
( , )
0
1
( , ) 1 1~ 1~ 1
1 ( , )~ ( , ),
iterp p x p p x p x p x
p p l p p lp p p p
n x
iter iter
l
x p
n x n x p
as Definition 3. Let . We define the functions
and as follows:
0
0
0
, ln ln .p p x
p p
x p p
0
, 0
lnln( , ) ln ln .
lnlnp x p p
xx p p
p
Theorem 3.1. We have
0( , )~ln (ln ).x p x o x
Proof. In view of the proof of Lemma 3.1, we have,
0 00 0
0 00 0
ln ln ( ) ~ ln ln ~ ,n n
i i
i i
x xx x x p x
p p
which is equivalent to
0
0 0
0
0
0
ln ln ln ln ( ) ln ln ln ln ( )
ln ln ~ ln .ln
n n
i i
i i
p p x
p p
x x x x
p x p
Finally, for all fixed and tend to infinity, we have
, then
0
0
ln ln ~ ln ln .p p x
p p
p x o x
i.e., Theorem 3.2. We have,
0
0
,1) , ~ ,
ln lniter
x px p x
x
0ln ln
2) , ~ ~ ,ln ln ln ln
iter iterx x
x p x o xx x
0
0
, ln3) , , .
ln ln ln lniter
x p xx p O x
x x
Proof.
1) For the proof of the first formula, we have, on the one
hand
0
0 0
0 0
0
, ln ln ln ln 1
, ln ln .
p p x p p x
p p p p
iter
x p p x
x p x
Then we obtain
0
0
,, .
ln lniter
x px p
x
On the other hand, for all with , we have
0
0
00
0 0
0
, ln 1 ln n ln 1
ln ln ln , ,
( , )ln
ln ln ln
p p xx p xp pp p
iter iter
x p x x
x x p x p
x px
x
Then
101 Turkish Journal of Analysis and Number Theory
0
0
,, ln .
ln δ ln lniter
x px p x
x
Now, according to Lemma 3.1,
( ) , and then for x sufficiently
large (depending on ), ,
and thus
0
0
,, .
ln δ ln lniter
x px p
x
Now, for all , we can choose more near to 1.
For this , so that
, and for sufficiently large,
we have
0
0
,, 1 .
ln lniter
x px p
x
2) According to Theorem 3.1, we have
00
, ln ln, ~ ~
ln ln ln ln ln ln
~ , .
iter
iter
x p x xx p o
x x x
x x
3) Concerning the third equality, we have
0 0, , ln ln .iterx p x p x
Evaluate now the difference
0 , 0
lnln, ln ln ln ln .
lnlniter p x p p
xx p x p
p
Then
0
, 0
, 0
, 0
, 0
, 0
lnln, ln ln ln ln
lnln
lnlnln ln ln ln
lnln
lnlnln ln ln ln
lnln
lnln+ ln ln ln ln
lnln
lnln
lnln
iter
p x p p
p x p p
p x p p
x p x p p
p x p p
xx p x p
p
xx p
p
xx p
p
xx p
p
x
, 0
0
, 0
0
, 0
ln ln
ln ln ln ln
,ln
, 1ln
x p x p p
x
px p x p p
x
p txx p x p p
pp
x p
dtx p
t t
dtx p
t t
However, for [√ ]
,, ,00 0
1 1 1p t p t p tp x p px p x p p p x p p
Then,
0 0 01 1 , , , ,iter iter iter
p t p x
t p x p t p
thus
0
00
0
0
, ln ln
,,
ln ln
2ln,
ln ln ln
2, .
ln ln
iter
x iter
x
x
x
x
x
x p x x
t px p dt
t t
tx p dt
t t t
x p dtt t
By using , since
0 0ln
, , ln ln ln ln ln .ln ln
iterx
x p x p x x xx
we obtain
0 0ln
, ln ln , lnln ln
iterx
x p x x p O x Ox
Then,
(
)
3.2. Definition and Estimate of and
3.2.1. Definition and Estimate of
Definition 4.
1. Let be a positive real number. We denote by
the number of different classes such that .
Precisely,
0 0
0
( ) : 1, .c
p x
x p
2. We denote by the function defined by
0 0
0
( ) n .l
p x
x p
Example 3. In , the value represents the origin
of the class but the values do not, since they
belong to the same class . The value represent the
origin of the class . Thus in this case, we have .
We have the following result:
Theorem 3.3. We have,
lim ( ) .cx
x
Proof. Suppose that the number of different classes is
finite as . We know that
0
1.
iip
(13)
Next, let , where is finite by hypothesis. We
obtain
0 0
1 1.
kii k i i
p p
Therefore, we have
0
1
ki ip
, and since the second
sum has a finite number of terms, we deduce
Turkish Journal of Analysis and Number Theory 102
0
1,
kk i ip
which contradicts formula (13).
Theorem 3.4. Let . Then
( ) ( ) ( ( )).c x x x (14)
Proof. To find the value of means that we estimate
the number of origins . Clearly, the prime number is
an origin that means is not a prime number. Then,
let be between and , and let be the
greatest prime number in , therefore is not a
prime number and for all integer ,
. So, we only have to search the numbers which
are not primes and less than . Thus, we have
1° The number of the even numbers less than or equal
to equal to
.
2° The number of the odd numbers less than or equal to
equal to
such that is
the number of the prime numbers less than or equal to .
Next, we add the two quantities, we obtain, since
, the quantity which is equal to
( ) ( ) ( ( )).c x x x
3.2.2. Estimate of
For all initial value , we define the following
sequence:
1 ln .n n ny y y (15)
It is not difficult to see that, this sequence is stationary
for and increasing divergent to infinite for
and as . It is clear that, inductively,
, then we have the
following consequence:
Lemma 3.2. We have,
0
0 0
2 2
ln ( )lnln lnln( ) ,
ln lnln ln
p xc
p xx x x xx
x xx x
where represents the origin of the classes .
Proof.. Since for all , we have ,
then
0 0ln ln 0ln , ,0 0 0 lnln 0
00 0
00 0
0
0
lnln lnln lnln
ln lnlnln ~ ln ln
lnln
l .n ln
n x pp x p p x n p np
p x p x
iter
p x
p p p
x pp x p
p
x x p
In the expression:
0
ln ,0 0
,ln lnnp p x n
p
the value of obviously depends on and if we denote
by , then, we have
0
0
~ ( ),p
p
n x
i.e., the number of elements of the form must
be equal to number of prime numbers .
According to the formula (3), we have
lnln
lnln ,ln
p x
x xp
x
therefore, the inequality is obtained directly by
substitution.
Proposition 3.1. We have,
0
1 lnln1
ln lnln ln
1( ) 1
ln lnln
xx
x x x
x xx x
(16)
0( )~ as ..x x x
Proof.
1. We have
00
2
0
0
lnlnln
( ) ( ( ))lnln
lnlnln ( ( ) ( ( ))) ln .
ln
p x
p x
px x
x xxx
x xp x x x
x
Moreover, since
0
0 0
ln ln 1 ( ) ln .c
p x p x
p x x x
And recalling that
( )~ .ln
xx
x
We obtain
0
0
lnlnln ln
ln ln (ln lnln ) ln
1 lnln1 .
ln lnln ln
p x
x x x xp x
x x x x x
xx
x x x
The second inequality is obtained in the same way,
0
0
ln lnln ln (ln lnln )
11 .
ln lnln
p x
x xp x
x x x x
xx x
2. It is enough to tend to in the inequality (16).
4 Conclusion and Future Work
There is a lot of results on properties of prime numbers.
There are innumerable ideas in this field regarding the
randomness of prime numbers. However, it turns out that
prime numbers do not appear absolutely randomly,
meaning, it is not entirely true that there is no way
whatsoever to see some relations and find some functions
103 Turkish Journal of Analysis and Number Theory
to generate a lenght of few prime numbers. Patterns do
emerge in the distribution of primes over varied ranges of
number sets. In this paper we have investigated sequences
of primes obtained by the equivalence relation , from
which we have associated various arithmetic functions. As
we have seen in the sections above, many classical results
stated in elementary number theory can be restated with
our sequences. Also, there are many other asymptotic
formulas can be deduced, we have just presented several
of them as examples.
As far as motivation, the concern relates to the
developing of new ideas for solving great unsolved
problems and conjectures encountered in this field.
A highly intriguing area in primes is the concept of twin
primes. These are prime numbers which differ by the
number for example: and , and , and , etc.
There is an attempt being made to prove that there are
infinitely many twin primes that exist in the natural
number system. These are the concepts that come to mind.
Others are pretty much minor results.
Yitang Zhang, in his paper [3], attacked the problem by
proving that the number of primes that are less than
million units apart is infinite. While million is a long,
long way away from , Zhang's work marked the first
time anyone was able to assign any specific proven
number to the gaps between primes.
In November , James Maynard, in [4] , introduced
a new refinement of the GPY sieve, allowing him to
reduce the bound to and show that for any there
exists a bounded interval containing prime numbers.
At the present time, let me explain and share some
general ideas and questions about my future work. Firstly,
the questions raised are very broad in scope and cannot be
addressed directly. This means that, it is preferable to
resort to a methodology (plan in stages) to tackle the great
problems in a structured manner. Secondly, for that reason,
we have proposed and developed the results of this paper
(see the problem 1 and 2).
Finally, here are just a few questions and conjectures a
little more direct, I think are important.
Problem 1 The twin prime conjecture is equivalent to
conjecturing the translates of the (the
values of the pair ) are simultaneously prime
values infinitely often. The question is: show that if the
are simultaneously prime values
infinitely often then the are
simultaneously twin primes infinitely often, and then the
twin prime conjecture is true. We have posed this question
from the perspective of finding recurrence relation
between primes.
Problem 2 Let and be twin primes. Show that
2
1 1 3 22
1
( ) ( ) ~ ( ln ).i n
i i
i
p p O n n
Problem 3
1. Does the set contain an infinity of twin primes?
2. Let fixed. Are there an infinity of primes of
the form such that is prime?
3. Study of
0
1, ,
11p
s
s is a complex number
p
There is continuing research to prove these conjectures
and questions rigorously, using the results of this paper
and advanced techniques in number theory in the next
work.
Acknowledgement
We sincerely thank the editor and the referees for their
valuable comments and suggestions.
References
[1] C. J. de la Vallée Poussin, “Recherche analytique sur la théorie
des nombres”, Ann. Soc. Sci. Bruxelle, 20. 183-256. 1896.
[2] J. Hadamard, “Sur la distribution des zéros de la fonction et
ses conséquences arithmétiques”, Bull. SOC. Math. France, 24.
199-220. 1896.
[3] Y. Zhang, “Bounded gaps between primes”, Ann. of Math, 179,
1121-1174. 2014.
[4] J. Maynard, “Small gaps between primes”, Ann. of Math, 181.
383-414. 2015.
Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 4, 104-107
Available online at http://pubs.sciepub.com/tjant/3/4/3
© Science and Education Publishing
DOI:10.12691/tjant-3-4-3
On the Generalized Degenerate Tangent Numbers and
Polynomials
Cheon Seoung Ryoo*
Department of Mathematics, Hannam University, Daejeon, Korea
*Corresponding author: [email protected]
Received May 24, 2015; Accepted August 01, 2015
Abstract In [7], Ryoo introduced the generalized tangent numbers and polynomials. In this paper, our goal is to
give generating functions of the degenerate generalized tangent numbers and polynomials. We also obtain some
explicit formulas for degenerate generalized tangent numbers and polynomials.
Keywords: generalized tangent numbers and polynomials, degenerate generalized tangent numbers and
polynomials
Cite This Article: Cheon Seoung Ryoo, “On the Generalized Degenerate Tangent Numbers and
Polynomials.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 104-107. doi:
10.12691/tjant-3-4-3.
1. Introduction
In [2], L. Carlitz introduced the degenerate Bernoulli
polynomials. Recently, Feng Qi et al. [3] studied the
partially degenerate Bernoull polynomials of the first kind
in p-adic field. In [9], Ryoo constructed the degenerate
tangent numbers and polynomials. In this paper, we
introduce the degenerate generalized tangent numbers and
polynomials. We also obtain some interesting properties
for degenerate generalized tangent numbers and
polynomials. Throughout this paper, we always make use
of the following notations: denotes the set of natural
numbers and 0 , denotes the set of
complex numbers. The tangent numbers are defined by
means of the following generating function:
2
0
2.
! 21
n
n tn
tT t
n e
(1.1)
Ryoo [6] defined tangent polynomials by multiplying xte on the right side of the Eq. (1.1) as follows:
2
0
2.
! 21
nxt
n tn
tT x e t
n e
(1.2)
Let be Dirichlet's character with conductor d
with 1 mod 2 .d Then the generalized tangent
numbers associated with associated with ,, nT , are
defined by the following generating function
1 2
0,2
0
2 1.
!1
d a at na
ndtn
a e tF t T
ne
(1.3)
We now consider the generalized tangent polynomials
associated with ,, ,nT x are also defined by
1 2
0
2
,0
2 1,
1
.!
d a atxta
dt
n
nn
a eF x t e
e
tT x
n
(1.4)
When 0 , above (1.3) and (1.2) will become the
corresponding definitions of the tangent numbers nT and
polynomials nT x (see [6]). For more theoretical
properties of the generalized tangent numbers and
polynomials, the readers may refer to [7].
For a variable t, we consider the degenerate tangent
polynomials which are given by the generating function to be
/
,2/0
21 .
!1 1
nx
nn
tt T x
nt
(1.5)
When 0,x , ,0n nT T are called the
degenerate tangent numbers(see [8]).
We recall that the classical Stirling numbers of the first
kind 1 ,S n k and 2 ,S n k are defined by the relations
(see [9,10])
1 20 0
, , ,n n
k nn k
k k
x S n k x and x S n k x
respectively. Here 1 1n
x x x x n denotes
the falling factorial polynomial of order .n The numbers
2 ,S n m also admit a representation in terms of a
generating function
2
1, .
! !
mt
n
n m
etS n m
n m
(1.6)
105 Turkish Journal of Analysis and Number Theory
We also have
1
log 1, .
! !
mn
n m
ttS n m
n m
(1.7)
The generalized falling factorial n
x with increment
is defined by
1
0
n
nk
x x k
(1.8)
for positive integer n, with the convention 0
1.x We
also need the binomial theorem:
for a variable x ,
/
0
1 .!
nx
nn
tt x
n
(1.9)
2. On the Degenerate Generalized
Tangent Polynomials
In this section, we define the degenerate generalized
tangent numbers and polynomials, and we obtain explicit
formulas for them. Let be Dirichlet's character with
conductor d with 1 mod 2 .d Then the
degenerate generalized tangent numbers associated with , , ,nT are defined by the following generating
function
1 2 //0
2 /
, ,
0
2 1 11
1 1
.!
d a axa
d
n
nn
a tt
t
tT x
n
(2.1)
When 0,x , , , ,0n nT T are called the
generalized degenerate tangent numbers.
From (2.1) and (1.4), we note that
, ,0
0
1 2 //0
2 /0
1 2
0
2
,0
lim!
2 1 1lim 1
1 1
2 1
1
.!
n
nn
d a axa
d
d a atxta
dt
n
nn
tT x
n
a tt
t
a ee
e
tT x
n
Thus, we get
0
, , ,l 0im , .n nT x T x n
From (2.1) and (1.5), we have
, ,0
1 2 //0
2 /
, ,0 0
,0 0
!
2 1 11
1 1
! !
.!
n
nn
d a axa
d
m l
m lm t
nn
l n ln l
tT x
n
a tt
t
t tT x
m l
n tT x
l n
(2.2)
Therefore, by (2.1) and (2.2), we obtain the following
theorem.
Theorem 1. For 0n , we have
, , , ,0
.n
n l n ll
nT x T x
l
For d with 1 mod 2 ,d we have
1/ 2 /
2 /0
, ,0
1 2 //0
2 /
1
, /0 0
21 1 1
1 1
!
2 1 11
1 1
21 .
!
dx l l
dl
n
nn
d a axa
d
ndln
n dn l
t l tt
tT x
n
a tt
t
l x td l T
d n
(2.3)
By comparing coefficients of !
mt
min the above
equation, we have the following theorem.
Theorem 2. Let be Dirichlet's character with
conductor d with 1 mod 2 .d Then we have
(1) 1
, , , /0
21 ,
dam
m m da
a xT x d a T
d
(2) 1
, , , /0
21 .
dam
m m da
aT d a T
d
From (2.1), we can derive the following relation:
, , , ,0
1 2 //0
2 /
1 2 /
0
2 /
12 /
0
1
0 0
2!
2 1 11
1 1
2 1 1
1 1
2 1 1
2 1 2 .!
m
m mm
d a axa
d
d a a
a
d
da a
a
mda
mm a
tT d T x
m
a tt
t
a t
t
a t
ta a
m
(2.4)
Turkish Journal of Analysis and Number Theory 106
By comparing of the coefficients !
mt
m on the both sides
of (2.4), we have the following theorem.
Theorem 3. For ,n we have
1
, , , ,0
2 2 1 2 / .d
am m m
a
T d T a a
By (2.1), we have
1
0 0 0
, , , ,0
12 / /
0
2 1 2 .!
2!
2 1 1 1
md ma
l m lm a l
m
m mm
da a x
a
m ta x a
l m
tT x d T x
m
a t t
(2.5)
By comparing of the coefficients !
mt
m on the both sides
of (2.5), we have the following theorem.
Theorem 4. For ,n we have
, , , ,
1
0 0
2
2 1 2 .
m m
d ma
l m la l
T x d T x
ma x a
l
From (2.1), we have
, ,0
1 2 //0
2 /
1 2 /
0
2 /
/ /
, ,0 0
, ,0 0
!
2 1 11
1 1
2 1 1
1 1
1 1
! !
.!
n
nn
d a ax ya
d
d a a
a
d
x y
n n
n nn n
nn
l n ln l
tT x y
n
a tt
t
a t
t
t t
t tT x y
n n
n tT y
l n
(2.6)
Therefore, by (2.6), we have the following theorem.
Theorem 5. For ,n we have
, , , ,0
.n
n k n kk
nT x y T x y
k
From Theorem 5, we note that , ,nT x is a Sheffer
sequence.
By replacing t by 1te
in (2.1), we obtain
1 2
0
2
, ,0
, , 20
, , 20 0
2 1
1
1 1
!
,!
, .!
d a atxta
dt
nt
nn
mn m
nn m n
mmm n
nm n
a ee
e
eT x
n
tT x S m n
m
tT x S m n
m
(2.7)
Thus, by (2.7) and (1.4), we have the following theorem.
Theorem 6. For ,n we have
, , , 20
, .m
m nm n
n
T x T x S m n
By replacing t by 1/
log 1 t
in (1.4), we have
1/,
0
1 2 //0
2 /
, ,0
1log 1
!
2 1 11
1 1
,!
n
nn
d a axa
d
m
mm
T tn
a tt
t
tT x
m
(2.8)
and
1/,
0
, 10 0
1log 1
!
, .!
n
nn
mm n
nm n
T tn
tT x S m n
m
(2.9)
Thus, by (2.8) and (2.9), we have the following theorem.
Theorem 7. For ,n we have
, , , 10
, .m nm n
n
T x T x S m n
References
[1] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers,
Utilitas Math. 15(1979), 51-88.
[2] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math.
(Basel) 7(1956), 28-33.
[3] F. Qi, D. V. Dolgy, T. Kim, C. S. Ryoo, On the partially
degenerate Bernoulli polynomials of the first kind, Global Journal
of Pure and Applied Mathematics, 11(2015), 2407-2412.
[4] T. Kim, Barnes' type multiple degenerate Bernoulli and Euler
polynomials, Appl. Math. Comput. 258(2015), 556-564.
[5] H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli
numbers associated with Daehee numbers, Adv. Stud. Contemp.
Math. 18(2009), no. 1, 41-48.
[6] C. S. Ryoo, A Note on the tangent numbers and polynomials, Adv.
Studies Theor. Phys., 7(2013), no. 9, 447-454.
[7] C. S. Ryoo, Generalized tangent numbers and polynomials
associated with p-adic integral on p, Applied Mathematical
Sciences, 7(2013), no. 99, 4929-4934.
107 Turkish Journal of Analysis and Number Theory
[8] C. S. Ryoo, Some identities on the (h; q)-tangent polynomials
and Bernstein Polynomials, Applied Mathematical Sciences,
8(2014), no. 75, 3747-3753.
[9] C. S. Ryoo, Notes on degenerate tangent polynomials, to appear in
Global Journal of Pure and Applied Mathematics, Volume 11,
number 5(2015), pp. 3631-3637.
[10] P. T. Young, Degenerate Bernoulli polynomials, generalized
factorial sums, and their applications, Journal of Number Theorey,
128(2008), 738-758.
Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 4, 108-110
Available online at http://pubs.sciepub.com/tjant/3/4/4
© Science and Education Publishing
DOI:10.12691/tjant-3-4-4
Generalized s-topological Groups
Rehman Jehangir1,*
, Moizud Din Khan2,*
1Department of Mathematics, Preston University Kohat (Islamabad), Pakistan 2Department of Mathematics, COMSATS institute of information technology, Chak Shehzad Islamabad, Pakistan
*Corresponding author: [email protected], [email protected]
Received June 04, 2015; Accepted August 09, 2015
Abstract In this paper, we explore the notion of generalized semi topological groups. This notion is based upon
the two ideas, generalized topological spaces introduced by Csaszar [2,3] and the semi open sets introduced by
Levine [7]. We investigate on the notion of generalized topological group introduced by Hussain [4]. We explore the
idea of Hussain by considering the generalized semi continuity upon the two maps of binary relation and inverse
function.
Keywords: generalized semi open sets, generalized compact sets, generalized continuity, generalized continuity,
generalized discrete sets
Cite This Article: Rehman Jehangir, and Moizud Din Khan, “Generalized s-topological Groups.” Turkish
Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 108-110. doi: 10.12691/tjant-3-4-4.
1. Introduction
Let denotes the generalized topological space ( , ).
In accordance with [3], let A be generalized semi open
if and only if there exists a generalized open set ( -open
set) such that , where
denotes the generalized closure of the set O in . For more
details on generalized topological spaces, we refer to [2,3].
In 2013, Murad et al. [4], defined and studied the concept
of generalized topological groups ( -topological groups).
This study was further extended and published in [5] and
[6]. In 2015, C. Selvi and R. Selvi [10] were motivated by
-topological groups [4] and S-topological groups [9], and
defined on new notion with the name of generalized S-
topological groups.
In this paper, we intend to generalized further the
notion of -topological groups and -S-topological groups
by using -semi continuity. -semi continuity is a
generalization of -continuity and it was defined by Á.
Császár in [3].
2. Generalized Semi Topological Group
In this section, we will explore the notion of
generalized semi topological group. Generalized semi
topological groups contains the structure of generalized
topology and groups. The whole idea is backed by the
generalized semi continuity, as the binary operation and
the inverse map undergo the process of generalized semi
continuity. We will study the basic definitions and gradual
development of the phenomenon.
Let ( ) and ( ) be generalized
topological spaces and is generalized semi
continuous, then for any subset of ,
( )
Let ( , ) and ( , ) be generalized topological spaces and let ( , ) = ( , ) be their product generalized topological space if is
generalized semi open set in and is generalized semi
open in then is generalized semi open
in ( ) Assume that = , where is
generalized open in and for
. This is nowhere dense set as well. Then,
1 2 2
1 2
1 2 1
1 2 2 1 21
A A O B O B
O O B O O B B B
But is generalized open set in and
1 1 2 1 2
1 2
1 2 2 1
2
2
1
B O O B B B cl O O
cl O O
Hence, 1 2 ( ) ( )
= ( )
1 2 1 2 1 2 1 2O O A A cl O O
This proves that 1 2 is generalized semi open set
in .
. Let: ( , ) ( , ) semi generalized
continuous map between two generalized topological
spaces. Let be semi generalized compact set relative to
( , ) then is semi generalized compact in ( , ).
Let be any collection of generalized
open set of ( , ), such that : i . Then
{ ( ): holds by hypothesis and there
exists a finite subset of of such that { ( ):
which shows that is semi generalized
compact in ).
109 Turkish Journal of Analysis and Number Theory
3. Semi Generalized Topological Group
In this section, we will define semi generalized
topological groups ( -s-topological groups) and
investigate its basic properties.
is said to be a - s-topological
group if
(1 is generalized topological space;
(2). is a group;
(3). The multiplication map , defined by
and the inverse map defined by
; are the generalized semi-continuous.
Equivalently, is semi generalized topological
group if and for each generalized open set
containing , there exist generalized semi
open sets containing and containing y, such that,
. Since every - is continuous therefore, every -topological group is
- -topological groups. And - -topological group may
not be a -topological group. Further, we note that, every
-topological group is -s-topological group and every -
s-topological group is - -topological group. However,
converses may not be true in general. It is evident from the
following example:
. Let = = {0 1} be the two-
element (cyclic) group with the multiplication mapping
= +2 the usual addition modulo 2. Equip with
the Sierpinski topology = { , {0} }. Then
the collection of all the semi open sets
SO
0,0 , 0,0 , 0,1 ,
0,0 , 1,0 , 0,0 , 0,1 , 1,0 ,
0,0 , 0,1 , 1,0 , 1,1 ,
0,0 , 1,1 , 0,0 , 0,1 , 1,1 ,
0,0 , 1,0 , 1,1
and that : is continuous at
(0 0) (1 0) (0 1), but not continuous at
(1 1). However, is semi-continuous at (1 1).
For this, let us take the open set = {0} in
containing (1 1) = 0. Then the semi-open set
= {(0 0) (1 1)} contains
(1 1) . The inverse mapping : is
continuous and hence semi-continuous. Therefore,
( +2 ) is a - -topological group which is
not a topological group. It was noticed in [1] that
( +2 ) is not a - -topological group.
Let , , ) be a generalized s-
topological group. Let be an inverse
mapping defined by = ; . Then is
generelized semi continuuous mapping.
Let . Let W be a generalized open set in
containing . Then by hypothesis, there exist
generalized semi open sets U and V containing e and x,
respectively, such that In
particular = e .
. If is semi generalized compact, then
is semi generalized compact in a semi generalized
topological group ( ).
Let { : I} be a cover of
⋃ This implies that ⋃ =
⋃ . This implies that ⋃
.
Since is semi generalized compact, then there
exists a finite set of such that
⋃ . This implies that y ⋃ That
is has a finite subcover of X. Hence is
semi generalized compact.
. A non empty subgroup of a semi
generalized topological group is semi open if and only if
its semi interior is non empty.
Assume that (semi generalized
interior). Then by definition there is a semi generalized
open set V such that : For every , we
have y Since V is semi generalized
open so is , we conclude that is a semi generalized open set as the union of semi
generalized open sets is semi generalized open. Converse
of this theorem is quite simple.
Let ( ) and ( ) be generalized
topological spaces and is generalized semi
continuous, then for any subset of ,
( ) Further theorem is the extension of the work presented
by Bohn Lee [1].
Let be a semi generalized
topological group. Then for each generalized open set
subset of ; is semi generalized open.
Let be generalized open in , there exists a
generalized open set in , such that,
( )U A cl U (By [5])
1 1 1[cl(U)]U A
Because, is semi generalized topological group.
Let be -topological space. If is -
semi open and , then is -semi open.
Let . Since is -open
therefore there exists a -open set such that .
implies that .
This proves that is semi -open in .
Let ) be a -s-topological group.
Then the multiplication mapping
: defined by
is semi -continuous for each Let and W be a -open set containing
, since X is -semi open sets and containing
and , such that
1
1
1
1 1
*
* *
, * *
*
U V W
x y U V W
m x y x y U V W
m U V U V W
Turkish Journal of Analysis and Number Theory 110
Since is -open set containing , therefore,
by Theorem-1.6, is -semi open set containing y.
Moreover, by Theorem 2, is -semi open set
containing . Hence „ ‟ is semi -continuous for
each By Theorems 2.5 and 2.7, it is clear that every -s-
topological group is - -topological group.
Let ) be a -s-topological group
and be any -open set in . Then, for each ,
both are -semi open in .
Proof: Let . This gives for some .
1 1.* * *y z x A x x
Since, is -s-topological group, therefore, for -open
set A containing , there exist -open set and
containing and respectively, such that
1* .U V A
Or
1 1 1* * * .z x U x U V A
This gives This proves that is -
semi open set.
Let . be semi -topological
group. If is generalized open and , then is generalized semi open in .
Let and
* z y x
or, for some = ( )
Now, , implies,
z
Where is generalized open set in , therefore, by the
hypothesis, i.e., is semi generalized topological group,
there exist generalized semi open set in X containing z and
containing such that,
1*VU A
or
1 1* *VU x U A
or
* .U A x
This implies that for each point z x, we can find a
generalized semi open set U containing z such that
x. This means x is generalized semi open. Since the
union of semi open sets is generalized semi open,
therefore,
* *xx BA B A
is generalized semi open.
Let be a semi -topological group
and let be the base at identity element e of . Then, for
every , there is an element ; so that
following holds,
1) U.
2) U.
3) .
. Let be a semi -topological
group. Then each left(right) translation :
( : ) is -semi homeomorphic.
It is obviously bijective map and : is
semi -continuous containing , there exists -semi
open set containing x such that Again, let
be a -open set in , then is semi -open.
That is the image of -open set is semi -open. This
proves that is -semi homeomorphic.
Let be semi -topological group and
. Then for any local base and e , then each of
the families and { is a semi -open neighborhood system of .
References
[1] E. Bohn, J. Lee, Semi-topological groups, Amer. Math. Monthly
72 (1965), 996.998.
[2] Á. Császár, Generalized open sets, Acta Math. Hungar., 75 (1997),
65-87.
[3] Á. Császár, Generalized topology, generalized continuity, Acta
Math.Hungar., 96 (2002), 351-357.
[4] M. Hussain, M. Khan and C. Ozel, On generalized topological
groups, Filomat, 2013 27(4):567-575.
[5] M. Hussain, M. Khan and C. Ozel, More on generalized
topological groups, Creative Math. & Inf. 22(2013)(1), 47-51.
[6] M. Hussain, M. Khan, and C. Ozel, Extension closed
properties on generalized topological groups, Arab J Math (2014)
3:341-347.
[7] N. Levine, Semi-open sets and semi-continuity in topological
spaces, Amer. Math. Monthly 70 (1963), 36.41.1.
[8] Michel Coste, Real Algebraic Sets, March 23, 2005.
[9] J. Cao, R. Drozdowski, Z. Piotrowski, Weak continuity properties
of topologized groups, Czech. Math. J., 60 (2010), 133-148.
[10] C. Silva and R. Silva, On generalized S topological group,
International journal of science and research, 6, (2015), 1-4.
Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 4, 111-115
Available online at http://pubs.sciepub.com/tjant/3/4/5
© Science and Education Publishing
DOI:10.12691/tjant-3-4-5
Further Inequalities Associated with the Classical Gamma Function
Kwara Nantomah*
Department of Mathematics, University for Development Studies, Navrongo Campus, Navrongo UE/R, Ghana
*Corresponding author: [email protected], [email protected]
Received June 15, 2015; Accepted August 20, 2015
Abstract In this paper, the authors present some double inequalities associated with certain ratios of the Gamma
function. The results are further generalizations of several previous results. The approach is based on some
monotonicity properties of some functions involving the generalized Gamma functions. At the end, some open
problems are posed.
Keywords: Gamma function, Psi function, inequality, generalization
Cite This Article: Kwara Nantomah, “Further Inequalities Associated with the Classical Gamma Function.”
Turkish Journal of Analysis and Number Theory, vol. 3, no. 4 (2015): 111-115. doi: 10.12691/tjant-3-4-5.
1. Introduction
Inequalities involving the classical Euler’s Gamma
function has gained the attention of researchers all over
the world. Recent advances in this area include those
inequalities involving ratios of the Gamma function. In
[1,5,6,10] and [11-17], the authors established some
interesting inequalities concerning such ratios, as well as
some generalizations. By utilizing similar techniques, this
paper seeks to present some new results generalizing the
results of [11-17]. At the end, we pose some open
problems involving the generalized Psi functions. In the
sequel, we recall some basic definitions concerning the
Gamma function and its generalizations. These definitions
are required in order to establish our results.
The well-known classical Gamma function, ( )t and
the classical Psi or Digamma function ( )t are usually
defined for 0t as:
1
0
( )( ) and ( ) .
( )
x t tt e x dx t
t
The p-Gamma function, ( )p t and the p-Psi function
( )p t are defined for p N and 0t as:
( )!
( ) and ( )( 1)...( ) ( )
tp
p pp
tp pt t
t t t p t
where ( ) ( )p t t and ( ) ( )p t t as p . For
more information on this function, see [9] and the
references therein.
Also, the q-Gamma function, ( )q t and the q-Psi
function ( )q t are defined for (0,1)q and 0t as:
1
1
( )1( ) (1 ) and ( ) .
( )1
nqt
q qt nqn
tqt q t
tq
where ( ) ( )q t t and ( ) ( )q t t as 1q .
See also [4,5] and the references therein.
Similarly, the k-Gamma function, ( )k t and the k-Psi
function ( )k t are defined for 0k and 0t as (see
[2,7]):
1
0
( )( ) and ( )
( )
kxt kk
k kk
tt e x dx t
t
where ( ) ( )k t t and ( ) ( )k t t as 1k .
Also, the (q,k)-Gamma function ( , ) ( )q k t and the
(q,k)-Psi function ( , ) ( )q k t are defined for (0,1)q ,
0k and 0t as [3]:
1
, ( , )( , ) ( , )
1 ( , )
(1 ) ( )( ) and ( )
( )(1 )
tk k
q k q kq k q kt
q kk
q tt t
tq
where 1
,
0
( ) ( )n
n k
j
t t jk
is the k-generalized Pochhammer
symbol and ( , ) ( ) ( )q k t t , ( , ) ( ) ( )q k t t as 1q ,
1k .
Furthermore, the (p,q)-Gamma function ( , ) ( )p q t and
the (p,q)-Psi function ( , ) ( )p q t are defined for p N ,
(0,1)q and 0t as [8]:
( , )
[ ] [ ] !( )
[ ] [ 1] [ ]
tq q
p qq q q
p pt
t t t p
Turkish Journal of Analysis and Number Theory 112
and
( , )
( , )( , )
( )( )
( )
p qp q
p q
tt
t
where 1
[ ]1
p
q
qp
q
, and ( , ) ( ) ( )p q t t , ( , ) ( ) ( )p q t t
as p , 1q .
As defined above, the generalized Psi functions: ( )p t ,
( )q t , ( )k t , ( , ) ( )q k t and ( , ) ( )p q t possess the
following series forms (see [16,17] and the references
therein):
0
1( ) In
p
pn
t pn t
(1)
1
( ) In(1 ) (In )1
nt
q nn
qt q q
q
(2)
1
In - 1( )
( )k
n
k tt
k t nk nk t
(3)
( , )1
( ) In[ ] (In )1
p nt
p q q nn
qt p q
q
(4)
( , )1
-In(1 )( ) (In )
1
nkt
q k nkn
q qt q
k q
(5)
with
1
1lim In 0.5721566...
n
nk
nk
denoting the
Euler-Mascheroni’s constant.
2. Results
We now present our results. Let us begin with the
following Lemmas pertaining to the results.
Lemma 2.1. Assume that 0 , p N , (0,1)q
and ( ) 0g t . Then,
( , )
In(1 ) In[ ]
( ( )) ( ( )) 0.
q
q p q
q p
g t g t
Proof. By using equations (2) and (4) we obtain,
( , )
( ) ( )
1 1
In(1 ) In[ ] ( ( )) ( ( ))
(In ) 0.1 1
q q p q
png t ng t
n nn n
q p g t g t
q qq
q q
concluding the proof.
Lemma 2.2. Assume that 0 , (0,1)q , 1k
and ( ) 0g t . Then,
( , )
In(1- )In(1 )
( ( )) ( ( )) 0.q q k
k
g t g t
Proof. By using equations (2) and (5) we obtain,
( , )
( ) ( )
1
In(1- )In(1 )
( ( )) ( ( ))
(In ) 01 1
q q k
ng t nkg t
n nkn
k
g t g t
q qq
q q
concluding the proof.
Lemma 2.3. Assume that 0 , 0 , 0k , p N ,
(0,1)q and ( ) 0g t . Then,
( , )
InIn[ ]
( )
( ( )) ( ( )) 0.
q
k p q
kp
k k g t
g t g t
Proof. By using equations (3) and (4) we obtain,
( , )
( )
1 1
InIn[ ]
( )
( ( )) ( ( ))
( )(In ) 0
( ( )) 1
q
k p q
p ng t
nn n
kp
k k g t
g t g t
g t qq
nk nk g t q
concluding the proof.
Lemma 2.4. Assume that 0 , 0 , (0,1)q ,
0k and ( ) 0g t . Then,
( , )
In( (1 ) )
( )
( ( )) ( ( )) 0.k q k
k q
k g t k
g t g t
Proof. By using equations (3) and (5) we obtain,
( , )
( )
1 1
In( (1 ) )
( )
( ( )) ( ( ))
( )(In ) 0
( ( )) 1
k q k
nkg t
nkn n
k q
k g t k
g t g t
g t qq
nk nk g t q
concluding the proof.
Theorem 2.5. Let ( )g t be a positive, increasing and
differentiable function, p N and (0,1)q . Then for
positive real numbers and such that , the
inequalities:
( (0) ( ))
( (0) ( ))( , )
( , )
( ( ) ( ))
( ( ) ( ))( , )
(1 ) ( (0))
[ ] ( (0))
( ( ))
( ( ))
(1 ) ( ( ))
[ ] ( ( ))
g g xq
g g xq p q
q
p q
g y g xq
g y g xq p q
q g
p g
g x
g x
q g y
p g y
(6)
hold true for 0 x y .
Proof. Define a function G for p N and (0,1)q
by
113 Turkish Journal of Analysis and Number Theory
( )
( )( , )
(1 ) ( ( ))( ) , (0, )
[ ] ( ( ))
g tq
g tq p q
q g tG t t
p g t
.
Let ( ) InG( )u t t . Then,
( )
( )( , )
( , )
(1 ) ( ( ))( ) In
[ ] ( ( ))
( )In(1 ) ( )In[ ]
In ( ( )) In ( ( )).
g tq
g tq p q
q
q p q
q g tu t
p g t
g t q g t p
g t g t
Then,
( , )
( , )
( ) ( )In(1 ) ( )In[ ]
( ) ( ( )) ( ) ( ( ))
( ) In(1 ) In[ ]
( ( )) ( ( )) 0
q
q p q
q
q p q
u t g t q g t p
g t g t g t g t
g t q p
g t g t
as a consequence of Lemma 2.1. That implies u is non-
increasing on (0, )t . Hence ( )u tG e is non-
increasing and for 0 x y we have,
(0) ( ) ( )G G x G y
establishing the inequalities in (6).
Theorem 2.6. Let ( )g t be a positive, increasing and
differentiable function, (0,1)q and 1k . Then for
positive real numbers and such that , the
inequalities:
( (0) ( ))
( (0) ( ))
( , )
( , )
( ( ) ( ))
( ( ) ( ))
( , )
(1 ) ( (0))
(1 ) ( (0))
( ( ))
( ( ))
(1 ) ( ( ))
(1 ) ( ( ))
g g xq
g g xk
q k
q
q k
g y g xq
g y g xk
q k
q g
q g
g x
g x
q g y
q g y
(7)
hold true for 0 x y .
Proof. Define a function H for (0,1)q and 1k by
( )
( )
( , )
(1 ) ( ( ))( ) , (0, )
(1 ) ( ( ))
g tq
g t
kq k
q g tH t t
q g t
.
Let ( ) In ( )v t H t . Then,
( )
( )
( , )
( , )
(1 ) ( ( ))( ) In
(1 ) ( ( ))
( ) ( )In(1 ) In(1- )
In ( ( )) In ( ( )).
g tq
g t
kq k
q q k
q g tv t
q g t
g tg t q q
k
g t g t
Then,
( , )
( , )
( )( ) ( )In(1 ) In(1 )
( ) ( ( )) ( ) ( ( ))
In(1 ) ( ) In(1 )
( ( )) ( ( )) 0
q q k
q q k
g tv t g t q q
k
g t g t g t g t
qg t q
k
g t g t
as a consequence of Lemma 2.2. That implies v is non-
increasing on (0, )t . Hence ( )v tH e is non-
increasing and for 0 x y we have,
(0) ( ) ( )H H x H y
establishing the inequalities in (7).
Theorem 2.7. Let ( )g t be a positive, increasing and
differentiable function, 0k , p N and (0,1)q . Then
for positive real numbers and , the inequalities:
( (0) ( )) ( (0) ( ))
( (0) ( ))( , )
( , )
( ( ) ( )) ( ( ) ( ))
( ( ) ( ))( , )
( (0)) ( (0))
( ( )) [ ] ( (0))
( ( ))
( ( ))
( ( )) ( ( ))
( ( )) [ ] ( ( ))
g g x g g xk k
k
g g xq p q
k
p q
g y g x g y g xk k
k
g y g xq p q
g k e g
g x p g
g x
g x
g y k e g y
g x p g y
(8)
hold true for 0 x y .
Proof. Define a function S for 0k , p N and
(0,1)q by
( ) ( )
( )( , )
( ( )) ( ( ))( ) , (0, )
[ ] ( ( ))
g t g t
k kk
g tq p q
g t k e g tS t t
p g t
.
Let ( ) In ( )w t S t . Then,
( ) ( )
( )( , )
( , )
( ( )) ( ( ))( ) In
[ ] ( ( ))
( ) ( )( )In[ ] In In( ( ))
In ( ( )) In ( ( )).
g t g t
k kk
g tq p q
q
k p q
g t k e g tw t
p g t
g t g tg t p k g t
k k
g t g t
Then,
( , )
( , )
( )In ( )( ) ( )In[ ]
( ) ( ) ( ( )) ( ) ( ( ))
( )
In ( ) In[ ]
( )
( ( )) ( ( )) 0
q
k p q
q
k p q
g t k g tw t g t p
k k
g tg t g t g t g t
g t
kg t p
k k g t
g t g t
as a result of Lemma 2.3. That implies w is increasing on
(0, )t . Hence ( )w tS e is increasing and for
0 x y we have,
Turkish Journal of Analysis and Number Theory 114
(0) ( ) ( )S S x S y
establishing the inequalities in (8).
Theorem 2.8. Let ( )g t be a positive, increasing and
differentiable function, 0k and (0,1)q . Then for
positive real numbers and , the inequalities:
( (0) ( ))
( (0) ( )) ( (0) ( ))
( , )
( , )
( ( ) ( ))
( ( ) ( )) ( ( ) ( ))
( , )
( ( )) ( (0)) ( (0))
(1 ) ( (0))
( ( ))
( ( ))
( ( )) ( ( )) ( ( ))
(1 ) ( ( ))
g g xk
k
g g x g g xk k
q k
k
q k
g y g xk
k
g y g x g y g xk k
q k
g x g e g
k q g
g x
g x
g x g y e g y
k q g y
(9)
hold true for 0 x y .
Proof. Define a function T for 0k and (0,1)q by
( )
( ) ( )
( , )
( ( )) ( ( ))( ) , (0, )
(1 ) ( ( ))
g t
kk
g t g t
k kq k
g t e g tT t t
k q g t
.
Let ( ) In ( )t T t . Then,
( )
( ) ( )
( , )
( , )
( ( )) ( ( ))( ) In
(1 ) ( ( ))
( ) ( ) ( )In ( ) In In(1 )
In ( ( )) In ( ( )).
g t
kk
g t g t
k kq k
k q k
g t e g tt
k q g t
g t g t g tg t k q
k k k
g t g t
Then,
( , )
( , )
( ) ( ) ( )In( (1 ) )( )
( )
( ) ( ( )) ( ) ( ( ))
In( (1 ) ) ( )
( )
( ( )) ( ( )) 0
k q k
k q k
g t g t g t k qt
k g t k
g t g t g t g t
k qg t
k g t k
g t g t
as a result of Lemma 2.4. That implies is -increasing on
(0, )t . Hence ( )tT e is increasing and for
0 x y we have,
(0) ( ) ( )T T x T y
establishing the inequalities in (9).
3. Concluding Remarks
In particular, if we let ( )g t t for 0 and
0 on the interval 0 1t , then we recover the entire
results of [17]. Also, by setting ( )g t t and
1 on the interval 0 1t , we obtain the results
of [16]. The results [11] – [17] are therefore special cases
of the results of this paper. For example, let ( )g t t
for , 0 on the interval
0 1t . Then;
(i) by allowing 1q in Theorem 2.5, we recover
Theorem 3.7 of [13].
(ii) by allowing 1k in Theorem 2.8, we recover
Theorem 3.8 of [13].
(iii) by allowing 1q in Theorem 2.6, we recover
Theorem 3.9 of [13].
(iv) by allowing 1k in Theorem 2.7, we recover
Theorem 3.1 of [15].
This paper is a slightly modified version of preprint
[18].
4. Open Problems
For 0k , p N and (0,1)q , let ( )p t , ( )q t ,
( , ) ( )p q t and ( , ) ( )q k t be the generalized Psi functions
as defined in equations (1) – (5).
Problem 1: Under what conditions will the statements:
0 1
In In(1 ) ( ) ( )
1(In ) ( )0
1
q p
p nt
nn n
p q t t
n t q
be valid?
Problem 2: Under what conditions will the statements:
( , ) ( , )
1 1
In(1 )In[ ] ( ) ( )
(In ) ( )01 1
q p q q k
p nt nkt
n nkn n
qp t t
k
q qq
q q
be valid?
Competing Interests
The authors declare that there is no competing interest.
Acknowlegement
The authors are very grateful to the anonymous
reviewers for their useful comments and suggestions
which helped in improving the quality of this paper.
References
[1] Y. C. Chen, T. Mansour and Q. Zou, On the complete
monotonicity of quotient of Gamma functions, Math. Ineq. & Appl.
15:2 (2012), 395-402.
[2] R. Diaz and E. Pariguan, On hypergeometric functions and
Pachhammer k-symbol, Divulgaciones Matematicas 15(2)(2007),
179-192.
[3] R. Diaz and C. Teruel, q,k-generalized gamma and beta functions,
J. Nonlin. Math. Phys. 12(2005), 118-134.
115 Turkish Journal of Analysis and Number Theory
[4] T. Mansour, Some inequalities for the q-Gamma Function, J. Ineq.
Pure Appl. Math. 9(1)(2008), Art. 18.
[5] T. Mansour and A.Sh. Shabani, Some inequalities for the q-
digamma function, J. Ineq. Pure and Appl. Math. 10:1 (2009),
Article 12.
[6] T. Mansour and A. Sh. Shabani, Generalization of some
inequalities for the (q1 ….. qs)-Gamma function, Le Matematiche
LXVII (2012), 119-130.
[7] F. Merovci, Power Product Inequalities for the k Function, Int.
Journal of Math. Analysis, 4(21)(2010), 1007-1012.
[8] V. Krasniqi and F. Merovci, Some Completely Monotonic
Properties for the (p,q)-Gamma Function, Mathematica Balkanica,
New Series 26(2012), 1-2.
[9] V. Krasniqi, T. Mansour and A. Sh. Shabani, Some Monotonicity
Properties and Inequalities for and Functions,
Mathematical Communications 15(2) (2010), 365-376.
[10] V. Krasniqi, T. Mansour, and A. Sh. Shabani, Some inequalities
for q-polygamma function and zeta q-Riemann zeta functions, Ann.
Math. Informaticae, 37 (2010), 95-100.
[11] K. Nantomah and M. M. Iddrisu, Some Inequalities Involving the
Ratio of Gamma Functions, Int. Journal of Math. Analysis
8(12)(2014), 555-560.
[12] K. Nantomah, M. M. Iddrisu and E. Prempeh, Generalization of
Some Inequalities for theRatio of Gamma Functions, Int. Journal
of Math. Analysis, 8(18)(2014), 895-900.
[13] K. Nantomah and E. Prempeh, Generalizations of Some
Inequalities for the p-Gamma, q-Gamma and k-Gamma Functions,
Electron. J. Math. Anal. Appl. 3(1)(2015),158-163.
[14] K. Nantomah and E. Prempeh, Some Sharp Inequalities for the
Ratio of Gamma Functions, Math. Aeterna, 4(5)(2014), 501-507.
[15] K. Nantomah and E. Prempeh, Generalizations of Some Sharp
Inequalities for the Ratio of Gamma Functions, Math. Aeterna,
4(5)(2014), 539-544.
[16] K. Nantomah, On Certain Inequalities Concerning the Classical
Euler's Gamma Function, Advances in Inequalities and
Applications, Vol. 2014 (2014) Article ID 42.
[17] K. Nantomah, Generalized Inequalities Related to the Classical
Euler's Gamma Function, Turkish Journal of Analysis and
Number Theory, 2(6)(2014), 226-229.
[18] K. Nantomah, Further Inequalities Associated with the Classical
Gamma Function, arXiv:1506.07393v1.