MODIFIED HADAMARD PRODUCT PROPERTIES OF CERTAIN … · kD2 k .n;k/C kja kb kj B A 1 kD2 k .n;k/C k...
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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 18 (2017), No. 1, pp. 397–406 DOI: 10.18514/MMN.2017.1997 MODIFIED HADAMARD PRODUCT PROPERTIES OF CERTAIN CLASS OF ANALYTIC FUNCTIONS WITH VARYING ARGUMENTS DEFINED BY RUSCHEWEYH DERIVATIVE P ´ ALL-SZAB ´ O ´ AGNES ORSOLYA Received 08 May, 2016 Abstract. In this paper we study the modified Hadamard product properties of certain class of analytic functions with varying arguments defined by Ruscheweyh derivative. The obtained results are sharp and they improve known results. 2010 Mathematics Subject Classification: 30C45 Keywords: analytic functions, modified Hadamard product, Ruscheweyh derivative 1. I NTRODUCTION Let A denote the class of functions of the form: f .´/ D ´ C 1 X kD2 a k ´ k ; (1.1) which are analytic and univalent in the open unit disc U Df´ 2 C Wj´j <1g. Let g 2 A where g.´/ D ´ C 1 X kD2 b k ´ k : (1.2) Ruscheweyh [5] defined that D f .´/ D ´ .1 ´/ C1 f .´/; . 1/ (1.3) which implies D n f .´/ D ´.´ n1 f .´// .n/ nŠ .n 2 N 0 Df0;1;2:::g/: (1.4) The symbol D n f .´/.n 2 N 0 / was called the n-th order Ruscheweyh derivative of f .´/ by Al-Amiri [1]. It is easy to see that D 0 f .´/ D f .´/; D 1 f .´/ D ´f 0 .´/ c 2017 Miskolc University Press
Transcript of MODIFIED HADAMARD PRODUCT PROPERTIES OF CERTAIN … · kD2 k .n;k/C kja kb kj B A 1 kD2 k .n;k/C k...
Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 397–406 DOI: 10.18514/MMN.2017.1997
MODIFIED HADAMARD PRODUCT PROPERTIES OF CERTAINCLASS OF ANALYTIC FUNCTIONS WITH VARYING
ARGUMENTS DEFINED BY RUSCHEWEYH DERIVATIVE
PALL-SZABO AGNES ORSOLYA
Received 08 May, 2016
Abstract. In this paper we study the modified Hadamard product properties of certain class ofanalytic functions with varying arguments defined by Ruscheweyh derivative.The obtained results are sharp and they improve known results.
2010 Mathematics Subject Classification: 30C45
Keywords: analytic functions, modified Hadamard product, Ruscheweyh derivative
1. INTRODUCTION
Let A denote the class of functions of the form:
f .´/D ´C
1XkD2
ak´k; (1.1)
which are analytic and univalent in the open unit disc U D f´ 2C W j´j< 1g. Letg 2A where
g.´/D ´C
1XkD2
bk´k : (1.2)
Ruscheweyh [5] defined that
D f .´/D´
.1�´/ C1�f .´/; . � �1/ (1.3)
which implies
Dnf .´/D´.´n�1f .´//
.n/
nŠ.n 2N0 D f0;1;2 : : :g/ : (1.4)
The symbol Dnf .´/.n 2N0/ was called the n-th order Ruscheweyh derivative off .´/ by Al-Amiri [1]. It is easy to see that
D0f .´/D f .´/;D1f .´/D ´f 0.´/
c 2017 Miskolc University Press
398 PALL-SZABO AGNES ORSOLYA
Dnf .´/D ´C
1XkD2
ı.n;k/ak´k (1.5)
where
ı.n;k/D
�nCk�1
n
�: (1.6)
Definition 1. Let f and g be analytic functions in U . We say that the function fis subordinate to the function g, if there exists a function w, which is analytic in Uand w.0/D 0I jw.´/j < 1I´ 2 U , such that f .´/D g.w.´//I 8´ 2 U: We denote by� the subordination relation.
Definition 2. [2,4] For �� 0I�1�A<B � 1I0<B � 1In2N0 letQ.n;�;A;B/denote the subclass of A which contain functions f .´/ of the form .1:1/ such that
.1��/.Dnf .´//0C�.DnC1f .´//0 �1CA´
1CB´: (1.7)
Definition 3 ([7]). A function f .´/ of the form .1:1/ is said to be in the classV.�k/ if f 2 A and arg.ak/D �k ,8k � 2. If 9ı 2 R such that�k C .k � 1/ı � �.mod2�/;8k � 2 then f .´/ is said to be in the class V.�k; ı/.The union of V.�k; ı/ taken over all possible sequences f�kg and all possible realnumbers ı is denoted by V . Let VQ.n;�;A;B/ denote the subclass of V consistingof functions f .´/ 2Q.n;�;A;B/.
Theorem 1 ([4]). Let the function f .´/ defined by .1:1/ be in V. Thenf .´/ 2 VQ.n;�;A;B/, if and only if
1XkD2
kı.n;k/Ck jakj � .B �A/.nC1/ (1.8)
whereCk D .1CB/ŒnC1C�.k�1/�:
The extremal functions are
fk.´/D ´C.B �A/.nC1/
kCkı.n;k/ei�k´k; .k � 2/:
The modified Hadamard product of two functions f and g of the form .1:1/ and.1:2/, and which belong to V.�k; ı/ is defined by (see also [3, 6, 8])
.f �g/.´/D ´�
1XkD2
akbk´kD .g �f /.´/: (1.9)
MODIFIED HADAMARD PRODUCT PROPERTIES... 399
2. MAIN RESULTS
Theorem 2. If f 2 VQ.n;�;A1;B/;g 2 VQ.n;�;A2;B/ thenf �g 2 VQ.n;�;A�;B/, where A� D B � .B�A1/.B�A2/.nC1/
2C2ı.n;2/.
The result is sharp.
Proof. Let f 2 VQ.n;�;A1;B/;g 2 VQ.n;�;A2;B/ and suppose they have theform .1:1/. Since f 2 VQ.n;�;A1;B/ we have
1PkD2
kı.n;k/Ck jakj
B �A1� nC1 (2.1)
and for g 2 VQ.n;�;A2;B/ we have1PkD2
kı.n;k/Ck jbkj
B �A2� nC1: (2.2)
We know from Theorem 1 that f �g 2 VQ.n;�;A�;B/ if and only if
1PkD2
kı.n;k/Ck jakbkj
B �A�� nC1: (2.3)
By using the Cauchy-Schwarz inequality for .2:1/ and .2:2/ we have1PkD2
kı.n;k/Ckpjakbkjp
.B �A1/.B �A2/� nC1:
We note that1PkD2
kı.n;k/Ck jakbkj
B �A��
1PkD2
kı.n;k/Ckpjakbkjp
.B �A1/.B �A2/
implies .2:3/. But this is equivalent to
jakbkj
B �A��
pjakbkjp
.B �A1/.B �A2/or p
jakbkj �B �A�p
.B �A1/.B �A2/: (2.4)
From Theorem 1 we have:
jakj �.B �A1/.nC1/
kCkı.n;k/and jbkj �
.B �A2/.nC1/
kCkı.n;k/; .k � 2/
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this implies that pjakbkj �
p.B �A1/.B �A2/.nC1/
kCkı.n;k/; .k � 2/: (2.5)
From .2:5/ we obtain that .2:4/ holds ifp.B �A1/.B �A2/.nC1/
kCkı.n;k/�
B �A�p.B �A1/.B �A2/
or equivalently
A� � B �.B �A1/.B �A2/.nC1/
kCkı.n;k/:
But kCkı.n;k/ < .kC1/CkC1ı.n;kC1/; .k � 2/ so
B �.B �A1/.B �A2/.nC1/
kCkı.n;k/� B �
.B �A1/.B �A2/.nC1/
2C2ı.n;2/; .k � 2/
) A� D B �.B �A1/.B �A2/.nC1/
2C2ı.n;2/:
The result is sharp, because if
f .´/D ´C.B �A1/.nC1/
2C2ı.n;2/ei�1´2 2 VQ.n;�;A1;B/
g.´/D ´C.B �A2/.nC1/
2C2ı.n;2/ei�2´2 2 VQ.n;�;A2;B/
f �g 2 VQ.n;�;A�;B/
and satisfy (8) with equality. Indeed,
2ı.n;2/C2.B �A1/.B �A2/.nC1/
2
22C 22 Œı.n;2/�2
D .B �A�/.nC1/
because
B �A� D.B �A1/.B �A2/.nC1/
2C2ı.n;2/:
�
Corollary 1. If f;g 2 VQ.n;�;A;B/ then f �g 2 VQ.n;�;A�;B/, where A� DB � .B�A/
2.nC1/2C2ı.n;2/
. The result is sharp.
Theorem 3. If f 2 VQ.n;�;A;B1/;g 2 VQ.n;�;A;B2/ thenf �g 2 VQ.n;�;A;B�/, where B� D AC .B1�A/.B2�A/.nC1/
2C2ı.n;2/.
The result is sharp.
MODIFIED HADAMARD PRODUCT PROPERTIES... 401
Proof. Let f 2 VQ.n;�;A;B1/;g 2 VQ.n;�;A;B2/ and suppose they have theform .1:1/. Since f 2 VQ.n;�;A;B1/ we have
1PkD2
kı.n;k/Ck jakj
B1�A� nC1 (2.6)
and for g 2 VQ.n;�;A;B2/ we have1PkD2
kı.n;k/Ck jbkj
B2�A� nC1: (2.7)
We know from Theorem 1 that f �g 2 VQ.n;�;A;B�/ if and only if
1PkD2
kı.n;k/Ck jakbkj
B��A� nC1: (2.8)
By using the Cauchy-Schwarz inequality for .2:6/ and .2:7/ we have1PkD2
kı.n;k/Ckpjakbkjp
.B1�A/.B2�A/� nC1:
We note that1PkD2
kı.n;k/Ck jakbkj
B��A�
1PkD2
kı.n;k/Ckpjakbkjp
.B1�A/.B2�A/
implies .2:8/. But this is equivalent to
jakbkj
B��A�
pjakbkjp
.B1�A/.B2�A/
or pjakbkj �
B��Ap.B1�A/.B2�A/
: (2.9)
From Theorem 1 we have:
jakj �.B1�A/.nC1/
kCkı.n;k/and jbkj �
.B2�A/.nC1/
kCkı.n;k/; .k � 2/
this implies that pjakbkj �
p.B1�A/.B2�A/.nC1/
kCkı.n;k/; .k � 2/: (2.10)
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from .2:10/ we obtain that .2:9/ holds ifp.B1�A/.B2�A/.nC1/
kCkı.n;k/�
B��Ap.B1�A/.B2�A/
or equivalently
B� � AC.B1�A/.B2�A/.nC1/
kCkı.n;k/:
But kCkı.n;k/ < .kC1/CkC1ı.n;kC1/; .k � 2/ so :
AC.B1�A/.B2�A/.nC1/
kCkı.n;k/� AC
.B1�A/.B2�A/.nC1/
2C2ı.n;2/; .k � 2/
) B� D AC.B1�A/.B2�A/.nC1/
2C2ı.n;2/:
The result is sharp, because if
f .´/D ´C.B1�A/.nC1/
2C2ı.n;2/ei�1´2 2 VQ.n;�;A;B1/
g.´/D ´C.B2�A/.nC1/
2C2ı.n;2/ei�2´2 2 VQ.n;�;A;B2/
f �g 2 VQ.n;�;A;B�/
and satisfy (8) with equality. Indeed,
2ı.n;2/C2.B1�A/.B2�A/.nC1/
2
22C 22 Œı.n;2/�2
D .B��A/.nC1/
because
B��AD.B1�A/.B2�A/.nC1/
2C2ı.n;2/:
�
Corollary 2. If f;g 2 VQ.n;�;A;B/ then f �g 2 VQ.n;�;A;B�/, where B�DAC .B�A/2.nC1/
2C2ı.n;2/. The result is sharp.
Theorem 4. If fj 2 VQ.n;�;Aj ;B/;j D 1;s; s 2 f2;3;4; : : :g then
f1 �f2 � : : :�fs 2 VQ.n;�;A.s�1/�;B/, where A.s�1/� DB�
.nC1/s�1sQ
jD1
.B�Aj /
2s�1C s�12 Œı.n;2/�s�1 .
The result is sharp.
Proof. For the proof we use the mathematical induction method and suppose thatfj ;8j have the form .1:1/.Let s D 2. If fj 2 VQ.n;�;Aj ;B/;j D 1;2 then f1 �f2 2 VQ.n;�;A�;B/ whereA� D B � .B�A1/.B�A2/.nC1/
kCkı.n;k/; from Theorem 2 is true.
MODIFIED HADAMARD PRODUCT PROPERTIES... 403
Assume, for s Dm, that the formula displayed below holds.If fj 2 VQ.n;�;Aj ;B/;j D 1;m;m 2 f2;3;4; : : :g thenf1 �f2 � : : :�fm 2 VQ.n;�;A
.m�1/�;B/, where
A.m�1/� D B �
.nC1/m�1mQ
jD1
.B�Aj /
km�1Cm�1k
Œı.n;k/�m�1 .
Let s DmC1: if f1 �f2 � : : :�fm 2 VQ.n;�;A.m�1/�;B/;m 2 f2;3;4; : : :g andfmC1 2 VQ.n;�;AmC1;B/ then we have to prove
f1�f2� : : :�fm�fmC1 2 VQ.n;�;Am�;B/, whereAm�DB�
.nC1/mmC1QjD1
.B�Aj /
kmCmkŒı.n;k/�m
.For the proof we use the result of Theorem 2:
Am� � B �.B �A.m�1/�/.B �AmC1/.nC1/
kCkı.n;k/
Am� � B �
.nC1/m�1mQ
jD1
.B�Aj /
km�1Cm�1k
Œı.n;k/�m�1 .B �AmC1/.nC1/
kCkı.n;k/
Am� � B �
.nC1/mmC1QjD1
.B �Aj /
kmCmkŒı.n;k/�m
:
But kCkı.n;k/ < .kC1/CkC1ı.n;kC1/; .k � 2/ so:
B �
.nC1/mmC1QjD1
.B �Aj /
kmCmkŒı.n;k/�m
� B �
.nC1/mmC1QjD1
.B �Aj /
2mCm2 Œı.n;2/�m ; .k � 2/)
Am� D B �
.nC1/mmC1QjD1
.B �Aj /
2mCm2 Œı.n;2/�m :
The result is sharp, because if
f .´/D ´C.B �A.s�1/�/.nC1/
2C2ı.n;2/ei�1´2 2 VQ.n;�;A.s�1/�;B/
g.´/D ´C.B �As/.nC1/
2C2ı.n;2/ei�2´2 2 VQ.n;�;As;B/
f �g 2 VQ.n;�;As�;B/
and satisfy (8) with equality. Indeed,
404 PALL-SZABO AGNES ORSOLYA
2ı.n;2/C2.B �A.s�1/�/.B �As/.nC1/
2
22C 22 Œı.n;2/�2
D .B �As�/.nC1/
because
B �As� D.B �A.s�1/�/.B �As/.nC1/
2C2ı.n;2/,
, B �As� D
.nC1/ssC1QjD1
.B �Aj /
2sC s2 Œı.n;2/�s :
�
Theorem 5. If fj 2 VQ.n;�;A;Bj /;j D 1;s; s 2 f2;3;4; : : :g then
f1�f2� : : :�fs 2 VQ.n;�;A;B.s�1/�/, where B.s�1/� DAC
.nC1/s�1sQ
jD1
.Bj�A/
2s�1C s�12 Œı.n;2/�s�1 .
The result is sharp.
Proof. For the proof we use the mathematical induction method and suppose thatfj ;8j have the form .1:1/.Let s D 2. If fj 2 VQ.n;�;A;Bj /;j D 1;2 then f1 �f2 2 VQ.n;�;A;B�/ whereB� D AC .B1�A/.B2�A/.nC1/
kCkı.n;k/; from Theorem 3 is true.
Assume, for s Dm, that the formula displayed below holds.If fj 2 VQ.n;�;A;Bj /;j D 1;m;m 2 f2;3;4; : : :g then
f1�f2� : : :�fm 2VQ.n;�;A;B.m�1/�/, whereB.m�1/�DAC
.nC1/m�1mQ
jD1
.Bj�A/
km�1Cm�1k
Œı.n;k/�m�1 .
Let s DmC1: if f1 �f2 � : : :�fm 2 VQ.n;�;A;B.m�1/�/;m 2 f2;3;4; : : :g andfmC1 2 VQ.n;�;A;BmC1/ then we have to prove
f1�f2� : : :�fm�fmC1 2 VQ.n;�;A;Bm�/, whereBm�DAC
.nC1/mmC1QjD1
.Bj�A/
kmCmkŒı.n;k/�m
.For the proof we use the result of Theorem 3:
Bm� � AC.B.m�1/��A/.BmC1�A/.nC1/
kCkı.n;k/
Bm� � AC
.nC1/m�1mQ
jD1
.Bj�A/
km�1Cm�1k
Œı.n;k/�m�1 .BmC1�A/.nC1/
kCkı.n;k/
Bm� � AC
.nC1/mmC1QjD1
.Bj �A/
kmCmkŒı.n;k/�m
:
MODIFIED HADAMARD PRODUCT PROPERTIES... 405
But kCkı.n;k/ < .kC1/CkC1ı.n;kC1/; .k � 2/ so :
AC
.nC1/mmC1QjD1
.Bj �A/
kmCmkŒı.n;k/�m
� AC
.nC1/mmC1QjD1
.Bj �A/
2mCm2 Œı.n;2/�m ; .k � 2/
) Bm� D AC
.nC1/mmC1QjD1
.Bj �A/
2mCm2 Œı.n;2/�m :
The result is sharp, because if
f .´/D ´C.B.s�1/��A/.nC1/
2C2ı.n;2/ei�1´2 2 VQ.n;�;A;B.s�1/�/
g.´/D ´C.Bs �A/.nC1/
2C2ı.n;2/ei�2´2 2 VQ.n;�;A;Bs/
f �g 2 VQ.n;�;A;Bs�/
and satisfy (8) with equality. Indeed,
2ı.n;2/C2.B.s�1/��A/.Bs �A/.nC1/
2
22C 22 Œı.n;2/�2
D .Bs��A/.nC1/
because
Bs��AD.B.s�1/��A/.Bs �A/.nC1/
2C2ı.n;2/,
, Bs��AD
.nC1/ssC1QjD1
.Bj �A/
2sC s2 Œı.n;2/�s :
�
REFERENCES
[1] H. S. Al-Amiri, “On Ruscheweyh derivatives,” Ann. Polon. Math.., vol. 38, pp. 87–94, 1980.[2] A. A. Attiya and M. K. Aouf, “A study on certain class of analytic functions defined by Ruscheweyh
derivative,” Soochow J. Math., vol. 33, no. 2, pp. 273–289, 2007.[3] H. M. Hossen, G. S. Salagean, and M. K. Aouf, “Notes on certain classes of analytic functions with
negative coefficients,” Mathematica., vol. 39, no. 2, pp. 165–179, 1997.[4] R. M.El-Ashwah, M. K.Aouf, A. A. Hassan, and A. H. Hassan , “Certain Class of Analytic Func-
tions Defined by Ruscheweyh Derivative with Varying Arguments,” Kyungpook Math. J., vol. 54,no. 3, pp. 453–461, 2014, doi: 10.5666/KMJ.2014.54.3.453.
[5] S. Ruscheweyh, “New criteria for univalent functions,” Proc. Amer. Math. Soc.., vol. 49, pp. 109–115, 1975, doi: 10.2307/2039801.
[6] A. Schild and H. Silverman, “Convolution of univalent functions with negative coefficients,” Ann.Univ. Mariae Curie-Sklodowska Sect. A, vol. 29, pp. 99–107, 1975.
406 PALL-SZABO AGNES ORSOLYA
[7] H. Silverman, “Univalent functions with varying arguments,” Houston J. Math., vol. 17, pp. 283–287, 1981.
[8] G. S. Salagean, Convolution properties of some classes of analytic functions with negative coef-ficients in Proc. Int. Symposium New. Develop. Geometric Function Th. and its Appl. GFTA, Ke-bangsaan Malaysia. Kebangsaan Malaysia, 2008.
Author’s address
Pall-Szabo Agnes OrsolyaBabes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, RomaniaE-mail address: [email protected]
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