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Inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric function
Applied Mathematics and Computation 187 (2007) 115–121 www.elsevier.com/locate/amc
Inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric function Nak Eun Cho *, In Hwa Kim Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, 599-1 Daeyeon-dong, Nam-gu, Pusan 608-737, South Korea
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract The purpose of the present paper is to introduce several new classes of meromorphic functions defined by using a meromorphic analogue of the Choi–Saigo–Srivastava operator for the generalized hypergeometric function and investigate various inclusion properties of these classes. Some interesting applications involving these and other classes of integral operators are also considered. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Meromorphic functions; Differential subordination; Meromorphic starlike functions; Meromorphic convex functions; Meromorphic close-to-convex functions; Generalized hypergeometric function; Choi–Saigo–Srivastava operator; Integral operator
1. Introduction Let M denote the class of functions of the form 1 1 X ak zk ; f ðzÞ ¼ þ z k¼0 which are analytic in the punctured open unit disk D ¼ fz 2 C : 0 < jzj < 1g. If f and g are analytic in U ¼ D [ f0g, we say that f is subordinate to g, written f g or f(z) g(z), if there exists a Schwarz function w in U such that f(z) = g(w(z)). For 0 6 g, b < 1, we denote by MSðgÞ, MKðgÞ and MCðg; bÞ the subclasses of M consisting of all meromorphic functions which are, respectively, starlike of order g, convex of order g and close-to-convex of order b and type g in U (cf. e.g., [8,9,16]).
*
Corresponding author. E-mail addresses: [email protected] (N.E. Cho), [email protected] (I.H. Kim).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.109
116
N.E. Cho, I.H. Kim / Applied Mathematics and Computation 187 (2007) 115–121
Let N be the class of all functions / which are analytic and univalent in U and for which /ðUÞ is convex with /(0) = 1 and Ref/ðzÞg > 0 ðz 2 UÞ. Making use of the principle of subordination between analytic functions, we introduce the subclasses MSðg; /Þ, MKðg; /Þ and MCðg; b; /; wÞ of the class M for 0 6 g, b < 1 and /; w 2 N, which are defined by 1 zf 0 ðzÞ g /ðzÞ in U ; MSðg; /Þ :¼ f 2 M : 1g f ðzÞ 1 zf 00 ðzÞ MKðg; /Þ :¼ f 2 M : 1þ 0 g /ðzÞ in U 1g f ðzÞ and
MCðg; b; /; wÞ :¼
1 zf 0 ðzÞ b wðzÞ f 2 M : 9g 2 MSðg; /Þ s:t: 1b gðzÞ
in U :
We note that the classes mentioned above is the familiar classes which have been used widely on the space of analytic and univalent functions in U [2,14] and for special choices for the functions / and w involved in these definitions, we can obtain the well-known subclasses of M. For examples, we have 1þz 1þz MS g; ¼ MSðgÞ; MK g; ¼ MKðgÞ 1z 1z and
1þz 1þz ; MC g; b; ¼ MCðg; bÞ: 1z 1z
For complex parameters a1 ; . . . ; aq and b1 ; . . . ; bs ðbj 2 C n Z 0 ; Z0 :¼ f0; 1; 2; . . .g; j ¼ 1; . . . ; sÞ; we now define the generalized hypergeometric function qFs(a1, . . . , aq; b1, . . . , bq; z) [19,20] as follows: 1 X ða1 Þk ðaq Þk zk ; q F s ða1 ; . . . ; aq ; b1 ; . . . ; bq ; zÞ :¼ ðb1 Þk ðbs Þk k! k¼0 ðq 6 s þ 1; q; s 2 N0 :¼ N [ f0g; N :¼ f1; 2; . . .g; z 2 UÞ; where (m)k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by 1 if k ¼ 0 and m 2 C n f0g; Cðm þ kÞ ¼ ðmÞk :¼ CðmÞ mðm þ 1Þ ðm þ k 1Þ if k 2 N and m 2 C: Corresponding to a function Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ :¼ z1 q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ:
ð1:1Þ
Liu and Srivastava [13] considered a linear operator H ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ : M ! M defined by the following Hadamard product (or convolution): H ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ :¼ Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ f ðzÞ:
ð1:2Þ
We note that the linear operator H(a1, . . . , aq; b1, . . . , bs) was motivated essentially by Dziok and Srivastava [3]. Some interesting developments associated with the generalized hypergeometric function were considered recently by Dziok and Srivastava [4,5] and Liu and Srivastava [11,12]. Corresponding to the function Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by (1.1), we introduce a function Fk ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ given by Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ Fk ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ ¼
1 k
zð1 zÞ
ðk > 0Þ:
ð1:3Þ
N.E. Cho, I.H. Kim / Applied Mathematics and Computation 187 (2007) 115–121
117
Analogous to H(a1, . . . , aq; b1, . . . , bs) defined by (1.2), we now define the linear operator Hk(a1, . . . , aq; b1, . . . , bs) on R as follows: H k ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼ Fk ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ f ðzÞ ðai ; bj 2 C n
Z 0;
ð1:4Þ
i ¼ 1; . . . ; q; j ¼ 1; . . . ; s; k > 0; z 2 D; f 2 MÞ:
For convenience, we write H k;q;s ða1 Þ :¼ H k ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ: It is easily verified from the definition (1.3) and (1.4) that zðH k;q;s ða1 þ 1Þf ðzÞÞ0 ¼ a1 H k;q;s ða1 Þf ðzÞ ða1 þ 1ÞH k;q;s ða1 þ 1Þf ðzÞ
ð1:5Þ
and 0
zðH k;q;s ða1 Þf ðzÞÞ ¼ kH kþ1;q;s ða1 Þf ðzÞ ðk þ 1ÞH k;q;s ða1 Þf ðzÞ:
ð1:6Þ
We note that the operator Hk,q,s(a1) is closely related to the Choi–Saigo–Srivastava operator [2] for analytic functions, which includes the integral operator studied by Liu [10] and Noor et al. [17,18]. Next, by using the operator Hk,q,s(a1), we introduce the following classes of meromprphic functions for /; w 2 N, k > 0 and 0 6 g, b < 1: MSk;a1 ðq; s; g; /Þ :¼ ff 2 M : H k;q;s ða1 Þf 2 MSðg; /Þg; MKk;a1 ðq; s; g; /Þ :¼ ff 2 M : H k;q;s ða1 Þf 2 MKðg; /Þg and MCk;a1 ðq; s; g; b; /; wÞ :¼ ff 2 M : H k;q;s ða1 Þf 2 MCðg; b; /; wÞg: We also note that f ðzÞ 2 MKk;a1 ðq; s; g; /Þ () zf 0 ðzÞ 2 MSk;a1 ðq; s; g; /Þ:
ð1:7Þ
In particular, we set 1 þ Az MSk;a1 q; s; g; ¼: MSk;a1 ðq; s; g; A; BÞ ð1 < B < A 6 1Þ 1 þ Bz and
1 þ Az MKk;a1 q; s; g; ¼: MKk;a1 ðq; s; g; A; BÞ 1 þ Bz
ð1 < B < A 6 1Þ:
In this paper, we investgate several inclusion properties of the classes MSk;a1 ðq; s; g; /Þ, MKk;a1 ðq; s; g; /Þ and MCk;a1 ðq; s; g; b; /; wÞ associated with the operator Hk,q,s(a1). Some applications involving integral operators are also considered. 2. Inclusion properties involving the operator Hk,q,s(a1) The following results will be required in our investigation. Lemma 1 [6]. Let / be convex univalent in U with /(0) = 1 and Refj/ðzÞ þ mg > 0 ðj; m 2 CÞ. If p is analytic in U with p(0) = 1, then pðzÞ þ
zp0 ðzÞ /ðzÞ ðz 2 UÞ jpðzÞ þ m
implies pðzÞ /ðzÞ ðz 2 UÞ:
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Lemma 2 [15]. Let / be convex univalent in U and x be analytic in U with Re{x(z)} P 0. If p is analytic in U and p(0) = /(0), then pðzÞ þ xðzÞzp0 ðzÞ /ðzÞ
ðz 2 UÞ
implies pðzÞ /ðzÞ
ðz 2 UÞ:
At first, with the help of Lemma 1, we obtain the following: Theorem 1. Let / 2 N with maxz2U Ref/ðzÞg < minfðk þ 1 gÞ=ð1 gÞ; ða1 þ 1 gÞ=ð1 gÞg ðk; a1 > 0; 0 6 g < 1Þ. Then MSkþ1;a1 ðq; s; g; /Þ MSk;a1 ðq; s; g; /Þ MSk;a1 þ1 ðq; s; g; /Þ: Proof. To prove the first part of Theorem 1, let f 2 MSkþ1;a1 ðq; s; g; /Þ and set 1 zðH k;q;s ða1 Þf ðzÞÞ0 pðzÞ ¼ g ; 1g H k;q;s ða1 Þf ðzÞ where p is analytic in U with p(0) = 1. Applying (1.6) and (2.1), we obtain 0 1 zðH kþ1;q;s ða1 Þf ðzÞÞ zp0 ðzÞ ðz 2 UÞ: g ¼ pðzÞ þ 1g ð1 gÞpðzÞ þ k þ 1 g H kþ1;q;s ða1 Þf ðzÞ
ð2:1Þ
ð2:2Þ
Since maxz2U Ref/ðzÞg < ðk þ 1 gÞ=ð1 gÞ, we see that Refð1 gÞ/ðzÞ þ k þ 1 gg > 0ðz 2 UÞ: Applying Lemma 1 to (2.3), it follows that p /, that is, f 2 MSk;a1 ðq; s; g; /Þ. Moreover, by using the arguments similar to those detailed above with (1.5), we can prove the second part of Theorem 1. Therefore we complete the proof of Theorem 1. h Theorem 2. Let / 2 N with maxz2U Ref/ðzÞg < minfðk þ 1 gÞ=ð1 gÞ; ða1 þ 1 gÞ=ð1 gÞg ðk; a1 > 0; 0 6 g < 1Þ. Then MKkþ1;a1 ðq; s; g; /Þ MKk;a1 ðq; s; g; /Þ MKk;a1 þ1 ðq; s; g; /Þ: Proof. Applying (1.7) and Theorem 1, we observe that f ðzÞ 2 MKkþ1;a1 ðq; s; g; /Þ () zf 0 ðzÞ 2 MSkþ1;a1 ðq; s; g; /Þ ) zf 0 ðzÞ 2 MSk;a1 ðq; s; g; /Þ () f ðzÞ 2 MKk;a1 ðq; s; g; /Þ; and f ðzÞ 2 MKk;a1 ðq; s; g; /Þ () zf 0 ðzÞ 2 MSk;a1 ðq; s; g; /Þ ) zf 0 ðzÞ 2 MSk;a1 þ1 ðq; s; g; /Þ () f ðzÞ 2 MKk;a1 þ1 ðq; s; g; /Þ; which evidently proves Theorem 2. h Taking /ðzÞ ¼
1 þ Az 1 þ Bz
ð1 < B < A 6 1; z 2 UÞ
in Theorems 1 and 2, we have Corollary 1. Let (1 + A)/(1 + B) < min{(k + 1 g)/(1 g), (a1 + 1 g)/(1 g)} (k, a1 > 0; 0 6 g < 1; 1 < B < A 6 1). Then MSkþ1;a1 ðq; s; g; A; BÞ MSk;a1 ðq; s; g; A; BÞ MSk;a1 þ1 ðq; s; g; A; BÞ
N.E. Cho, I.H. Kim / Applied Mathematics and Computation 187 (2007) 115–121
119
and MKkþ1;a1 ðq; s; g; A; BÞ MKk;a1 ðq; s; g; A; BÞ MKk;a1 þ1 ðq; s; g; A; BÞ: Next, by using Lemma 2, we obtain the following inclusion relation for the class MCk;a1 ðq; s; g; b; /; wÞ. Theorem 3. Let /; w 2 N with maxz2U Ref/ðzÞg < minfðk þ 1 gÞ=ð1 gÞ; ða1 þ 1 gÞ=ð1 gÞg ðk; a1 > 0; 0 6 g < 1Þ. Then MCkþ1;a1 ðq; s; g; b; /; wÞ MCk;a1 ðq; s; g; b; /; wÞ MCk;a1 þ1 ðq; s; g; b; /; wÞ: Proof. To prove the first inclusion of Theorem 3, let f 2 MCkþ1;a1 ðq; s; g; b; /; wÞ. Then, from the definition of MCkþ1;a1 ðq; s; g; b; /; wÞ, there exists a function g 2 MSkþ1;a1 ðq; s; g; /Þ such that 1 zðH kþ1;q;s ða1 Þf ðzÞÞ0 b wðzÞ ðz 2 UÞ: 1b H kþ1;q;s ða1 ÞgðzÞ Now let 0 1 zðH k;q;s ða1 Þf ðzÞÞ b ; pðzÞ ¼ 1b H k;q;s ða1 ÞgðzÞ
ð2:3Þ
where p is analytic in U with p(0) = 1. Using (1.6), we obtain 0zðH ða Þðzf 0 ðzÞÞÞ0 1 H k;q;s ða1 Þðzf 0 ðzÞÞ k;q;s 1 0 1 zðH kþ1;q;s ða1 Þf ðzÞÞ 1 @ H k;q;s ða1 ÞgðzÞ þ ðk þ 1Þ H k;q;s ða1 ÞgðzÞ b ¼ bA: zðH k;q;s ða1 ÞgðzÞÞ0 1b 1b H kþ1;q;s ða1 ÞgðzÞ þkþ1
ð2:4Þ
H k;q;s ða1 ÞgðzÞ
Since g 2 MSkþ1;a1 ðq; s; g; /Þ MSk;a1 ðq; s; g; /Þ, by Theorem 1, we set 1 zðH k;q;s ða1 ÞgðzÞÞ0 g ; qðzÞ ¼ 1g H k;q;s ða1 ÞgðzÞ
ð2:5Þ
where q / in U with the assumption for / 2 N. Then, by virtue of (2.3), (2.4) and (2.5), we observe that 0 1 zðH kþ1;q;s ða1 Þf ðzÞÞ zp0 ðzÞ wðzÞ ðz 2 UÞ: ð2:6Þ b ¼ pðzÞ þ 1b ð1 gÞqðzÞ þ k þ 1 g H kþ1;q;s ða1 ÞgðzÞ Since k > 0 and q / in U with maxz2U Ref/ðzÞg < ðk þ 1 gÞ=ð1 gÞ, Refð1 gÞqðzÞ þ k þ 1 gg > 0
ðz 2 UÞ:
Hence, by taking xðzÞ ¼
1 ð1 gÞqðzÞ þ k þ 1 g
in (2.6), and applying Lemma 2, we can show that p w in U, so that f 2 MCk;a1 ðq; s; g; b; /; wÞ. Moreover, we have the second inclusion by using arguments similar to those detailed above with (1.5). Therefore we complete the proof of Theorem 3. h 3. Inclusion properties involving the integral operator Fl In this section, we consider the integral operator Fl (see, e.g., [8]) defined by Z z l F l ðf Þ :¼ F l ðf ÞðzÞ ¼ lþ1 tl f ðtÞ dt ðf 2 M; l > 0Þ: z 0
ð3:1Þ
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From the definition of Fl defined by (3.1), we observe that 0
zðH k;q;s ða1 ÞF l ðf ÞðzÞÞ ¼ lH k;q;s ða1 Þf ðzÞ ðl þ 1ÞH k;q;s ða1 ÞF l ðf ÞðzÞ: We first state Theorem 4 below, the proof of which is much akin to that of Theorem 1. Theorem 4. Let / 2 N with maxz2U Ref/ðzÞg < ðl þ 1 gÞ=ð1 gÞ ðl > 0; 0 6 g < 1Þ. If f 2 MSk;a1 ðq; s; g; /Þ, then F l ðf Þ 2 MSk;a1 ðq; s; g; /Þ. Next, we derive an inclusion property involving Fl, which is obtained by applying (1.7) and Theorem 4. Theorem 5. Let / 2 N with maxz2U Ref/ðzÞg < ðl þ 1 gÞ=ð1 gÞ ðl > 0; 0 6 g < 1Þ. If f 2 MKk;a1 ðq; s; g; /Þ, then F l ðf Þ 2 MKk;a1 ðq; s; g; /Þ. From Theorems 4 and 5, we have Corollary 2. Let (1 + A)/(1 + B) < (l + 1 g)/(1 g) (l > 0; 1 < B < A 6 1; 0 6 g < 1). Then if f 2 MSk;a1 ðq; s; g; A; BÞ ðorMKk;a1 ðq; s; g; A; BÞÞ, then F c ðf Þ 2 MSk;a1 ðq; s; g; A; BÞ ðorMKk;a1 ðq; s; g; A; BÞÞ. Finally, we obtain Theorem 6 below by using the same techniques as in the proof of Theorem 3. Theorem 6. Let /; w 2 N with maxz2U Ref/ðzÞg < ðl þ 1 gÞ=ð1 gÞ ðl > 0; 0 6 g < 1Þ. If f 2 MCk;a1 ðq; s; g; b; /; wÞ, then F c ðf Þ 2 MCk;a1 ðq; s; g; b; /; wÞ. Remark 1. If we take k = a1 = 2, b1 = 1, ai = bi (i = 2, . . . , s) and as+1 = 1 in all theorems of this section, then we extend the results by Goel and Sohi [7], which reduce the results earlier obtained by Bajpai [1]. Acknowledgement This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-521-C00008). References [1] S.K. Bajpai, A note on a class of meromorphic univalent functions, Rev. Roumaine Math. Pures Appl. 22 (1977) 295–297. [2] J.H. Choi, M. Sagio, H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002) 432–445. [3] J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13. [4] J. Dziok, H.M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5 (2002) 115–125. [5] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Trans. Spec. Funct. 14 (2003) 7–18. [6] P. Eenigenberg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot–Bouquet differential subordination, General Inequalities 3 (1983) 339–348. [7] R.M. Goel, N.S. Sohi, On a class of meromorphic functions, Glas. Mat. 17 (37) (1982) 19–28. [8] V. Kumar, S.L. Shukla, Certain integrals for classes of p-valent meromorphic functions, Bull. Austral. Math. Soc. 25 (1982) 85– 97. [9] R.J. Libera, M.S. Robertson, Meromorphic close-to-convex functions, Michigan Math. J. 8 (1961) 167–176. [10] J.-L. Liu, The Noor integral and strongly starlike functions, J. Math. Anal. Appl. 261 (2001) 441–447. [11] J.-L. Liu, H.M. Srivastava, A linear operator and associated families of meromorphically multivalent functions, J. Math. Anal. Appl. 259 (2001) 566–581. [12] J.-L. Liu, H.M. Srivastava, Certain properties of the Dziok–Srivastava operator, Appl. Math. Comput. 159 (2004) 485–493. [13] J.-L. Liu, H.M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Modell. 39 (2004) 21–34. [14] W.C. Ma, D. Minda, An internal geometric characterization of strongly starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 45 (1991) 89–97. [15] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981) 157–171. [16] S.S. Miller, P.T. Mocanu, Differential Subordination, Marcel Dekker, Inc., New York–Basel, 2000.
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K.I. Noor, On new classes of integral operators, J. Natur. Geom. 16 (1999) 71–80. K.I. Noor, M.A. Noor, On integral operators, J. Math. Anal. Appl. 238 (1999) 341–352. S. Owa, H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987) 1057–1077. H.M. Srivastava, S. Owa, Some characterizations and distortions theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions, Nagoya Math. J. 106 (1987) 1–28.
Inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric function Nak Eun Cho *, In Hwa Kim Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, 599-1 Daeyeon-dong, Nam-gu, Pusan 608-737, South Korea
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract The purpose of the present paper is to introduce several new classes of meromorphic functions defined by using a meromorphic analogue of the Choi–Saigo–Srivastava operator for the generalized hypergeometric function and investigate various inclusion properties of these classes. Some interesting applications involving these and other classes of integral operators are also considered. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Meromorphic functions; Differential subordination; Meromorphic starlike functions; Meromorphic convex functions; Meromorphic close-to-convex functions; Generalized hypergeometric function; Choi–Saigo–Srivastava operator; Integral operator
1. Introduction Let M denote the class of functions of the form 1 1 X ak zk ; f ðzÞ ¼ þ z k¼0 which are analytic in the punctured open unit disk D ¼ fz 2 C : 0 < jzj < 1g. If f and g are analytic in U ¼ D [ f0g, we say that f is subordinate to g, written f g or f(z) g(z), if there exists a Schwarz function w in U such that f(z) = g(w(z)). For 0 6 g, b < 1, we denote by MSðgÞ, MKðgÞ and MCðg; bÞ the subclasses of M consisting of all meromorphic functions which are, respectively, starlike of order g, convex of order g and close-to-convex of order b and type g in U (cf. e.g., [8,9,16]).
*
Corresponding author. E-mail addresses: [email protected] (N.E. Cho), [email protected] (I.H. Kim).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.109
116
N.E. Cho, I.H. Kim / Applied Mathematics and Computation 187 (2007) 115–121
Let N be the class of all functions / which are analytic and univalent in U and for which /ðUÞ is convex with /(0) = 1 and Ref/ðzÞg > 0 ðz 2 UÞ. Making use of the principle of subordination between analytic functions, we introduce the subclasses MSðg; /Þ, MKðg; /Þ and MCðg; b; /; wÞ of the class M for 0 6 g, b < 1 and /; w 2 N, which are defined by 1 zf 0 ðzÞ g /ðzÞ in U ; MSðg; /Þ :¼ f 2 M : 1g f ðzÞ 1 zf 00 ðzÞ MKðg; /Þ :¼ f 2 M : 1þ 0 g /ðzÞ in U 1g f ðzÞ and
MCðg; b; /; wÞ :¼
1 zf 0 ðzÞ b wðzÞ f 2 M : 9g 2 MSðg; /Þ s:t: 1b gðzÞ
in U :
We note that the classes mentioned above is the familiar classes which have been used widely on the space of analytic and univalent functions in U [2,14] and for special choices for the functions / and w involved in these definitions, we can obtain the well-known subclasses of M. For examples, we have 1þz 1þz MS g; ¼ MSðgÞ; MK g; ¼ MKðgÞ 1z 1z and
1þz 1þz ; MC g; b; ¼ MCðg; bÞ: 1z 1z
For complex parameters a1 ; . . . ; aq and b1 ; . . . ; bs ðbj 2 C n Z 0 ; Z0 :¼ f0; 1; 2; . . .g; j ¼ 1; . . . ; sÞ; we now define the generalized hypergeometric function qFs(a1, . . . , aq; b1, . . . , bq; z) [19,20] as follows: 1 X ða1 Þk ðaq Þk zk ; q F s ða1 ; . . . ; aq ; b1 ; . . . ; bq ; zÞ :¼ ðb1 Þk ðbs Þk k! k¼0 ðq 6 s þ 1; q; s 2 N0 :¼ N [ f0g; N :¼ f1; 2; . . .g; z 2 UÞ; where (m)k is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by 1 if k ¼ 0 and m 2 C n f0g; Cðm þ kÞ ¼ ðmÞk :¼ CðmÞ mðm þ 1Þ ðm þ k 1Þ if k 2 N and m 2 C: Corresponding to a function Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ :¼ z1 q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ:
ð1:1Þ
Liu and Srivastava [13] considered a linear operator H ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ : M ! M defined by the following Hadamard product (or convolution): H ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ :¼ Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ f ðzÞ:
ð1:2Þ
We note that the linear operator H(a1, . . . , aq; b1, . . . , bs) was motivated essentially by Dziok and Srivastava [3]. Some interesting developments associated with the generalized hypergeometric function were considered recently by Dziok and Srivastava [4,5] and Liu and Srivastava [11,12]. Corresponding to the function Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by (1.1), we introduce a function Fk ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ given by Fða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ Fk ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ ¼
1 k
zð1 zÞ
ðk > 0Þ:
ð1:3Þ
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Analogous to H(a1, . . . , aq; b1, . . . , bs) defined by (1.2), we now define the linear operator Hk(a1, . . . , aq; b1, . . . , bs) on R as follows: H k ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼ Fk ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ f ðzÞ ðai ; bj 2 C n
Z 0;
ð1:4Þ
i ¼ 1; . . . ; q; j ¼ 1; . . . ; s; k > 0; z 2 D; f 2 MÞ:
For convenience, we write H k;q;s ða1 Þ :¼ H k ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ: It is easily verified from the definition (1.3) and (1.4) that zðH k;q;s ða1 þ 1Þf ðzÞÞ0 ¼ a1 H k;q;s ða1 Þf ðzÞ ða1 þ 1ÞH k;q;s ða1 þ 1Þf ðzÞ
ð1:5Þ
and 0
zðH k;q;s ða1 Þf ðzÞÞ ¼ kH kþ1;q;s ða1 Þf ðzÞ ðk þ 1ÞH k;q;s ða1 Þf ðzÞ:
ð1:6Þ
We note that the operator Hk,q,s(a1) is closely related to the Choi–Saigo–Srivastava operator [2] for analytic functions, which includes the integral operator studied by Liu [10] and Noor et al. [17,18]. Next, by using the operator Hk,q,s(a1), we introduce the following classes of meromprphic functions for /; w 2 N, k > 0 and 0 6 g, b < 1: MSk;a1 ðq; s; g; /Þ :¼ ff 2 M : H k;q;s ða1 Þf 2 MSðg; /Þg; MKk;a1 ðq; s; g; /Þ :¼ ff 2 M : H k;q;s ða1 Þf 2 MKðg; /Þg and MCk;a1 ðq; s; g; b; /; wÞ :¼ ff 2 M : H k;q;s ða1 Þf 2 MCðg; b; /; wÞg: We also note that f ðzÞ 2 MKk;a1 ðq; s; g; /Þ () zf 0 ðzÞ 2 MSk;a1 ðq; s; g; /Þ:
ð1:7Þ
In particular, we set 1 þ Az MSk;a1 q; s; g; ¼: MSk;a1 ðq; s; g; A; BÞ ð1 < B < A 6 1Þ 1 þ Bz and
1 þ Az MKk;a1 q; s; g; ¼: MKk;a1 ðq; s; g; A; BÞ 1 þ Bz
ð1 < B < A 6 1Þ:
In this paper, we investgate several inclusion properties of the classes MSk;a1 ðq; s; g; /Þ, MKk;a1 ðq; s; g; /Þ and MCk;a1 ðq; s; g; b; /; wÞ associated with the operator Hk,q,s(a1). Some applications involving integral operators are also considered. 2. Inclusion properties involving the operator Hk,q,s(a1) The following results will be required in our investigation. Lemma 1 [6]. Let / be convex univalent in U with /(0) = 1 and Refj/ðzÞ þ mg > 0 ðj; m 2 CÞ. If p is analytic in U with p(0) = 1, then pðzÞ þ
zp0 ðzÞ /ðzÞ ðz 2 UÞ jpðzÞ þ m
implies pðzÞ /ðzÞ ðz 2 UÞ:
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Lemma 2 [15]. Let / be convex univalent in U and x be analytic in U with Re{x(z)} P 0. If p is analytic in U and p(0) = /(0), then pðzÞ þ xðzÞzp0 ðzÞ /ðzÞ
ðz 2 UÞ
implies pðzÞ /ðzÞ
ðz 2 UÞ:
At first, with the help of Lemma 1, we obtain the following: Theorem 1. Let / 2 N with maxz2U Ref/ðzÞg < minfðk þ 1 gÞ=ð1 gÞ; ða1 þ 1 gÞ=ð1 gÞg ðk; a1 > 0; 0 6 g < 1Þ. Then MSkþ1;a1 ðq; s; g; /Þ MSk;a1 ðq; s; g; /Þ MSk;a1 þ1 ðq; s; g; /Þ: Proof. To prove the first part of Theorem 1, let f 2 MSkþ1;a1 ðq; s; g; /Þ and set 1 zðH k;q;s ða1 Þf ðzÞÞ0 pðzÞ ¼ g ; 1g H k;q;s ða1 Þf ðzÞ where p is analytic in U with p(0) = 1. Applying (1.6) and (2.1), we obtain 0 1 zðH kþ1;q;s ða1 Þf ðzÞÞ zp0 ðzÞ ðz 2 UÞ: g ¼ pðzÞ þ 1g ð1 gÞpðzÞ þ k þ 1 g H kþ1;q;s ða1 Þf ðzÞ
ð2:1Þ
ð2:2Þ
Since maxz2U Ref/ðzÞg < ðk þ 1 gÞ=ð1 gÞ, we see that Refð1 gÞ/ðzÞ þ k þ 1 gg > 0ðz 2 UÞ: Applying Lemma 1 to (2.3), it follows that p /, that is, f 2 MSk;a1 ðq; s; g; /Þ. Moreover, by using the arguments similar to those detailed above with (1.5), we can prove the second part of Theorem 1. Therefore we complete the proof of Theorem 1. h Theorem 2. Let / 2 N with maxz2U Ref/ðzÞg < minfðk þ 1 gÞ=ð1 gÞ; ða1 þ 1 gÞ=ð1 gÞg ðk; a1 > 0; 0 6 g < 1Þ. Then MKkþ1;a1 ðq; s; g; /Þ MKk;a1 ðq; s; g; /Þ MKk;a1 þ1 ðq; s; g; /Þ: Proof. Applying (1.7) and Theorem 1, we observe that f ðzÞ 2 MKkþ1;a1 ðq; s; g; /Þ () zf 0 ðzÞ 2 MSkþ1;a1 ðq; s; g; /Þ ) zf 0 ðzÞ 2 MSk;a1 ðq; s; g; /Þ () f ðzÞ 2 MKk;a1 ðq; s; g; /Þ; and f ðzÞ 2 MKk;a1 ðq; s; g; /Þ () zf 0 ðzÞ 2 MSk;a1 ðq; s; g; /Þ ) zf 0 ðzÞ 2 MSk;a1 þ1 ðq; s; g; /Þ () f ðzÞ 2 MKk;a1 þ1 ðq; s; g; /Þ; which evidently proves Theorem 2. h Taking /ðzÞ ¼
1 þ Az 1 þ Bz
ð1 < B < A 6 1; z 2 UÞ
in Theorems 1 and 2, we have Corollary 1. Let (1 + A)/(1 + B) < min{(k + 1 g)/(1 g), (a1 + 1 g)/(1 g)} (k, a1 > 0; 0 6 g < 1; 1 < B < A 6 1). Then MSkþ1;a1 ðq; s; g; A; BÞ MSk;a1 ðq; s; g; A; BÞ MSk;a1 þ1 ðq; s; g; A; BÞ
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and MKkþ1;a1 ðq; s; g; A; BÞ MKk;a1 ðq; s; g; A; BÞ MKk;a1 þ1 ðq; s; g; A; BÞ: Next, by using Lemma 2, we obtain the following inclusion relation for the class MCk;a1 ðq; s; g; b; /; wÞ. Theorem 3. Let /; w 2 N with maxz2U Ref/ðzÞg < minfðk þ 1 gÞ=ð1 gÞ; ða1 þ 1 gÞ=ð1 gÞg ðk; a1 > 0; 0 6 g < 1Þ. Then MCkþ1;a1 ðq; s; g; b; /; wÞ MCk;a1 ðq; s; g; b; /; wÞ MCk;a1 þ1 ðq; s; g; b; /; wÞ: Proof. To prove the first inclusion of Theorem 3, let f 2 MCkþ1;a1 ðq; s; g; b; /; wÞ. Then, from the definition of MCkþ1;a1 ðq; s; g; b; /; wÞ, there exists a function g 2 MSkþ1;a1 ðq; s; g; /Þ such that 1 zðH kþ1;q;s ða1 Þf ðzÞÞ0 b wðzÞ ðz 2 UÞ: 1b H kþ1;q;s ða1 ÞgðzÞ Now let 0 1 zðH k;q;s ða1 Þf ðzÞÞ b ; pðzÞ ¼ 1b H k;q;s ða1 ÞgðzÞ
ð2:3Þ
where p is analytic in U with p(0) = 1. Using (1.6), we obtain 0zðH ða Þðzf 0 ðzÞÞÞ0 1 H k;q;s ða1 Þðzf 0 ðzÞÞ k;q;s 1 0 1 zðH kþ1;q;s ða1 Þf ðzÞÞ 1 @ H k;q;s ða1 ÞgðzÞ þ ðk þ 1Þ H k;q;s ða1 ÞgðzÞ b ¼ bA: zðH k;q;s ða1 ÞgðzÞÞ0 1b 1b H kþ1;q;s ða1 ÞgðzÞ þkþ1
ð2:4Þ
H k;q;s ða1 ÞgðzÞ
Since g 2 MSkþ1;a1 ðq; s; g; /Þ MSk;a1 ðq; s; g; /Þ, by Theorem 1, we set 1 zðH k;q;s ða1 ÞgðzÞÞ0 g ; qðzÞ ¼ 1g H k;q;s ða1 ÞgðzÞ
ð2:5Þ
where q / in U with the assumption for / 2 N. Then, by virtue of (2.3), (2.4) and (2.5), we observe that 0 1 zðH kþ1;q;s ða1 Þf ðzÞÞ zp0 ðzÞ wðzÞ ðz 2 UÞ: ð2:6Þ b ¼ pðzÞ þ 1b ð1 gÞqðzÞ þ k þ 1 g H kþ1;q;s ða1 ÞgðzÞ Since k > 0 and q / in U with maxz2U Ref/ðzÞg < ðk þ 1 gÞ=ð1 gÞ, Refð1 gÞqðzÞ þ k þ 1 gg > 0
ðz 2 UÞ:
Hence, by taking xðzÞ ¼
1 ð1 gÞqðzÞ þ k þ 1 g
in (2.6), and applying Lemma 2, we can show that p w in U, so that f 2 MCk;a1 ðq; s; g; b; /; wÞ. Moreover, we have the second inclusion by using arguments similar to those detailed above with (1.5). Therefore we complete the proof of Theorem 3. h 3. Inclusion properties involving the integral operator Fl In this section, we consider the integral operator Fl (see, e.g., [8]) defined by Z z l F l ðf Þ :¼ F l ðf ÞðzÞ ¼ lþ1 tl f ðtÞ dt ðf 2 M; l > 0Þ: z 0
ð3:1Þ
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From the definition of Fl defined by (3.1), we observe that 0
zðH k;q;s ða1 ÞF l ðf ÞðzÞÞ ¼ lH k;q;s ða1 Þf ðzÞ ðl þ 1ÞH k;q;s ða1 ÞF l ðf ÞðzÞ: We first state Theorem 4 below, the proof of which is much akin to that of Theorem 1. Theorem 4. Let / 2 N with maxz2U Ref/ðzÞg < ðl þ 1 gÞ=ð1 gÞ ðl > 0; 0 6 g < 1Þ. If f 2 MSk;a1 ðq; s; g; /Þ, then F l ðf Þ 2 MSk;a1 ðq; s; g; /Þ. Next, we derive an inclusion property involving Fl, which is obtained by applying (1.7) and Theorem 4. Theorem 5. Let / 2 N with maxz2U Ref/ðzÞg < ðl þ 1 gÞ=ð1 gÞ ðl > 0; 0 6 g < 1Þ. If f 2 MKk;a1 ðq; s; g; /Þ, then F l ðf Þ 2 MKk;a1 ðq; s; g; /Þ. From Theorems 4 and 5, we have Corollary 2. Let (1 + A)/(1 + B) < (l + 1 g)/(1 g) (l > 0; 1 < B < A 6 1; 0 6 g < 1). Then if f 2 MSk;a1 ðq; s; g; A; BÞ ðorMKk;a1 ðq; s; g; A; BÞÞ, then F c ðf Þ 2 MSk;a1 ðq; s; g; A; BÞ ðorMKk;a1 ðq; s; g; A; BÞÞ. Finally, we obtain Theorem 6 below by using the same techniques as in the proof of Theorem 3. Theorem 6. Let /; w 2 N with maxz2U Ref/ðzÞg < ðl þ 1 gÞ=ð1 gÞ ðl > 0; 0 6 g < 1Þ. If f 2 MCk;a1 ðq; s; g; b; /; wÞ, then F c ðf Þ 2 MCk;a1 ðq; s; g; b; /; wÞ. Remark 1. If we take k = a1 = 2, b1 = 1, ai = bi (i = 2, . . . , s) and as+1 = 1 in all theorems of this section, then we extend the results by Goel and Sohi [7], which reduce the results earlier obtained by Bajpai [1]. Acknowledgement This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-521-C00008). References [1] S.K. Bajpai, A note on a class of meromorphic univalent functions, Rev. Roumaine Math. Pures Appl. 22 (1977) 295–297. [2] J.H. Choi, M. Sagio, H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002) 432–445. [3] J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13. [4] J. Dziok, H.M. Srivastava, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5 (2002) 115–125. [5] J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Trans. Spec. Funct. 14 (2003) 7–18. [6] P. Eenigenberg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Briot–Bouquet differential subordination, General Inequalities 3 (1983) 339–348. [7] R.M. Goel, N.S. Sohi, On a class of meromorphic functions, Glas. Mat. 17 (37) (1982) 19–28. [8] V. Kumar, S.L. Shukla, Certain integrals for classes of p-valent meromorphic functions, Bull. Austral. Math. Soc. 25 (1982) 85– 97. [9] R.J. Libera, M.S. Robertson, Meromorphic close-to-convex functions, Michigan Math. J. 8 (1961) 167–176. [10] J.-L. Liu, The Noor integral and strongly starlike functions, J. Math. Anal. Appl. 261 (2001) 441–447. [11] J.-L. Liu, H.M. Srivastava, A linear operator and associated families of meromorphically multivalent functions, J. Math. Anal. Appl. 259 (2001) 566–581. [12] J.-L. Liu, H.M. Srivastava, Certain properties of the Dziok–Srivastava operator, Appl. Math. Comput. 159 (2004) 485–493. [13] J.-L. Liu, H.M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Modell. 39 (2004) 21–34. [14] W.C. Ma, D. Minda, An internal geometric characterization of strongly starlike functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 45 (1991) 89–97. [15] S.S. Miller, P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981) 157–171. [16] S.S. Miller, P.T. Mocanu, Differential Subordination, Marcel Dekker, Inc., New York–Basel, 2000.
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