DIFFERENTIAL OPERATOR GENERALIZED BY FRACTIONAL DERIVATIVES

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1 Miskolc Mathematical Notes HU e-issn Vol. 12 (2011), No. 2, pp DIFFERENTIAL OPERATOR GENERALIZED BY FRACTIONAL DERIVATIVES RABHA W. IBRAHIM AND M. DARUS Received March 30, 2010 Abstract. By using the fractional derivative operator of Owa and Srivastava, we define a new linear multiplier fractional differential operator. Some generalized classes of analytic functions containing this multiplier are introduced. Basic properties of these classes are studied, such as inclusion relations and coefficient bounds. Some well known subclasses are pointed out as new special cases of our results. Moreover, the Cesáro partial sums m of functions f are considered, and sharp lower bounds for the ratios of real part of f and m (and also of f 0 and 0 m ) are determined in the unit open disk Mathematics Subject Classification: 30C45 Keywords: starlike function, convex function, close to convex function, fractional derivative, inclusion relations, subordination, superordination, Cesáro partial sums 1. INTRODUCTION Let H be the class of analytic functions in U WD f 2 C W j j < 1g and HŒa;n be the subclass of H consisting of functions of the form D aca n n Ca nc1 nc1 C :::: Let A be the class of functions of the form D C a n n; (1.1) which are analytic in the unit disk U: Given two functions f;g 2 A; D C P 1 a n n and D C P 1 their convolution or Hadamard product is defined by D C a n b n n;. 2 U /: b n n The work here is fully supported by MOHE: UKM-ST-06-FRGS c 2011 Miskolc University Press

2 168 RABHA W. IBRAHIM AND M. DARUS For several functions f 1. /;:::;f m. / 2 A, f 1. / ::: f m. / D C.a 1n :::a mn / n;. 2 U /: A function f 2 A is called starlike of order if it satisfies the following inequality <f f 0. / g > ;. 2 U / for some 0 < 1: We denoted this by by class S./: A function f 2 A is called convex of order if it satisfies the following inequality <f f 00. / f 0 C 1g > ;. 2 U /. / for some 0 < 1: We denoted this class C./: Note that f 2 C./ if and only if f 0 2 S./: The following definitions are required in our present investigation. Definition 1. (see [8] ) (Subordination Principal). For two functions f and g analytic in U; we say that the function is subordinated to in U and write. 2 U /; if there exists a Schwarz function w. / analytic in U with w.0/ D 0; and jw. /j < 1; such that D g.w. //; 2 U: In particular, if the function is univalent in U; the above subordination is equivalent to f.0/ D g.0/ and f.u / g.u /: Definition 2. (see [9]) (Differential subordination ) Let W C 2! C and let h be univalent in U: If p is analytic in U and satisfies the differential subordination.p. //; p 0. // h. / then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, p q: If p and.p. //; p 0. // are univalent in U and satisfy the differential superordination h. /.p. //; p 0. //, then p is called a solution of the differential superordination. An analytic function q is called subordinant of the solution of the differential superordination if q p: Definition 3. (see [10]) The fractional derivative of order is defined, for a function, by Z D WD 1 d f./ di 0 < 1;.1 / d 0. / where the function is analytic in simply-connected region of the complex z- plane C containing the origin and the multiplicity of. / is removed by requiring log. / to be real when. / > 0: Definition 4. [15] A function f 2 S (the class of univalent functions in U ) is said to be in the class SH./ if it satisfies j f 0. / 2. p 2 1/j < <f p 2 f 0. / g C 2.p 2 1/

3 for some > 0 and for all 2 U: DIFFERENTIAL OPERATOR 169 Remark 1. Geometric interpretation: Let./ WD f f 0. / ; f 2 HS./g then./ D f! D u C iv W v 2 < 4u C u 2 Iu > 0g: Note that./ is the interior of a hyperbola in the right half-plane which is symmetric with respect to the real axis and has vertex at the origin. Define the function '.a;ci / by.a/ n '.a;ci / WD n;. 2 U I a 2 R; c 2 Rnf0;.c/ n 1; 2;:::g/; nd0 where.a/ n is the Pochhammer symbol defined by.a/ n WD.a C n/.a/ D 1;.n D 0/I a.a C 1/.a C 2/:::.a C n 1/;.n 2 N/: Corresponding to the function '.a;ci /; Carlson and Shaffer [5] introduced a linear operator L.a;c/ by L.a;c/ WD '.a;ci / ; f 2 A: Note that L.a;a/ is the identity operator. In [11], Owa and Srivastava introduced the operator W A! A; that is known as an extension of fractional derivative and fractional integral, as follows WD.2 / D.n C 1/.2 / D C a n n.n C 1 / D '.2;2 I / D L.2;2 I /: (1.2) Note that 0 D : We define the linear multiplier fractional differential operator as follows D 0 D D 1; D C a n n; D.ˇ / C. / 0 C.1 ˇ/.n C 1/.2 / D C Œ Œ.n 1/ C ˇ a n n.n C 1 / D D./;

4 170 RABHA W. IBRAHIM AND M. DARUS D 2; D D.D1; /.n C 1/.2 / D C fœ Œ.n.n C 1 / 1/ C ˇ g 2 a n n; : D D D k 1;.n C 1/.2 / D C fœ Œ.n.n C 1 / WD C n;k. ;/a n n; 1/ C ˇ g k a n n for 0 < 1;ˇ 1; 0 and k 2 N 0 D N [ f0g with f.0/ D 0: (1.3) Remark 2. (1) When D 0;ˇ D 1; we have Al-Oboudi s differential operator (see [3]). (2) When D 0;ˇ D 1 and D 1; we get Sǎlǎgean s differential operator (see [16]). (3) When k D 1;ˇ D 1 and D 0; we obtain the Owa-Srivastava fractional differential operator (see [11]). (4) When ˇ D 1; we get the linear multiplier fractional differential operator that introduced in [4]. Using the operator ; we define the following classes. Definition 5. Let SH k; ;./; k 2 N be the class of all functions f 2 A and univalent in U satisfying ˇ p 2. 2 ˇDkC1; for some ;. > 0/ and for all 2 U: np D kc1; o 1/ˇˇˇ < < 2 C 2. p 2 1/ Remark 3. Geometric interpretation: If we denote with p the analytic and univalent functions with the properties p.0/ D 1;p 0.0/ > 0 and p.u / D./ (see Remark 1), then f 2 SH k; ;./ if and only if D kc1; p. /; 2 U;

5 q 1Cb such that p. / D.1 C 2/ 1 2; where b D the square root p w is chosen so that I m p w 0: We can observe that DIFFERENTIAL OPERATOR 171 SH k;0;1;1./ SH k./ 1C4 42.1C2/ 2 where the class SH k./ was introduced in [1]. Also we have SH 0;0;1;1./ SH./; where the class SH./ was proposed in [15]. and the branch of Definition 6. Let 0 < 1;ˇ 0; 0; and f 2 A; 2 Œ0;1/: We define the class.;k/ ; S by requiring that f 2 S and Note that < ˇ > ˇDk;.;k/ 0;1;1 S.;k/ S; where the class.;k/ S is defined and studied in [2]. if ˇ 1. 2 U /: Remark 4. (Geometric interpretation): A function f 2.;k/ ; p. /;. 2 Œ0;1/; 2 U / S if and only where p denoted the function which maps the unit disk conformally onto the region ; such that 1 2 D fu C iv W u 2 D 2.u 1/ 2 C 2 v 2 g: The domain is elliptic for > 1; is hyperbolic when 0 < < 1; parabolic for D 1; and is the right half-plane when D 0: Definition 7. Let 0 < 1;ˇ 0; 0; and f 2 A; 2 Œ0;1/: We define the class.;k/ ; CC with respect to the function g 2.;k/ ; S Note that < ˇ > ˇDk;.;k/ 0;1;1 CC.;k/ CC; where the class.;k/ CC is defined and studied in [1]. ˇ 1. 2 U /:

6 172 RABHA W. IBRAHIM AND M. DARUS CC with re- Remark 5. (Geometric interpretation): A function f 2.;k/ ; spect to g 2.;k/ ; S if and only if p. /;. 2 Œ0;1/; 2 U / Dk; where p is defined in Remark 4. Or take all values in the domain : We need the following preliminaries in the sequel. The Libera-Pascu integral operator L a W A! A is defined by Z WD L a F. / D 1 C a F.t/t a 1 dt; a 2 C; <.a/ 0: a 0 For a D 1 we obtain the Libera integral operator, for a D 0 we obtain the Alexander integral operator and in the cases a D 1;2;3;::: we obtain the Bernardi integral operator. Lemma 1. ( see [7]) Let be convex in U with <f. / C g > 0 for ; 2 C; also let p. / 2 H.U / with p.0/ D.0/ and assume that the Briot-Bouquet differential subordination p. / C p0. / p. / C. /; 2 U is satisfied. These imply that p. /. /; 2 U: Lemma 2. ( see [7]) Let q be convex in U and W U! C with <f. /g > 0: If p. / 2 H.U / and assume that the subordination is satisfied, then p. / C. /: p 0. / q. /; 2 U p. / q. /; 2 U: 2. THE CLASS SH k; ;./ In this section we study the inclusion property of functions in the class SH k; ;./: Further we show that if F 2 SH k; ;./ implies that for the Libera integral operator L a F. / 2 SH k; ;./: Theorem 1. Let > 0; 0 < 1;ˇ 1; > 0: We have SH kc1; ;./ SH k; ;./: Proof. Let f 2 SH kc1; ;./: Denote DkC1; p. / WD ; p.0/ D 1;. 2 U /:

7 Then we have D kc2; D kc1; D D kc2; : DIFFERENTIAL OPERATOR 173 D kc1; D Now, by using the definition of the operator (1.3), we verify hd kc1; p 2. / C p 0. / D i 2 1 D kc2; p. / : (2.1) h C :.DkC1; / 0 D kc1; :. /0 i. /2 hd kc1; D Putting (2.2) in (2.1), we obtain i 2 h C :.DkC2;.ˇ /D kc1; /.1 ˇ/ i. /2 hd kc1; :.D kc1;.ˇ / /.1 ˇ/ i. /2.DkC1; / 2 D. f C DkC2; D kc1; 2. //2 2 D DkC2; : (2.2) D kc2; D kc1; D p. / C Since f 2 SH kc1; ;./, then in view of Remark 3, we have D kc2; D kc1; D p. / C p 0. / ;. 2 U /: (2.3) p. / p 0. / p. / p. /; with p.0/ D p.0/ D 1; > 0; and <.p. // > 0: Hence, from Lemma 1, we obtain p. / p. /;. 2 U /;

8 174 RABHA W. IBRAHIM AND M. DARUS or D kc1; p. /;. 2 U /: Now Remark 3 gives 2 SH k; ;./: The next results can be found in [1, 2] respectively. Corollary 1. Let > 0: We have Proof. By letting k D 0 in Theorem 1. Corollary 2. Let > 0: We have SH./ S: SH nc1;./ SH n;./: Proof. By assuming D 0 and ˇ D 1 in Theorem 1. Theorem 2. Let > 0; 0 < 1;ˇ 1; > 0: If F 2 SH k; ;./ with.1 C a/ > then the Libera-Pascu integral operator WD L a F. / is in SH k; ;./: Proof. Let F 2 SH k; ;./: Denote DkC1; p. / WD ; p.0/ D 1;. 2 U / (2.4) and assume that 0 < <fp. /g 1: From the definition of the Libera-Pascu integral operator we have.1 C a/f. / D a C f 0. /; by using the linear operator D kc1; ; we have.1 C a/d kc1; F. / D ad kc1; C D kc1;. f 0. // D ad kc1; C.D kc1; / 0 DkC2; D ad kc1;.ˇ /D kc1;.1 ˇ/ C ; or equivalently,.1 C a/d kc1; F. / D Œ.1 C a/ ˇ D kc1; C D kc2;.1 ˇ/ : Similarly, we obtain.1 C a/ F. / D Œ.1 C a/ ˇ Dk; C DkC1;.1 ˇ/ :

9 Then by using (2.4) and (2.3) we have D kc1; F. / F. / Œ.1 C a/ D DIFFERENTIAL OPERATOR 175 DkC1; ˇ C D kc2; : D kc1;.1 ˇ/ D kc1; Œ.1 C a/ ˇ C DkC1;.1 ˇ/ DkC2; Œ.1 C a/ ˇ p. / C :p. / C '. / D kc1; WD Œ.1 C a/ ˇ C p. / C '. / Œ.1 C a/ D ˇ p. / C p2. / C p 0. / C '. / Œ.1 C a/ ˇ C p. / C '. / D p. /fœ.1 C a/ ˇ C p. / C '. /g C p0. / C '. /.1 p. // Œ.1 C a/ ˇ C p. / C '. / '. /.1 p. // C D p. / C p 0 p. /: 0. / Œ.1 C a/ ˇ C p. / C '. / WD p. / C p 0. /:. /; where <.. // > 0 as! 1 C : From F 2 HS k; ;./; we have thus in view of Lemma 2, we obtain p. / C p 0. /:. / p. /; p. / p. /;. 2 U / or D kc1; p. /; p.0/ D 1;. 2 U /: Implies f 2 HS k; ;./: This completes the proof. The next results can be found in [2]. Corollary 3. Let > 0: If F 2 SH n;./: Then the f 2 SH n;./ where f is the Libera-Pascu integral operator. Proof. Assume D 0 and ˇ D 1: Hence. / D 3. THE CLASS.;k/ ; CC in Theorem 2. Œ.1Ca/ 1 Cp. / In this section, we show that for any function F 2 A in the classes.;k/ ; S and.;k/ ; CC; the Libera-Pascu integral operator acting on F is also belongs to these classes. Moreover, some well known subclasses are obtained as special cases of our results.

10 176 RABHA W. IBRAHIM AND M. DARUS Theorem 3. If F 2.;k/ ; S; 2 Œ0;1/ and 0 < 1;ˇ > 0; > 0; then Libera-Pascu integral operator D L a F. / 2.;k/ ; S: Proof. Let F 2.;k/ ; S: Consider Dk; p. / WD ; p.0/ D 1;. 2 U /: (3.1) From the definition of the Libera-Pascu integral operator we obtain By using the linear operator ; we have.1 C a/f. / D a C f 0. /: (3.2).1 C a/ F. / D adk; C Dk;. f 0. // From (3.2) and (3.3) we obtain D a C.Dk; /0 DkC1; D a.ˇ /.1 ˇ/ C D Œa C ˇ Dk; C 1 DkC1;.ˇ 1/ C WD A C BDkC1; C '. /: (3.3) F. / F. / D ADk; D C BDkC1; C '. / ha Dk; D A a C f 0. / C B DkC1; Œa C f 0. / DkC1; C B C '. / a C f 0. / i C '. / : (3.4) From (3.1) we have

11 DIFFERENTIAL OPERATOR 177 p 0. / D ŒDk; 0 where C WD D ŒDk; 0 D f 2. / :f 0. / : f 0. / D kc1;.ˇ /.1 ˇ/ WD BDkC1; D BDkC1;.ˇ / : Hence CD į ; Cp. / C '. / C '. / p. /: f 0. / ; BD kc1; D p 0. / C p. /ŒC C f 0. / Substitute (3.6) in (3.4), then we obtain F. / F. / From the hypothesis, we have Consequently, we have p. / C D Ap. / C B D kc1; C '. / a C f 0. / D p0. / C p. /ŒA C C C f 0. / a C f 0. / D p0. / C p. /Œa C f 0. / a C f 0. / 1 D p. / C a C f 0. / p 0. /: F. / p. /;. 2 Œ0;1/; 2 U /: F. / 1 a C f 0. / : f 0. / p. /: f 0. / p 0. / p. /;. 2 Œ0;1/; 2 U /: (3.5) '. / : (3.6) (3.7)

12 178 RABHA W. IBRAHIM AND M. DARUS Now in view of Lemma 2, we get Equivalently p. / p. /;. 2 Œ0;1/; 2 U /: p. /;. 2 Œ0;1/; 2 U /; which implies D L a F. / 2.;k/ ; S: The next result can be found in [2]. Corollary 4. If F 2.;n/ D L a F. / 2.;n/ S: S; 2 Œ0;1/; then Libera-Pascu integral operator Proof. Let D 0;ˇ D D 1 in Theorem 3. Corollary 5. If F 2.;k/ D L a F. / 2.;k/ S: S; 2 Œ0;1/; then Libera-Pascu integral operator Proof. Put D 0;ˇ D 1 in Theorem 3. Note that in this case we have Al-Oboudi s differential operator. Theorem 4. If F 2.;k/ ; CC; 2 Œ0;1/ and 0 < 1;ˇ > 0; > 0; with respect to the function G. / 2.;k/ ; S, then the Libera-Pascu integral operator D L a F. / 2.;k/ ; CC with respect to the function D L a G. / 2.;k/ ; S: Proof. Let F 2.;k/ ; CC and Dk; p. / WD ; p.0/ D 1;. 2 U /: (3.8) Differentiating the Libera-Pascu integral operator D L a F. / and operating by ; we have.1 C a/ F. / D ADk; C BDkC1; C '. /; (3.9) where A;B and '. / are defined in Theorem 3. We also observe that.1 C a/g. / D a C g 0. /: (3.10)

13 By using (3.9) and (3.10) we get F. / G. / From (3.8) we have DIFFERENTIAL OPERATOR 179 D ADk; D C BDkC1; C '. / ha Dk; D A p 0. / D ŒDk; 0 D ŒDk; 0 D DkC1; g 2. / a C g 0. / C B DkC1; Œa C g0. / DkC1; C B C '. / a C g0. / :g0. / : g0. /.ˇ /.1 ˇ/ WD BDkC1; D BDkC1; C C '. / i C '. / : p. /: g0. / : g0. / (3.11) Cp. / p. /: g0. / C '. / ; (3.12) where C defined in Theorem 3. Hence BD kc1; D p 0. / C p. /ŒC C g0. / Substitute (3.13) in (3.11) we obtain F. / 1 D p. / C p 0. / G. / a C g0. / WD p. / C. / p 0. /; where <fg > 0; 2 U: Thus in view of Lemma 2, we obtain p. / p. /;. 2 Œ0;1/; 2 U / '. / : (3.13) (3.14)

14 180 RABHA W. IBRAHIM AND M. DARUS or equivalently p. /;. 2 Œ0;1/; 2 U /: Hence D L a F. / 2.;k/ ; CC with respect to the function D L a G. / 2.;k/ ; S: The next result can be found in [2]. Corollary 6. If F 2.;k/ CC; 2 Œ0;1/ with respect to the function G. / 2.;k/ S then Libera-Pascu integral operator D L a F. / 2.;n/ CC with respect to the function D L a G. / 2.;k/ S: Proof. Let D 0;ˇ D D 1 in Theorem 4. Corollary 7. If F 2.;k/ CC; 2 Œ0;1/ and > 0; with respect to the function G. / 2.;k/ S, then Libera-Pascu integral operator D L a F. / 2.;k/ CC with respect to the function D L a G. / 2.;k/ S: Proof. Put D 0;ˇ D 1 in Theorem 4. Note that in this case we have Al-Oboudi s differential operator. 4. THE CLASS.;k/ ; S In this section, we determine coefficient bounds for functions of the form (1.1), with positive and negative coefficients, in the class.;k/ ; S: This study is required in the next section. Theorem 5. A sufficient condition for a function of the form (1.1) to be in S is ja n jœ. n;k. ;/ 1/.1 C / C 1 < 1; (4.1) class.;k/ ; where 2 Œ0;1/ and 0 < 1;ˇ > 0; > 0: Proof. It suffices to show that We obtain ˇ ˇDk; ˇ ˇDk; 1ˇ < 1 1: 1ˇ < ˇ 1.1 C / ˇDk;.1 C /P 1 ja nj. n;k. ;/ 1/j j n 1 P 1 1 ja njj j n 1 1ˇ

15 DIFFERENTIAL OPERATOR C /P 1 ja nj. n;k. ;/ 1/ P 1 1 ja : nj This last expression is bounded above by 1 if ja n jœ. n;k. ;/ 1/.1 C / C 1 < 1; and the proof is complete. Now we prove that the condition (4.1) is also necessary for f with negative coefficients. Let T be the class of all analytic functions of the form D a n n;.a n 0; 2 U /: Then we have the following result. Theorem 6. A sufficient and necessary condition for f 2 T to be in the class.;k/ ; T S D.;k/ ; S \T (here 2 Œ0;1/ and 0 < 1;ˇ > 0; > 0), is that a n Œ. n;k. ;/ 1/.1 C / C 1 < 1: Proof. In view of Theorem 1, we need to prove only the necessity. If f 2.;k/ ; T S and is real, then ˇ 1 > D ˇDk; 1ˇ <.1 C /< P 1.1 C /< 1 1 a n. n;k. ;/ 1/ n 1 P 1 1 : a n n 1 Letting! 1 along the real axis, we obtain the desired inequality a n Œ. n;k. ;/ 1/.1 C / C 1 < 1: Corollary 8. The extreme points of.;k/ ; T S are the functions given by 1 f 1. / D 1; and f n. / D Œ. n;k. ;/ 1/.1 C / C 1 n; where n D 2;3;:::; 2 Œ0;1/ and 0 < 1;ˇ > 0; > 0:

16 182 RABHA W. IBRAHIM AND M. DARUS 5. CESÁRO SUM Recently, Silverman [14] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums: n o < ;< f N. / n fn. / o n f 0. / o n f 0 ;< fn 0. / ; and < N. / f 0. / In this section, we consider the Cesáro sum m of functions in the class.;k/ ; S and obtain sharp lower bounds for the ratios of real part of and m, and also of f 0. / and 0 m : Let us first construct the Cesáro sum m. / of functions f 2 A by m. / D 1 m C 1 mx s n. / nd0 D 1 s0. / C s 1. / C :: C s m. / m C 1 D 1 a0 C.a 0 C a 1 / C :: C.a 0 C ::: C a m m/ m C 1 D 1.m C 1/a0 C ma 1 C :: C a m m m C 1 mx m n C 1 D am n m C 1 nd0 D mx nd0 WD g m. / m n C 1 n m C 1 o : (5.1) where s 0. / D 0; s 1. / D : Note that the classical Cesáro means play an important role in geometricn function otheory (see [6, 12, 13]). In the sequel, we will make use of the result that <.1Cw. // > 0;. 2 U / if and only if w. / D P 1.1 w. // nd1 c n n satisfies the inequality jw. /j < j j. Theorem 7. Let the function of the form (1.1) satisfies condition (4.1), then <f m. / g. mc1;k. ;/ 1/.1 C / ;. 2 U /;. mc1;k. ;/ 1/.1 C / C 1 where m. / defined in (5.1). Proof. Assume that f 2 A and satisfies (4.1). By setting d m WD Œ. m;k. ;/ 1/.1 C / C 1 ; C.n;m/ WD m n C 1 m C 1

17 and w. / D d mc1 n m. / DIFFERENTIAL OPERATOR o d mc1 D 1 C d mc1œ P 1 a n n 1 P m C.n;m/a n n 1 1 C P m C.n;m/a n n 1 when C.n;m/! 1; we find that ˇ w. / 1 P d 1 mc1 ndmc1 ˇ ja nj w. / C P m ja P nj d 1 mc1 ndmc1 ja nj 1;. 2 U / if and only if which is equivalent to In order to see that 2d mc1 1 X ndmc1 mx ja n j C d mc1 ja n j X ndmc1 mx ja n j mc1 D C ;. 2 U / d mc1 gives sharp result, we observe that for D re i m m m. / D 1 C! 1 d mc1 Hence the proof is complete. ja n j 1: (5.2) 1 d mc1 as! 1 : In the same manner of Theorem 7, we can verify the following result Theorem 8. Let the function of the form (1.1) satisfies condition (4.1), then <f f 0. / m 0. /g. mc1;k. ;/ 1/.1 C / m ;. 2 U /;. mc1;k. ;/ 1/.1 C / C 1 where m. / defined in (5.1). Proof. Assume that f 2 A and satisfies (4.1). By letting n f 0. / m C 1 o w. / D d mc1 m 0. / 1 d mc1 mc1 d D 1 C mc1 Œ P 1 na n n 1 P m C.n;m/na n n 1 1 C P m C.n;m/na n n 1 ; where d m is defined in Theorem 7.

18 184 RABHA W. IBRAHIM AND M. DARUS REFERENCES [1] M. Acu, On a subclass of n-starlike functions associated with some hyperbola, Gen. Math., vol. 13, no. 1, pp , [2] M. Acu, On a subclass of n-uniformly close to convex functions, Gen. Math., vol. 14, no. 1, pp , [3] F. M. Al-Oboudi, On univalent functions defined by a generalized Sǎlǎgean operator, Int. J. Math. Math. Sci., no , pp , [4] F. M. Al-Oboudi and K. A. Al-Amoudi, On classes of analytic functions related to conic domains, J. Math. Anal. Appl., vol. 339, no. 1, pp , [5] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., vol. 15, no. 4, pp , [6] M. Darus and R. W. Ibrahim, On Cesáro means for Fox-Wright functions, J. Math. Stat., vol. 4, no. 3, pp , [7] S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., vol. 28, no. 2, pp , [8] S. S. Miller and P. T. Mocanu, Differential subordinations. Theory and applications, ser. Monographs and Textbooks in Pure and Applied Mathematics. New York: Marcel Dekker, Inc., 2000, vol [9] S. S. Miller and P. T. Mocanu, Subordinants of differential superordinations, Complex Var. Theory Appl., vol. 48, no. 10, pp , [10] S. Owa, On the distortion theorems. I, Kyungpook Math. J., vol. 18, no. 1, pp , [11] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., vol. 39, no. 5, pp , [12] S. Ruscheweyh, Geometric properties of the Cesàro means, Results Math., vol. 22, no. 3-4, pp , [13] S. Ruscheweyh and L. C. Salinas, Subordination by Cesàro means, Complex Variables Theory Appl., vol. 21, no. 3-4, pp , [14] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl., vol. 209, no. 1, pp , [15] J. Stankiewicz and A. Wiśniowska, Starlike functions associated with some hyperbola, Zeszyty Nauk. Politech. Rzeszowskiej Mat., no. 19, pp , [16] G. c. Sǎlǎgean, Subclasses of univalent functions, in Complex analysis fifth Romanian-Finnish seminar, Part 1(Bucharest, 1981), ser. Lecture Notes in Math., vol Berlin: Springer, 1983, pp Authors addresses Rabha W. Ibrahim University Malaya, Institute of Mathematical Sciences, Kuala Lumpur, 50603, Malaysia address: rabhaibrahim@yahoo.com M. Darus Universiti Kebangsaan Malaysia, School of Mathematical Sciences, Bangi 43600, Malaysia address: maslina@ukm.my