Some inclusion relationships associated with Dziok–Srivastava operator

Applied Mathematics and Computation 216 (2010) 431–437

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Some inclusion relationships associated with Dziok–Srivastava operator M.K. Aouf Faculty of Science, Mansoura University, Mansoura 35516, Egypt

a r t i c l e

i n f o

a b s t r a c t In this paper we introduce and study some new subclasses of p-valent starlike, convex, close-to-convex and quasi-convex functions defined by Dziok–Srivastava operator. Inclusion relationships are established and integral operator of functions in these subclasses is discussed. Ó 2010 Elsevier Inc. All rights reserved.

Keywords: p-Valent Starlike Convex Close-to-convex Quasi-convex Dziok–Srivastava operator

1. Introduction Let AðpÞ denote the class of functions of the form:

f ðzÞ ¼ zp þ

1 X

apþk zpþk

ðp 2 N ¼ f1; 2; . . .gÞ;

ð1:1Þ

k¼1

which are analytic and p-valent in the unit disc U ¼ fz : jzj < 1g. For functions f 2 AðpÞ, given by (1.1), and g 2 AðpÞ given by

gðzÞ ¼ zp þ

1 X

bpþk zpþk

ðp 2 NÞ;

ð1:2Þ

k¼1

the Hadamard product (or convolution) of f and g is defined by

ðf  gÞðzÞ ¼ zp þ

1 X

apþk zpþk ¼ ðg  f ÞðzÞ ðp 2 NÞ:

ð1:3Þ

k¼1

A function f ðzÞ 2 AðpÞ is said to be in the class Sp ðcÞ of p-valently starlike of order c, if it satisfies

 0  zf ðzÞ Re > c ð0 6 c < p; z 2 UÞ: f ðzÞ

ð1:4Þ

We write Sp ð0Þ ¼ Sp , the class of p-valent starlike in U. Also a function f ðzÞ 2 AðpÞ is said to be in the class K p ðcÞ of p-valently convex of order c, if it satisfies

  00 zf ðzÞ > c ð0 6 c < p; z 2 UÞ: Re 1 þ 0 f ðzÞ The class of p-valently convex functions in U is denoted by K p ¼ K p ð0Þ. It follows from (1.4) and (1.5) that

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.034

ð1:5Þ

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M.K. Aouf / Applied Mathematics and Computation 216 (2010) 431–437 0

f ðzÞ 2 K p ðcÞ if and only if

zf ðzÞ 2 Sp ðcÞ ð0 6 c < p; z 2 UÞ: p

ð1:6Þ

The class Sp ðcÞ was introduced by Patil and Thakare [18] and the class K p ðcÞ was introduced by Owa [17]. Also we note that the classes Sp and K p were introduced by Goodman [6]. Furthermore, a function f ðzÞ 2 AðpÞ is said to be p-valently close-to-convex of order h and type c in U, if there exists a function gðzÞ 2 Sp ðcÞ such that

 0  zf ðzÞ Re > h ð0 6 h; c < p; z 2 UÞ: gðzÞ

ð1:7Þ

We denote by C p ðh; cÞ, the subclass of AðpÞ consisting of all such functions. The class C p ðh; cÞ was studied by Aouf [1]. We note that C 1 ðh; cÞ ¼ Cðh; cÞ, is the class of close-to-convex functions of order h and type cð0 6 h; c < 1Þ, was studied by Libera [9]. Also a function f ðzÞ 2 AðpÞ is called p-valently quasi-convex of order h and type c if there exists a function gðzÞ 2 K p ðcÞ such that

 0  ðzf ðzÞÞ0 > h ð0 6 h; c < p; z 2 UÞ: Re 0 g ðzÞ

ð1:8Þ

We denote this class by C p ðh; cÞ. We note that C 1 ðh; cÞ ¼ C  ðh; cÞ, is the class of quasi-convex of order h and type cð0 6 h; c < 1Þ, was introduced and studied by Noor [14,15]. It follows from (1.7) and (1.8) that 0

f ðzÞ 2 C p ðh; cÞ if and only if

zf ðzÞ 2 C p ðh; cÞ: p

ð1:9Þ

For complex parameters a1 ; . . . ; aq and b1 ; . . . ; bs ðbj R Z  0 ¼ f0; 1; 2; . . .g; j ¼ 1; . . . ; sÞ, we define the generalized hypergeometric function q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ by

a

qFsð 1; . . . ;

aq ; b1 ; . . . ; bs ; zÞ ¼

1 X ða1 Þk    ðaq Þk zk  ðb1 Þk    ðbs Þk k! k¼0

ð1:10Þ

ðq 6 s þ 1; q; s 2 N0 ¼ N [ f0g; z 2 UÞ; where ðhk Þ is the Pochhammer symbol defined, in terms the Gamma function C, by

ðhk Þ ¼

Cðh þ kÞ ¼ CðhÞ



1

ðk ¼ 0Þ;

hðh þ 1Þ . . . ðh þ k  1Þ ðk 2 NÞ:

ð1:11Þ

Corresponding to a function hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ defined by

hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ ¼ zp q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ; we consider a linear operator

Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ : AðpÞ ! AðpÞ; defined by the convolution

Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼ hp ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ  f ðzÞ:

ð1:12Þ

If, for convenience, we write

Hp;q;s ða1 Þ ¼ Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ;

ð1:13Þ

then one can easily verify from the definition (1.12) that

zðHp;q;s ða1 Þf ðzÞÞ0 ¼ a1 Hp;q;s ða1 þ 1Þf ðzÞ  ða1  pÞHp;q;s ða1 Þf ðzÞ:

ð1:14Þ

The linear operator hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ was introduced and studied by Dziok and Srivastava [5]. It should be remarked that the linear operator Hp;q;s ða1 Þ is a generalization of many other linear operators considered earlier. In particular, for f ðzÞ 2 AðpÞ we have: (i) Hp;2;1 ða; 1; cÞf ðzÞ ¼ Lp ða; cÞf ðzÞða > 0; c > 0Þ (Saitoh [19]); (ii) Hp;2;1 ðm þ p; 1; m þ p þ 1Þf ðzÞ ¼ J m;p ðf ÞðzÞ, where J m;p ðf ÞðzÞ is the generalization Bernardi–Libera–Livingston operator (see [2,10,12]) defined by

J m;p ðf ÞðzÞ ¼

mþp zm

Z

z

t m1 f ðtÞdt

ðm > p; p 2 NÞ;

ð1:15Þ

0

(iii) Hp;2;1 ðl þ p; 1; 1Þf ðzÞ ¼ Dlþp1 f ðzÞðl > pÞ, where Dlþp1 f ðzÞ is the ðl þ p  1Þ-th order Ruscheweyh derivative of a function f ðzÞ 2 AðpÞ (see Kumar and Shukla [7,8]);

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M.K. Aouf / Applied Mathematics and Computation 216 (2010) 431–437

(iv) Hp;2;1 ð1 þ p; 1; 1 þ p  lÞf ðzÞ ¼ Xzðl;pÞ f ðzÞ, where Xzðl;pÞ f ðzÞ is defined by (see Srivastava and Aouf [21])

Xðzl;pÞ f ðzÞ ¼

Cð1 þ p  lÞ l l z Dz f ðzÞ ð0 6 l < 1; p 2 NÞ; Cð1 þ pÞ

ð1:16Þ

where Xlz is the fractional derivative operator (see, for details [16]); (v) H1;2;1 ðl; 1; k þ 1Þ ¼ Ik;l f ðzÞðk > 1; l > 0Þ, where Ik;l f ðzÞ is the Choi–Saigo–Srivastava operator [4]; (vi) Hp;2;1 ðp þ 1; 1; n þ pÞf ðzÞ ¼ In;p f ðzÞðn 2 Z; n > pÞ, where the operator In;p f ðzÞ is considered by Liu and Noor [11]; k (vii) Hp;2;1 ðk þ p; c; aÞf ðzÞ ¼ Ikp ða; cÞf ðzÞða; c 2 R n Z  0 ; k > pÞ, where I p ða; cÞf ðzÞ is the Cho–Kwon–Srivastava operator [3]. Using the linear operator Hp;q;s ða1 Þ, we now introduce the following classes:

n o Sp;q;s ða1 ; cÞ ¼ f 2 AðpÞ : Hp;q;s ða1 Þf 2 Sp ðcÞ; 0 6 c < p ;

ð1:17Þ

  K p;q;s ða1 ; cÞ ¼ f 2 AðpÞ : Hp;q;s ða1 Þf 2 K p ðcÞ; 0 6 c < p ;   C p;q;s ða1 ; h; cÞ ¼ f 2 AðpÞ : Hp;q;s ða1 Þf 2 C p ðh; cÞ; 0 6 h; c < p ;

ð1:18Þ ð1:19Þ

and

n o C p;q;s ða1 ; h; cÞ ¼ f 2 AðpÞ : Hp;q;s ða1 Þf 2 C p ðh; cÞ; 0 6 h; c < p :

ð1:20Þ

In this paper, we shall establish the various inclusion relationships for these classes and investigate integral operator in these classes. 2. The main inclusion relationships In order to prove our main results, we shall require the following lemma. Lemma 1 [13]. Let uðu; v Þ be a complex-valued function such that

u : D ! C; ðD  C  CÞ; C being (as usual) the complex plane and let u ¼ u1 þ iu2 and v ¼ v 1 þ iv 2 . Suppose that the function uðu; v Þ satisfies each of the following conditions: (i) uðu; v Þ is continuous in D; (ii) ð1; 0Þ 2 D and Re fuð1; 0Þg > 0; (iii) Re fuðiu2 ; v 1 Þg 6 0 for all ðiu2 ; v 1 Þ 2 D such that

v 1 6  12 ð1 þ u22 Þ.

Let pðzÞ ¼ 1 þ p1 z þ p2 z2 þ    be regular in U, such that ðpðzÞ; zp0 ðzÞÞ 2 D for all z 2 U. If

Re fuðpðzÞ; zp0 ðzÞÞg > 0 ðz 2 UÞ; then Re fpðzÞg > 0ðz 2 UÞ. Theorem 1. Sp;q;s ða1 þ 1; cÞ  Sp;q;s ða1 ; cÞðf ðzÞ 2 AðpÞ; a1 þ c > p; 0 6 c < p; p 2 NÞ: Proof. Let f ðzÞ 2 Sp;q;s ða1 þ 1; cÞ and assume that

zðHp;q;s ða1 Þf ðzÞÞ0 ¼ c þ ðp  cÞhðzÞ; Hp;q;s ða1 Þf ðzÞ

ð2:1Þ

where hðzÞ ¼ 1 þ c1 z þ c2 z2 þ    Using the identity (1.14), we have

Hp;q;s ða1 þ 1Þf ðzÞ 1 ¼ ½ða1 þ c  pÞ þ ðp  cÞhðzÞ: Hp;q;s ða1 Þf ðzÞ a1

ð2:2Þ

Now, we logarithmically differentiate both sides of (2.2) with respect to z, and thus, we find that 0

zðHp;q;s ða1 þ 1Þf ðzÞÞ0 zðHp;q;s ða1 Þf ðzÞÞ0 ðp  cÞzh ðzÞ ¼ þ ; ða1 þ c  pÞ þ ðp  cÞhðzÞ Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ which, in view of (2.1), yields 0

zðHp;q;s ða1 þ 1Þf ðzÞÞ0 ðp  cÞzh ðzÞ  c ¼ ðp  cÞhðzÞ þ : ða1 þ c  pÞ þ ðp  cÞhðzÞ Hp;q;s ða1 þ 1Þf ðzÞ

ð2:3Þ

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M.K. Aouf / Applied Mathematics and Computation 216 (2010) 431–437

Let

uðu; v Þ ¼ ðp  cÞu þ

ðp  cÞv ða1 þ c  pÞ þ ðp  cÞu

ð2:4Þ

0

with hðzÞ ¼ u ¼ u1 þ iu2 and zh ðzÞ ¼ v ¼ v 1 þ iv 2 . Then cp gÞ  C; (i) uðu; v Þ is continuous in D ¼ ðC n fa1cþp (ii) ð1; 0Þ 2 D and Re fuð1; 0Þg ¼ p  c > 0; (iii) for all ðiu2 ; v 1 Þ 2 D such that v 1 6  12 ð1 þ u22 Þ,

Re fuðiu2 ; v 1 g ¼ Re



ðp  cÞv 1 ða1 þ c  pÞ þ ðp  cÞiu2

 ¼

ðp  cÞða1 þ c  pÞv 1 ða1 þ c  pÞ2 þ ðp  cÞ2 u22

<0

for m1 < 0. Therefore, the function uðu; v Þ satisfies the conditions of Lemma 1. Thus we have Re fhðzÞg > 0ðz 2 UÞ, that is, f ðzÞ 2 Sp;q;s ða1 ; cÞ. This completes the proof of Theorem 1. h

Theorem 2. K p;q;s ða1 þ 1; cÞ  K p;q;s ða1 ; cÞðf ðzÞ 2 AðpÞ; a1 þ c > p; 0 6 c < p; p 2 NÞ: Proof. f ðzÞ 2 K p;q;s ða1 þ 1; cÞ () Hp;q;s ða1 þ 1Þf ðzÞ 2 K p ðcÞ () pz ðHp;q;s ða1 Þf ðzÞÞ0 0 0 2 Sp ðcÞ () Hp;q;s ða1 þ 1Þðzf pðzÞÞ 2 Sp ðcÞ () zf pðzÞ 2 Sp;q;s ða1 þ 1; cÞ

)

 0  0 zf ðzÞ zf ðzÞ z 2 Sp ðcÞ () ðHp;q;s ða1 Þf ðzÞÞ0 2 Sp ðcÞ () Hp;q;s ða1 Þf ðzÞ 2 K p ðcÞ () f ðzÞ 2 Sp;q;s ða1 ; cÞ () Hp;q;s ða1 Þ p p p 2 K p;q;s ða1 ; cÞ;

which evidently proves Theorem 2. h Theorem 3. C p;q;s ða1 þ 1; h; cÞ  C p;q;s ða1 ; h; cÞðf ðzÞ 2 AðpÞ; a1 þ c > p; 0 6 h; c < p; p 2 NÞ. Proof. Let f ðzÞ 2 C p;q;s ða1 þ 1; h; cÞ. Then, by (1.19), there exists a function kðzÞ 2 Sp ðcÞð0 6 c < pÞ such that

Re

  zðHp;q;s ða1 þ 1Þf ðzÞÞ0 > h ðz 2 U; 0 6 h < pÞ: kðzÞ

ð2:5Þ

Taking

kðzÞ ¼ Hp;q;s ða1 þ 1ÞgðzÞ 2 Sp ðcÞð0 6 c < pÞ; we find from (1.17) and (2.5) that gðzÞ 2

ð2:6Þ

Sp;q;s ð 1

a þ 1; cÞ and

  zðHp;q;s ða1 þ 1Þf ðzÞÞ0 > h ðz 2 U; 0 6 h < pÞ; Re Hp;q;s ða1 þ 1ÞgðzÞ

ð2:7Þ

respectively. We now let

zðHp;q;s ða1 Þf ðzÞÞ0 ¼ h þ ðp  hÞhðzÞ; Hp;q;s ða1 ÞgðzÞ

ð2:8Þ

where hðzÞ ¼ 1 þ c1 z þ c2 z2 þ    From (2.8), we have

zðHp;q;s ða1 Þf ðzÞÞ0 ¼ ½h þ ðp  hÞhðzÞHp;q;s ða1 ÞgðzÞ

ð2:9Þ

so from (2.9) and the identity (1.14), we have

a1 zðHp;q;s ða1 Þf ðzÞÞ0 ¼ zðHp;q;s ða1 ÞgðzÞÞ0 ½h þ ðp  hÞhðzÞ þ Hp;q;s ða1 ÞgðzÞ  ðp  hÞzh0 ðzÞ þ ða1  pÞzðHp;q;s ða1 Þf ðzÞÞ0 :

ð2:10Þ

Now apply the identity (1.14) for the function gðzÞ and use (2.10) to get 0

zðHp;q;s ða1 þ 1Þf ðzÞÞ0 Hp;q;s ða1 ÞgðzÞ ðp  hÞzh ðzÞ ¼ ½h þ ðp  hÞhðzÞ þ  : Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 þ 1ÞgðzÞ a1 Since gðzÞ 2 Sp;q;s ða1 þ 1; cÞ and Sp;q;s ða1 þ 1; cÞ  Sp;q;s ða1 ; cÞ, we let

zðHp;q;s ða1 ÞgðzÞÞ0 ¼ c þ ðp  cÞHðzÞ; Hp;q;s ða1 ÞgðzÞ

ð2:11Þ

M.K. Aouf / Applied Mathematics and Computation 216 (2010) 431–437

435

where Re HðzÞ > 0ðz 2 UÞ. Thus (2.8) can be written as : 0

zðHp;q;s ða1 þ 1Þf ðzÞÞ0 ðp  hÞzh ðzÞ  h ¼ ðp  hÞhðzÞ þ : ða1 þ c  pÞ þ ðp  cÞHðzÞ Hp;q;s ða1 þ 1ÞgðzÞ Now we form the function Wðu; v Þ by taking u ¼ hðzÞ and

Wðu; v Þ ¼ ðp  hÞu þ

ð2:12Þ

v ¼ zh0 ðzÞ in (2.12) as:

ðp  hÞv : ða1 þ c  pÞ þ ðp  cÞHðzÞ

It is easy to see that the function uðu; v Þ satisfies the conditions (i) and (ii) of Lemma 1 in D ¼ C  C. To verify the condition (iii), we proceed as follows:

Re Wðiu2 ; v 1 Þ ¼

ðp  hÞv 1 ½a1 þ c  p þ ðp  cÞh1 ðx; yÞ 2

½a1 þ c  p þ ðp  cÞh1 ðx; yÞ þ ½ðp  cÞh2 ðx; yÞ

2

<0

for m1 < 0, where HðzÞ ¼ h1 ðx; yÞ þ ih2 ðx; yÞ; h1 ðx; yÞ and h2 ðx; yÞ being functions of x and y and Re HðzÞ ¼ h1 ðx; yÞ > 0. Hence Re hðzÞ > 0ðz 2 UÞ and f ðzÞ 2 C p;q;s ða1 ; h; cÞ. This completes the proof of Theorem 3. h Theorem 4. C p;q;s ða1 þ 1; h; cÞ  C p;q;s ða1 ; h; cÞðf ðzÞ 2 AðpÞ; a1 þ c > p; 0 6 h; c < p; p 2 NÞ: Proof. Just as we derived Theorem 2 as a consequence of Theorem 1 by means of the equivalence (1.6), we can prove Theorem 4 by appealing analogously to Theorem 3 and the equivalene (1.9). h

3. Integral operator For c > p and f ðzÞ 2 AðpÞ, the integral operator Jc;p f ðzÞ : AðpÞ ! AðpÞ is defined by

cþp J c;p f ðzÞ ¼ c z

Z

z

t

c1

f ðtÞdt ¼

p

z þ

0

1 X k¼1

c þ p pþk z cþpþk

!

 f ðzÞ ðc > p; z 2 UÞ:

ð3:1Þ

The operator J c;1 ðc 2 NÞ was introduced by Bernardi [2]. In particular, the operator J 1;1 was studied earlier by Libera [9] and Livingston [11]. Some results for the operator J c;p were showed by Saitoh [19] and Saitoh et al. [20]. Theorem 5. Let c > p; 0 6 c < p: If f ðzÞ 2 Sp;q;s ða1 ; cÞ; then J c;p f ðzÞ 2 Sp;q;s ða1 ; cÞ. Proof. Let f ðzÞ 2 Sp;q;s ða1 ; cÞ. From (3.1), we have

zðHp;q;s ða1 ÞJ c;p f ðzÞÞ0 ¼ ðc þ pÞHp;q;s ða1 Þf ðzÞ  cHp;q;s ða1 ÞJ c;p f ðzÞ:

ð3:2Þ

zðHp;q;s ða1 ÞJ c;p f ðzÞÞ0 ¼ c þ ðp  cÞhðzÞ; Hp;q;s ða1 ÞJ c;p f ðzÞ

ð3:3Þ

Put

where hðzÞ ¼ 1 þ c1 z þ c2 z2 þ   , using the identity (3.2), we have

Hp;q;s ða1 Þf ðzÞ 1 ¼ fc þ c þ ðp  cÞhðzÞg: Hp;q;s ða1 ÞJ c;p f ðzÞ c þ p

ð3:4Þ

Differentiating (3.4) logarithmically with respect to z, we obtain 0

zðHp;q;s ða1 Þf ðzÞÞ0 ðp  cÞzh ðzÞ  c ¼ ðp  cÞhðzÞ þ : c þ c þ ðp  cÞhðzÞ Hp;q;s ða1 Þf ðzÞ Now we form the function uðu; v Þ by taking u ¼ hðzÞ and

uðu; v Þ ¼ ðp  cÞu þ

ð3:5Þ

v ¼ zh0 ðzÞ in (3.5) as:

ðp  cÞv : c þ c þ ðp  cÞu

It is easy to see that the function uðu; v Þ satisfies the conditions (i) and (ii) of Lemma 1 in D ¼ ðC  fccþpcgÞ  C. To verify the condition (iii), we proceed as follows:

Re uðiu2 ; v 1 Þ ¼ Re



ðp  cÞv 1 c þ cðp  cÞiu2

 ¼

ðp  cÞðc þ cÞv 1 2

Þ2 u22

ðc þ cÞ þ ðp  c

ðp  cÞðc þ cÞð1 þ u22 Þ i < 0; 6 h 2 ðc þ cÞ2 þ ðp  cÞ2 u22

where v 1 6  12 ð1 þ u22 Þ and ðiu2 ; v 1 Þ 2 D. Therefore the function uðu; v Þ satisfies the conditions of Lemma 1. This shows that 0 if Re fuðhðzÞ; zh ðzÞÞg > 0 ðz 2 UÞ, then Re hðzÞ > 0 ðz 2 UÞ, that is, if f ðzÞ 2 Sp;q;s ða1 ; cÞ, then J c;p f ðzÞ 2 Sp;q;s ða1 ; cÞ. This completes the proof of Theorem 5. h

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M.K. Aouf / Applied Mathematics and Computation 216 (2010) 431–437

Theorem 6. Let c > p; 0 6 c < pÞ:If f ðzÞ 2 K p;q;s ða1 ; cÞ; then J c;p f ðzÞ 2 K p;q;s ða1 ; cÞ. 0

0

Proof. f ðzÞ 2 K p;q;s ða1 ; cÞ ) zf pðzÞ 2 Sp;q;s ða1 ; cÞ ) J c;p ðzf pðzÞÞ 2 Sp;q;s ða1 ; cÞ () pz ðJ c;p f ðzÞÞ0 2 Sp;q;s ða1 ; cÞ () J c;p f ðzÞ 2 K p;q;s ða1 ; cÞ. This completes the proof of Theorem 6. h Theorem 7. Let c > c; 0 6 c < p: If f ðzÞ 2 C p;q;s ða1 ; h; cÞ: Then J c;p f ðzÞ 2 C p;q;s ða1 ; h; cÞ. Proof. Let f ðzÞ 2 C p;q;s ða1 ; h; cÞ. Then by (1.19) there exists a function gðzÞ 2 Sp;q;s ða1 ; cÞ such that

Re

  zðHp;q;s ða1 Þf ðzÞÞ0 > h ðz 2 UÞ: Hp;q;s ða1 ÞgðzÞ

Put

zðHp;q;s ða1 ÞJ c;p f ðzÞÞ0 ¼ h þ ðp  hÞhðzÞ; Hp;q;s ða1 ÞJ c;p gðzÞ

ð3:6Þ

where hðzÞ ¼ 1 þ c1 z þ c2 z2 þ    From (3.2) and (3.6), we have 0

ðc þ pÞzðHp;q;s ða1 Þf ðzÞÞ0 ¼ zðHp;q;s ða1 ÞJ c;p gðzÞÞ0 ½h þ ðp  hÞhðzÞ þ Hp;q;s ða1 ÞJ c;p gðzÞ  ðp  hÞzh ðzÞ þ czðHp;q;s ða1 ÞJ c;p f ðzÞÞ0

ð3:7Þ

now apply (3.2) for the function gðzÞ and use (3.7), we obtain 0

Hp;q;s ða1 ÞJ c;p gðzÞ ðp  hÞzh ðzÞ zðHp;q;s ða1 Þf ðzÞÞ0 : ¼ h þ ðp  hÞhðzÞ þ  cþp Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 ÞgðzÞ

ð3:8Þ

Since gðzÞ 2 Sp;q;s ða1 ; cÞ, then from Theorem 5, we have J c;p gðzÞ 2 Sp;q;s ða1 ; cÞ, we let

zðHp;q;s ða1 ÞJ c;p gðzÞÞ0 ¼ c þ ðp  cÞHðzÞ; Hp;q;s ða1 ÞJ c;p gðzÞ where Re HðzÞ > 0 ðz 2 UÞ. Thus (3.8) can be written as 0

zðHp;q;s ða1 Þf ðzÞÞ0 ðp  hÞzh ðzÞ  h ¼ ðp  hÞhðzÞ þ : c þ c þ ðp  cÞHðzÞ Hp;q;s ða1 ÞgðzÞ Now we form the function uðu; v Þ by taking u ¼ hðzÞ and

uðu; v Þ ¼ ðp  hÞu þ

ð3:9Þ

v ¼ zh0 ðzÞ in (3.9) as:

ðp  hÞv : c þ c þ ðp  cÞHðzÞ

It is easy to see that the function uðu; v Þ satisfies the conditions (i) and (ii) of Lemma 1, in D ¼ C  C. To verify the condition (iii), we proceed as follows:

Re uðiu2 ; v 1 Þ ¼

ðp  hÞv 1 ½c þ c þ ðp  cÞh1 ðx; yÞ 2

½c þ c þ ðp  cÞh1 ðx; yÞ þ ½ðp  cÞh2 ðx; yÞ

2

;

where HðzÞ ¼ h1 ðx; yÞ þ ih2 ðx; yÞ; h1 ðx; yÞ and h2 ðx; yÞ being functions of x and y and Re HðzÞ ¼ h1 ðx; yÞ > 0. By putting v 1 6  12 ð1 þ u22 Þ, we have

ðp  hÞð1 þ u22 Þ½c þ c þ ðp  cÞh1 ðx; yÞ o < 0: Re uðiu2 ; v 1 Þ 6  n 2 2 2 ½c þ c þ ðp  cÞh1 ðx; yÞ þ ½ðp  cÞh2 ðx; yÞ Hence Re hðzÞ > 0 ðz 2 UÞ and J c;p f ðzÞ 2 C p;q;s ða1 ; h; cÞ. This completes the proof of Theorem 7. h Similarly we can prove Theorem 8. Let c > c; 0 6 c < p: Iff ðzÞ 2 C p;q;s ða1 ; h; cÞ: Then J c;p f ðzÞ 2 C p;q;s ða1 ; h; cÞ. Acknowledgements The author thank the referees for their valuable suggestions to improve the paper.

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