Some properties of certain subclasses of meromorphically multivalent functions

Applied Mathematics and Computation 204 (2008) 824–832

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Some properties of certain subclasses of meromorphically multivalent functions R.M. El-Ashwah Department of Mathematics, 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, the author investigates interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of certain linear operator. Ó 2008 Elsevier Inc. All rights reserved.

Keywords: Analytic functions Meromorphic functions Linear operator

1. Introduction Let

P

p

be the class of functions of the form:

f ðzÞ ¼ zp þ

1 X

akp zkp

ðp 2 N ¼ f1; 2; . . . :gÞ;

ð1:1Þ

k¼1

which are analytic and p-valent in the punctured unit disc U  ¼ fz : z 2 C and 0 < jzj < 1g ¼ U n f0g. For functions f ðzÞ 2 P given by (1.1) and gðzÞ 2 p given by

gðzÞ ¼ zp þ

1 X

bkp zkp

ðp 2 NÞ;

P

p

ð1:2Þ

k¼1

then the Hadamard product (or convolution) of f ðzÞ and gðzÞ is given by

ðf  gÞðzÞ ¼ zp þ

1 X

akp bkp zkp ¼ ðg  f ÞðzÞ:

ð1:3Þ

k¼1

For complex numbers

a1 ; . . . ; aq and b1 ; . . . ; bs ðbj R Z 0 ¼ f0; 1; 2; . . .g; j ¼ 1; 2; . . . ; sÞ; we now define the generalized hypergeometric function q F s ða1 ; . . . ; aq ; b1 ; . . . ; bs ; zÞ by (see, for example [11, p. 19])

a

qFsð 1; . . . ;

aq ; b1 ; . . . ; bs ; zÞ ¼

1 X ða1 Þk . . . ðaq Þk zk : ðb1 Þk . . . ðbs Þk k! k¼1

ðq 6 s þ 1; q; s 2 N0 ¼ N [ f0g; z 2 UÞ;

ð1:4Þ

where ðhÞm is the Pochhammer symbol defined, in terms of the Gamma function C, by

ðhÞm ¼

Cðh þ mÞ ¼ CðhÞ



1

ðm ¼ 0; h 2 C n f0gÞ

hðh þ 1Þ . . . ðh þ m  1Þ ðm 2 N; h 2 CÞ:

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.07.032

ð1:5Þ

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R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

Corresponding to the 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;

ð1:6Þ

we consider a linear operator

Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þ :

X p

!

X ; p

which is defined by the following Hadamard product (or convolution):

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

ð1:7Þ

We observe that, for a function f ðzÞ of the form (1.1), we have

Hp ða1 ; . . . ; aq ; b1 ; . . . ; bs Þf ðzÞ ¼ zp þ

1 X ða1 Þk . . . ðaq Þk akp kp : z : ðb1 Þk . . . ðbs Þk k! k¼1

ð1:8Þ

If, for convenience, we write

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

ð1:9Þ

then one can easily verify from the definition (1.8) that

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

ð1:10Þ

The linear operator Hp;q;s ða1 Þ was investigated recently by Liu and Srivastava [7] and Aouf [1]. In particular, for q ¼ 2; s ¼ 1 and a2 ¼ 1, we obtain the linear operator

Hp;2;1 ða1 ; 1; b1 Þf ðzÞ ¼ ‘p ða1 ; b1 Þf ðzÞ

f ðzÞ 2

! X

;

p

which was introduced and studied by Liu and Srivastava [8]. Also we note that, for any integer n > p and f ðzÞ 2

Hp;2;1 ðn þ p; 1; 1Þf ðzÞ ¼ Dnþp1 f ðzÞ ¼

P

p,

we have

1  f ðzÞ; zp ð1  zÞnþp

where Dnþp1 f ðzÞ is the differential operator studied by Uralegaddi and Somanatha [12]. Some interesting subclasses of analytic functions associated with the generalized hypergeometric function, were considered recently by (for example) Dziok and Srivastava [2,3], Gangadharan et al. [4] and Liu [6]. P P1 ða; d; l; kÞ the class of all functions f ðzÞ 2 p such that We denote by ap;q;s

(



Hp;q;s ða1 Þf ðzÞ Re ð1  kÞ Hp;q;s ða1 ÞgðzÞ where gðzÞ 2

P

p

l

 l1 ) Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ > a; þk Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ

ð1:11Þ

satisfies the following condition:

  Hp;q;s ða1 ÞgðzÞ > d ð0 6 d < 1; z 2 UÞ; Re Hp;q;s ða1 þ 1ÞgðzÞ

ð1:12Þ

where a and l are real numbers such that 0 6 a < 1; l > 0 and k 2 C with Refkg > 0. We note that: (i) For q ¼ 2; s ¼ 1; a1 ¼ a > 0; b1 ¼ c > 0 and a2 ¼ 1, we have the following condition:

(



Lp ða; cÞf ðzÞ Re ð1  kÞ Lp ða; cÞgðzÞ where gðzÞ 2

Re



P

p

l

P

a;c;p ð

a; d; l; kÞ, is the class of functions f ðzÞ 2

 l1 ) Lp ða þ 1; cÞf ðzÞ Lp ða; cÞf ðzÞ > a; þk Lp ða þ 1; cÞgðzÞ Lp ða; cÞgðzÞ

P

p

satisfying

ð1:13Þ

satisfies the following condition:

Lp ða; cÞgðzÞ Lp ða þ 1; cÞgðzÞ



> d ð0 6 d < 1; z 2 UÞ;

ð1:14Þ

where 0 6 a < 1; l > 0, and k 2 C, with Refkg > 0. P (ii) For q ¼ 2; s ¼ 1; a1 ¼ n þ pðn > p; p 2 NÞ; b1 ¼ 1, and a2 ¼ 1;, we have n;p ða; d; l; kÞ, is the class of functions P f ðzÞ 2 p satisfying the following condition:

8 !l !l1 9 < = Dnþp1 f ðzÞ Dnþp f ðzÞ Dnþp1 f ðzÞ > a; Re ð1  kÞ nþp1 þ k nþp nþp1 : ; D gðzÞ D gðzÞ gðzÞ D

ð1:15Þ

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R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

where gðzÞ 2

P

p

satisfies the following condition:

( ) Dnþp1 gðzÞ Re > d ð0 6 d < 1; z 2 UÞ; Dnþp gðzÞ where 0 6 d < 1; l > 0, and k 2 C, with Refkg > 0. The class

ð1:16Þ P

n;p ð

a; d; l; kÞ was studied by Kwon et al. [5].

To establish our main results we need the following lemmas. Lemma 1. [9] Let X be a set in the complex plane C and let the function W : C 2 ! C satisfy the condition Wðir 2 ; s1 Þ R X for all real 1þr 2 r2 ; s1 6  2 2 . If qðzÞ is analytic in U with qð0Þ ¼ 1 and WðqðzÞ; zq0 ðzÞÞ 2 X; z 2 U, then RefqðzÞg > 0ðz 2 UÞ. Lemma 2. [10] If qðzÞ is analytic in U with qð0Þ ¼ 1, and if k 2 C n f0g with Refkg P 0, then RefqðzÞ þ kzq0 ðzÞg > a ð0 6 a < 1Þ implies RefqðzÞg > a þ ð1  aÞð2c  1Þ, where c is given by

c ¼ cðRekÞ ¼

Z

1

ð1 þ tRefkg Þ1 dt;

0

which is increasing function of Refkg and

1 2

6 c < 1. The estimate is sharp in the sense that the bound cannot be improved.

For real or complex numbers a; b; cðc R Z  0 Þ, the Gauss hypergeometric function is defined by 2 F 1 ða; b; c; zÞ

¼1þ

ab z aða þ 1Þbðb þ 1Þ z2 þ þ  c 1! cðc þ 1Þ 2!

We note that the above series converges absolutely for z 2 U and hence represents an analytic function in the unit disc U (see, for details [13, Chapter 14]). Each of the identities (asserted by Lemma 3) is fairly well known (cf., e.g. [13, Chapter 14]). Lemma 3. For real or complex parameters a; b; cðc R Z  0 Þ,

Z

1

tb1 ð1  tÞcb1 ð1  tzÞa dt ¼

0

CðbÞCðc  bÞ ðReðcÞ > ReðbÞ > 0Þ; 2 F 1 ða; b; c; zÞ CðcÞ

2 F 1 ða; b; c; zÞ

¼ 2 F 1 ðb; a; c; zÞ;  a 2 F 1 ða; b; c; zÞ ¼ ð1  zÞ 2 F 1 a; c  b; c;

z  ; z1

ð1:17Þ ð1:18Þ ð1:19Þ

and 2F1

  1 1; 1; 2; ¼ 2‘n2: 2

ð1:20Þ

2. Main results Theorem 1. Let f ðzÞ 2

Pa1

a; d; l; kÞ; a1 2 R n f0g and k P 0. Then

p;q;s ð

 l Hp;q;s ða1 Þf ðzÞ 2a1 al þ dk Re > Hp;q;s ða1 ÞgðzÞ 2a1 l þ dk where the function gðzÞ 2

P

p

ð0 6 a < 1; l > 0; z 2 UÞ;

ð2:1Þ

satisfies the condition (1.12).

þdk , and we define the function qðzÞ by Proof. Let c ¼ 22aa11allþdk

qðzÞ ¼

1 ð1  cÞ



Hp;q;s ða1 Þf ðzÞ Hp;q;s ða1 ÞgðzÞ

l

 c :

ð2:2Þ

Then qðzÞ is analytic in U and qð0Þ ¼ 1. If we set

hðzÞ ¼

Hp;q;s ða1 ÞgðzÞ ; Hp;q;s ða1 þ 1ÞgðzÞ

ð2:3Þ

then by the hypothesis, RefhðzÞg > d. Differentiating (2.2) and using the identity (1.10), we have

 ð1  kÞ

Hp;q;s ða1 Þf ðzÞ Hp;q;s ða1 ÞgðzÞ

l þk

 l1 Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ kð1  cÞ ¼ ½c þ ð1  cÞqðzÞ þ hðzÞzq0 ðzÞ: Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ a1 l

Let us define the function Wðr; sÞ by

ð2:4Þ

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R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

kð1  cÞ

Wðr; sÞ ¼ c þ ð1  cÞr þ

la1

Using (2.5) and the fact that f ðzÞ 2

hðzÞs:

Pa1

p;q;s ð

ð2:5Þ

a; d; l; kÞ, we obtain

0

fWðqðzÞ; zq ðzÞÞ; z 2 Ug  X ¼ fw 2 C : ReðwÞ > ag: Now for all real r2 ; s1 6 

1þr 22 , 2

RefWðir 2 ; s1 Þg ¼ c þ

we have

kð1  cÞs1

la1

RefhðzÞg 6 c 

kð1  cÞdð1 þ r 22 Þ kð1  cÞd 6c ¼ a: 2la1 2la1

Hence for each z 2 U; Wðir2 ; s1 Þ R XÞ, Thus by Lemma 1, we have RefqðzÞg > 0ðz 2 UÞ and hence

 Re

l Hp;q;s ða1 Þf ðzÞ > c ðz 2 UÞ: Hp;q;s ða1 ÞgðzÞ

This proves Theorem 1. h Corollary 1. Let the functions f ðzÞ and gðzÞ be in

then

P

p

and let gðzÞ satisfy the condition (1.12). If a1 2 R n f0g; k P 1 and

  Hp;q;s ða1 Þf ðzÞ Hp;q;s ða1 þ 1Þf ðzÞ Re ð1  kÞ þk > a ð0 6 a < 1; z 2 UÞ; Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 þ 1ÞgðzÞ

ð2:6Þ

  Hp;q;s ða1 þ 1Þf ðzÞ að2a1 þ dÞ þ dðk  1Þ Re >c¼ Hp;q;s ða1 þ 1ÞgðzÞ 2a1 þ kd

ð2:7Þ

ðz 2 UÞ:

Proof. We have

k

 Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ þ ðk  1Þ ¼ ð1  kÞ þk Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ

Since k P 1, making use of (2.6) and (2.1) (for

ðz 2 UÞ:

l ¼ 1), we deduce that

  Hp;q;s ða1 þ 1Þf ðzÞ að2a1 þ dÞ þ dðk  1Þ Re >c¼ : Hp;q;s ða1 þ 1ÞgðzÞ 2a1 þ kd



Corollary 2. Let k 2 C n f0g with Refkg P 0 and a1 2 R n f0g. If f ðzÞ 2

P

p

satisfies the following condition:

Refð1  kÞðzp Hp;q;s ða1 Þf ðzÞÞl þ kzp Hp;q;s ða1 þ 1Þf ðzÞ  ðzp Hp;q;s ða1 Þf ðzÞÞl1 g > að0 6 a < 1; l > 0; p 2 N; z 2 UÞ; then

Refzp Hp;q;s ða1 Þf ðzÞgl >

2la1 a þ Refkg 2la1 þ Refkg

Further, if k P 1; a1 2 R n f0g and f ðzÞ 2

P

p

ðz 2 UÞ:

ð2:8Þ

satisfies

Refð1  kÞzp Hp;q;s ða1 Þf ðzÞ þ kzp Hp;q;s ða1 þ 1Þf ðzÞg > a ðz 2 UÞ; then

Refzp Hp;q;s ða1 þ 1Þf ðzÞg >

ð2a1 þ 1Þa þ k  1 2a1 þ k

ð0 6 a < 1; p 2 N; z 2 UÞ:

ð2:9Þ

Proof. The results (2.8) and (2.9) follows by putting gðzÞ ¼ z1p in Theorem 1 and Corollary 1, respectively. h Remark 1. Choosing q; s; a1 ; a2 ; b1 ; l and k appropriately in Corollary 2, we obtain the following results.

(i) For k ¼ 1; q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ and bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ in Corollary 2, we have

Re

   0 zf ðzÞ 2þ ðzp f ðzÞÞl > a ð0 6 a < 1; l > 0; z 2 UÞ pf ðzÞ

implies

Refzp f ðzÞgl >

2lap þ 1 2l p þ 1

ðz 2 UÞ:

ð2:10Þ

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R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

(ii) For k 2 C n f0g with Refkg P 0; Corollary 2, we have

l ¼ 1; q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ and bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ in

  k Re ð1 þ kÞzp f ðzÞ þ zpþ1 f 0 ðzÞ > a ð0 6 a < 1; l > 0; z 2 UÞ p implies

Refzp f ðzÞg >

2ap þ Refkg 2p þ Refkg

ðz 2 UÞ:

0

(iii) Replacing f(z) by  zf pðzÞ in the result (ii), we have

Re

   k zpþ1 f 0 ðzÞ k pþ2 00 1þkþ þ 2 z f ðzÞ > a ð0 6 a < 1; z 2 UÞ p p p

implies

Re

 pþ1  z 2ap þ Refkg f 0 ðzÞ > 2p þ Refkg p

(iv) For k 2 R with k P 1; 2, we have

ðz 2 UÞ:

l ¼ 1; q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ and bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ in Corollary

  k Re ð1 þ kÞzp f ðzÞ þ zpþ1 f 0 ðzÞ > a ð0 6 a < 1; z 2 UÞ p implies

Refzp f ðzÞg >

að2p þ 1Þ þ k  1 2p þ k

ðz 2 UÞ:

(v) Putting k ¼ 1; q ¼ 2; s ¼ 1; a1 ¼ a > 0; b1 ¼ c > 0 and a2 ¼ 1, in Corollary 2, we have

Refzp Lp ða þ 1; cÞf ðzÞðzp Lp ða; cÞf ðzÞÞl1 g > a ð0 6 a < 1; l > 0; p 2 N; z 2 UÞ implies

Refzp Lp ða; cÞf ðzÞgl >

2laa þ 1 2la þ 1

(vi) For k 2 C n f0g with Refkg P 0;

ðz 2 UÞ:

l ¼ 1; q ¼ 2; s ¼ 1; a1 ¼ a > 0; b1 ¼ c > 0 and a2 ¼ 1 in Corollary 2, we have

Refð1  kÞzp Lp ða; cÞf ðzÞ þ kzp Lp ða þ 1; cÞf ðzÞg > a ð0 6 a < 1; p 2 N; z 2 UÞ implies

Refzp Lp ða; cÞf ðzÞg >

2aa þ Refkg 2a þ Refkg

ðz 2 UÞ:

Theorem 2. Let k 2 C with Refkg > 0 and a1 2 R n f0g. Let f ðzÞ 2

P

p

satisfy the following condition:

l

Refð1  kÞðzp Hp;q;s ða1 Þf ðzÞÞ þ kzp Hp;q;s ða1 þ 1Þf ðzÞ  ðzp Hp;q;s ða1 Þf ðzÞÞl1 g > a

ð2:11Þ

ð0 6 a < 1; l > 0; p 2 N; z 2 UÞ: Then

Refzp Hp;q;s ða1 Þf ðzÞgl > a þ ð1  aÞð2q  1Þ;

ð2:12Þ

where

q¼

  1 la1 1 þ 1; : 2 F 1 1; 1; 2 2 Refkg

ð2:13Þ

Proof. Let

qðzÞ ¼ ðzp Hp;q;s ða1 Þf ðzÞÞl :

ð2:14Þ

Then qðzÞ is analytic in U with qð0Þ ¼ 1. Differentiating qðzÞ with respect to z and using the identity (1.10), we obtain

ð1  kÞðzp Hp;q;s ða1 Þf ðzÞÞl þ kzp Hp;q;s ða1 þ 1Þf ðzÞðzp Hp;q;s ða1 Þf ðzÞÞl1 ¼ qðzÞ þ

kzq0 ðzÞ

la1

;

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R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

so that by the hypothesis (2.11), we have

  kzq0 ðzÞ Re qðzÞ þ > a ðz 2 UÞ:

la1

In view Lemma 2, this implies that

RefqðzÞg > a þ ð1  aÞð2q  1Þ; where

q ¼ qðRefkgÞ ¼

Z

1

 Refkg 1 1 þ t la1 dt:

0

Putting Refkg ¼ k1 > 0, we have

q¼

Z 0

1

 1 Z k1 la1 1 lka11 1 1 þ t la1 dt ¼ u ð1 þ uÞ1 du: k1 0

Using (1.17)–(1.20), we obtain



q ¼ 2 F 1 1;

la1 la1 k1

;

k1

   1 la1 1 þ 1; 1 ¼ 2 F 1 1; 1; þ 1; : 2 2 k1

Thus the proof of Theorem 2 is complete. h Corollary 3. Let k 2 R with k P 1. If f ðzÞ 2

P

p

satisfies

Refð1  kÞzp Hp;q;s ða1 Þf ðzÞ þ kzp Hp;q;s ða1 þ 1Þf ðzÞg > a ð0 6 a < 1; a1 2 R n f0g;p 2 N; z 2 UÞ;

ð2:15Þ

then

Refzp Hp;q;s ða1þ1 Þf ðzÞg > a þ ð1  aÞð2q  1Þð1  k1 Þ ðz 2 UÞ; where

q ¼

  1 a1 1 þ 1; : 2 F 1 1; 1; 2 2 k

Proof. The result follows by using the identity

 kzp Hp;q;s ða1 þ 1Þf ðzÞ ¼ ð1  kÞzp Hp;q;s ða1 Þf ðzÞ þ kzp Hp;q;s ða1 þ 1Þf ðzÞ þ ðk  1Þzp Hp;q;s ða1 Þf ðzÞ:



ð2:16Þ

Remark 2. (i) We note that if k ¼ l > 0; q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1ðj ¼ 2; 3; . . . ; s þ 1Þ, and bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ in Corollary 2, that is,

  kzpþ1 0 Re ð1 þ kÞðzp f ðzÞÞk þ f ðzÞðzp f ðzÞÞk1 > a ð0 6 a < 1; z 2 UÞ; p

ð2:17Þ

then (2.8) implies that

Refzp f ðzÞgk >

2ap þ 1 2p þ 1

whereas if f ðzÞ 2 p

P

p

ðz 2 UÞ;

satisfies the condition (2.17) then by using Theorem 2, we have

k

Refz f ðzÞg > 2ð1  ‘n2Þa þ ð2‘n2  1Þ ðz 2 UÞ; which is better than (2.18). (ii) Similarly, if q ¼ s þ 1; a1 ¼ 2; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 1; 2; . . . ; sÞ and k ¼ 2, in (2.9) and

Ref2zp Hp;sþ1;s ð3Þf ðzÞ  zp Hp;sþ1;s ð2Þf ðzÞg > a ð0 6 a < 1; z 2 UÞ; then we have

Refzp Hp;sþ1;s ð3Þf ðzÞg >

5a þ 1 6

ðz 2 UÞ;

ð2:18Þ

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R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

whereas by using (1.20), Corollary 3 implies that

Refzp Hp;sþ1;s ð3Þf ðzÞg >

1 5a þ 1 ½að3  2‘n2Þ þ ð2‘n2  1Þ > 2 6

ðz 2 UÞ:

(iii) We observe that if k 2 R, satisfying k > 0 and

kðzÞ ¼

  Hp;q;s ða1 þ 1Þf ðzÞ 1 Hp;q;s ða1 Þf ðzÞ 1 þ Hp;q;s ða1 þ 1ÞgðzÞ k Hp;q;s ða1 ÞgðzÞ

then from Theorem 1 (for

RefkðzÞg >

ðz 2 UÞ;

l ¼ 1), we have

a k

implies

  Hp;q;s ða1 Þf ðzÞ 2a1 a þ kd > Re Hp;q;s ða1 ÞgðzÞ 2a1 þ kd

ð2:19Þ

whenever

  Hp;q;s ða1 ÞgðzÞ Re > d ð0 6 d < 1; z 2 UÞ: Hp;q;s ða1 þ 1ÞgðzÞ Let k ! þ1, then from (2.19), we have

 RefkðzÞg P 0 ðz 2 UÞ implies Re whenever Re

n

Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 þ1ÞgðzÞ

o

Hp;q;s ða1 Þf ðzÞ Hp;q;s ða1 ÞgðzÞ

 P 1 ðz 2 UÞ;

> dð0 6 d < 1; z 2 UÞ.

In the following theorem, we shall extend the above results as follows. P Theorem 3. Suppose the functions f ðzÞ and gðzÞ are in p , and suppose gðzÞ, satisfies the condition (1.12). If

  Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ ð1  aÞd  > Re Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ 2a1

then

ð0 6 a < 1; 0 6 d < 1; a1 2 R n f0g; z 2 UÞ;

  Hp;q;s ða1 Þf ðzÞ Re > a ðz 2 UÞ Hp;q;s ða1 ÞgðzÞ

ð2:20Þ

ð2:21Þ

and

  Hp;q;s ða1 þ 1Þf ðzÞ ð2a1 þ dÞa  d > Re Hp;q;s ða1 þ 1ÞgðzÞ 2a1

ð0 6 a < 1; 0 6 d < 1; a1 2 R n f0g; z 2 UÞ:

ð2:22Þ

Proof. Let

qðzÞ ¼

  1 Hp;q;s ða1 Þf ðzÞ a : ð1  aÞ Hp;q;s ða1 ÞgðzÞ

ð2:23Þ

Then qðzÞ is analytic in U with qð0Þ ¼ 1. Putting

/ðzÞ ¼

Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 þ 1ÞgðzÞ

ðz 2 UÞ

we observe that by hypothesis, Ref/ðzÞg > dð0 6 d < 1Þ in U. A simple computation shows that

ð1  aÞzq0 ðzÞ/ðzÞ

a1

¼

Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ  ¼ WðqðzÞ; zq0 ðzÞÞ; Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ

where

Wðr; sÞ ¼

ð1  aÞ/ðzÞs

a1

ða1 2 R n f0gÞ:

Using the hypothesis (2.20), we obtain

fWðqðzÞ; zq0 ðzÞ; z 2 Ug  X ¼

  ð1  aÞd w 2 C : Rew >  : 2a1

ð2:24Þ

R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

Now, for all real r2 ; s1 6 

RefWðir 2 ; s1 Þg ¼

ð1þr 22 Þ , 2

831

we have

s1 ð1  aÞRef/ðzÞg

a1

6

ð1  aÞdð1 þ r 22 Þ ð1  aÞd 6 : 2a1 2a1

This shows that Wðir2 ; s1 Þ R X for each z 2 U. Hence by Lemma 1, we have RefqðzÞg > 0ðz 2 UÞ. This proves (2.21). The proof of (2.22) follows by using (2.21) and (2.22) in the identity:

      Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 þ 1Þf ðzÞ Hp;q;s ða1 Þf ðzÞ Hp;q;s ða1 Þf ðzÞ Re ¼ Re  þ Re : Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 þ 1ÞgðzÞ Hp;q;s ða1 ÞgðzÞ Hp;q;s ða1 ÞgðzÞ This completes the proof of Theorem 3. h Putting q ¼ 2; s ¼ 1; a1 ¼ a > 0; b1 ¼ c > 0 and a2 ¼ 1 in Theorem 3, we obtain P Corollary 4. Suppose the functions f ðzÞ and gðzÞ are in p and suppose gðzÞ satisfies



Lp ða; cÞgðzÞ Lp ða þ 1; cÞgðzÞ

Re



> d ð0 6 d < 1; z 2 UÞ:

ð2:25Þ

If

Re

  Lp ða þ 1; cÞgðzÞ Lp ða; cÞf ðzÞ ð1  aÞd  > Lp ða þ 1; cÞgðzÞ Lp ða; cÞgðzÞ 2a

ð0 6 a < 1; 0 6 d < 1; z 2 UÞ;

ð2:26Þ

then



Lp ða; cÞf ðzÞ Lp ða; cÞgðzÞ

Re



> a ðz 2 UÞ

ð2:27Þ

and

Re



Lp ða þ 1; cÞf ðzÞ Lp ða þ 1; cÞgðzÞ

 >

ð2a þ dÞa  d 2a

ð0 6 a < 1; 0 6 d < 1; z 2 UÞ:

ð2:28Þ

Remark 3 (i) Putting q ¼ s þ 1; a1 ¼ b1 ¼ p; aj ¼ 1 ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ and gðzÞ ¼ z1p in Theorem 3, we obtain:

Refzp f ðzÞ þ

zpþ1 0 ð1  aÞd f ðzÞg >  2p p

ðz 2 UÞ

implies

Refzp f ðzÞg > a ðz 2 UÞ and

  zpþ1 0 ð2p þ dÞa  d Re 2zp f ðzÞ þ f ðzÞ > 2p p

ðz 2 UÞ:

(ii) For q ¼ s þ 1; a1 ¼ p þ 1; b1 ¼ p; aj ¼ 1ðj ¼ 2; 3; . . . ; s þ 1Þ; bj ¼ 1 ðj ¼ 2; 3; . . . ; sÞ and gðzÞ ¼ z1p in Theorem 3, we have:

Refzp Hp;sþ1;s ðp þ 2Þf ðzÞ  zp Hp;sþ1;s ðp þ 1Þf ðzÞg > 

ð1  aÞd 2ðp þ 1Þ

ð0 6 a < 1; 0 6 d < 1; z 2 UÞ implies

Refzp Hp;sþ1;s ðp þ 1Þf ðzÞg > a ðz 2 UÞ and

Refzp Hp;sþ1;s ðp þ 2Þf ðzÞg >

ð2ðp þ 1Þ þ dÞa  d 2ðp þ 1Þ

ðz 2 UÞ:

Acknowledgements The author would like to thank the referees of the paper for their valuable suggestions.

832

R.M. El-Ashwah / Applied Mathematics and Computation 204 (2008) 824–832

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