Univalence of a New General Integral Operator Associated with the Hypergeometric Function

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Volume 2013, Article ID 769537,5pages http://dx.doi.org/10.1155/2013/769537

Research Article

Univalence of a New General Integral Operator Associated with

the

𝑞

-Hypergeometric Function

Huda Aldweby and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Correspondence should be addressed to Maslina Darus; [email protected]

Received 11 December 2012; Revised 3 February 2013; Accepted 17 February 2013

Academic Editor: Shyam Kalla

Copyright © 2013 H. Aldweby and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Motivated by the familiar𝑞-hypergeometric functions, we introduce a new family of integral operators and obtain new sufficient

conditions of univalence criteria. Several corollaries and consequences of the main results are also pointed out.

1. Introduction

LetAdenote the class of functions of the form

𝑓 (𝑧) = 𝑧 +∑∞

𝑛=2

𝑐_𝑛𝑧𝑛, 𝑐_𝑛≥ 0, (1)

which are analytic in the open unit diskU= {𝑧 ∈C: |𝑧| < 1}, andSthe class of functions𝑓 ∈Awhich are univalent inU. Let𝑓, 𝑔 ∈A, where𝑓is defined by (1) and𝑔is given by

𝑔 (𝑧) = 𝑧 +∑∞

𝑛=2

𝑏_𝑛𝑧𝑛, 𝑏_𝑛≥ 0. (2)

Then the Hadamard product (or convolution)𝑓 ∗ 𝑔of the functions𝑓and𝑔is defined by

(𝑓 ∗ 𝑔) (𝑧) = 𝑧 +∑∞

𝑛=2

𝑐_𝑛𝑏_𝑛𝑧𝑛. (3)

For complex parameters𝑎_𝑖, 𝑏_𝑗, and𝑞 (𝑖 = 1, . . . , 𝑟, 𝑗 =

1, . . . , 𝑠, 𝑏_𝑗 ∈ C\ {0, −1, −2, . . .}, |𝑞| < 1), we define the𝑞 -hypergeometric function _𝑟Φ_𝑠(𝑎₁, . . . , 𝑎_𝑟; 𝑏₁, . . . , 𝑏_𝑠; 𝑞, 𝑧)by

𝑟Φ𝑠(𝑎𝑖; 𝑏𝑗; 𝑞, 𝑧) = ∞

∑

𝑛=0

(𝑎₁, 𝑞)_𝑛⋅ ⋅ ⋅ (𝑎_𝑟, 𝑞)_𝑛

(𝑞, 𝑞)_𝑛(𝑏₁, 𝑞)_𝑛⋅ ⋅ ⋅ (𝑏_𝑠, 𝑞)_𝑛𝑧𝑛 (4)

(𝑟 = 𝑠 + 1; 𝑟, 𝑠 ∈ N₀ = N∪ {0}; 𝑧 ∈ U), whereNdenotes the set of positive integers and(𝑎, 𝑞)_𝑛is the𝑞-shifted factorial defined by

(𝑎, 𝑞)_𝑛={1,_{(1 − 𝑎) (1 − 𝑎𝑞) (1 − 𝑎𝑞}₂_{) ⋅ ⋅ ⋅ (1 − 𝑎𝑞}_𝑛−1_{) , 𝑛 ∈}𝑛 = 0;_N_.

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By using the ratio test, we should note that, if|𝑞| < 1, the series (4) converges absolutely for|𝑧| < 1if𝑟 = 𝑠 + 1. For more mathematical background of these functions, one may refer to [1].

Corresponding to the function defined by (4), consider

𝑟G𝑠(𝑎𝑖; 𝑏𝑗; 𝑞, 𝑧) = 𝑧 𝑟Φ𝑠(𝑎𝑖; 𝑏𝑗; 𝑞, 𝑧) . (6)

Recently, the authors [2] defined the linear operator

M(𝑎𝑖, 𝑏𝑗; 𝑞)𝑓 :A → Aby

M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧) = 𝑟G𝑠(𝑎𝑖; 𝑏𝑗; 𝑞, 𝑧) ∗ 𝑓 (𝑧)

= 𝑧 +∑∞

𝑛=2

Υ_𝑛𝑐_𝑛𝑧𝑛, (7)

where

Υ𝑛= _{(𝑞, 𝑞)}(𝑎1, 𝑞)𝑛−1⋅ ⋅ ⋅ (𝑎𝑟, 𝑞)𝑛−1

𝑛−1(𝑏1, 𝑞)𝑛−1⋅ ⋅ ⋅ (𝑏𝑠, 𝑞)𝑛−1, (󵄨󵄨󵄨󵄨𝑞󵄨󵄨󵄨󵄨 < 1).

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It should be remarked that the linear operator (7) is a generalization of many operators considered earlier. For𝑎_𝑖=

𝑞𝛼𝑖, 𝑏

𝑗 = 𝑞𝛽𝑗, 𝛼𝑖, 𝛽𝑗 ∈C, 𝛽𝑗 ̸= 0, −1, −2, . . . , (𝑖 = 1, . . . , 𝑟, 𝑗 =

1, . . . , 𝑠), and𝑞 → 1, we obtain the Dziok-Srivastava linear operator [3](for𝑟 = 𝑠 + 1), so that it includes (as its special cases) various other linear operators introduced and studied by Ruscheweyh [4], Carlson and Shaffer [5] and the Bernardi-Libera-Livingston operators [6–8].

The𝑞-difference operator is defined by

𝑑𝑞ℎ (𝑧) = ℎ (𝑞𝑧) − ℎ (𝑧)_{(𝑞 − 1) 𝑧} , 𝑞 ̸= 1, 𝑧 ̸= 0,

lim

𝑞 → 1𝑑𝑞ℎ (𝑧) = ℎ 󸀠_{(𝑧) ,}

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whereℎ󸀠(𝑧)is the ordinary derivative. For more properties of

𝑑_𝑞see [9,10].

Lemma 1 (see [2]). Let𝑓 ∈A; then

(i)for𝑟 = 1, 𝑠 = 0, and𝑎₁= 𝑞, one hasM(𝑞, −; 𝑞)𝑓(𝑧) =

𝑓(𝑧).

(ii)For 𝑟 = 1, 𝑠 = 0, and 𝑎₁ = 𝑞2, one

has M(𝑞2, −; 𝑞)𝑓(𝑧) = 𝑧𝑑_𝑞𝑓(𝑧) and lim_{𝑞 → 1}

M(𝑞2_{, −; 𝑞)𝑓(𝑧) = 𝑧𝑓}󸀠_(𝑧)_{, where}_𝑑

𝑞is the𝑞-derivative

defined by(9).

Definition 2. A function𝑓 ∈ Ais said to be in the class

B𝑟

𝑠(𝑎𝑖, 𝑏𝑗; 𝑞; 𝜇)if it is satisfying the condition

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑧2₍_M_(𝑎

𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧))󸀠

[M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧)]2 − 1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨< 1 − 𝜇 (𝑧 ∈U; 0 ≤ 𝜇 < 1) ,

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whereM(𝑎_𝑖, 𝑏_𝑗; 𝑞)𝑓is the operator defined by (7).

Note thatB1₀(𝑞, −; 𝑞; 𝜇) =B(𝜇), where the classB(𝜇)of analytic and univalent functions was introduced and studied by Frasin and Darus [11].

Using the operatorM(𝑎_𝑖, 𝑏_𝑗; 𝑞)𝑓(𝑧)𝑓, we now introduce the following new general integral operator.

For𝑚 ∈ N∪ {0},𝛾₁, 𝛾₂, . . . , 𝛾_𝑚, 𝛿 ∈ C\ {0, −1, −2, . . .}, and|𝑞| < 1, we define the integral operator𝐼_𝛾_𝑘_,𝛿(𝑎_𝑖, 𝑏_𝑗; 𝑞; 𝑧) :

A𝑛 _→ _A𝑛_by

𝐼_𝛾_𝑘_,𝛿(𝑎_𝑖, 𝑏_𝑗; 𝑞; 𝑧)

= (𝛿 ∫𝑧

0 𝑡 𝛿−1_∏𝑚

𝑘=1

(M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓_𝑡 𝑘(𝑡))

1/𝛾𝑘

𝑑𝑡)

1/𝛿

,

(11)

where𝑓_𝑘 ∈A.

Remark 3. It is interesting to note that the integral operator

𝐼𝛾𝑘,𝛿(𝑎𝑖, 𝑏𝑗; 𝑞; 𝑧)generalizes many operators introduced and

studied by several authors, for example,

(1) for𝑟 = 𝑠 + 1, 𝑎_𝑖 = 𝑞𝛼𝑖, 𝑏

𝑗 = 𝑞𝛽𝑗, 𝑖 = 1, . . . , 𝑟, 𝑗 =

1, . . . , 𝑠, 𝑞 → 1, 𝛾_𝑘 = 1/(𝛼 − 1), and𝛿 = 1 + 𝑚(𝛼 − 1), where𝛼 ∈CandR(𝛼) > 0, we obtain the following integral operator introduced and studied by Selvaraj and Karthikeyan [12]:

𝐹_𝛼(𝛼₁, 𝛽₁; 𝑧)

= (1 + 𝑚 (𝛼 − 1) ∫𝑧

0 (𝐻 𝑟

𝑠(𝛼1, 𝛽1) 𝑓1(𝑡))𝛼−1

⋅ ⋅ ⋅ (𝐻_𝑠𝑟(𝛼₁, 𝛽₁) 𝑓_𝑚(𝑡))𝛼−1𝑑𝑡)1/(1+𝑚(𝛼−1)),

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where for convenience 𝐻_𝑠𝑟(𝛼₁, 𝛽₁)𝑓 := 𝐻(𝛼₁, . . . , 𝛼_𝑟; 𝛽₁,

. . . , 𝛽_𝑠; 𝑧)𝑓(𝑧), and 𝐻_𝑠𝑟(𝛼₁, 𝛽₁)𝑓(𝑧) = 𝑧 + ∑∞_𝑛=2((𝛼₁)_𝑛−1

⋅ ⋅ ⋅ (𝛼_𝑟)_𝑛−1/(𝛽₁)_𝑛−1⋅ ⋅ ⋅ (𝛽_𝑠)_𝑛−1(𝑛 − 1)!)𝑎_𝑛𝑧𝑛 _{is the}

Dziok-Sriv-astava operator [3].

(2) For𝑟 = 1, 𝑠 = 0, 𝑎₁ = 𝑞, 𝛾_𝑘 = 1/(𝛼 − 1), and𝛿 =

1 + 𝑚(𝛼 − 1), we obtain the integral operator

𝐹_𝑚,𝛼(𝑧) = (1 + 𝑚 (𝛼 − 1)

× ∫𝑧

0 (𝑓1(𝑡))

𝛼−1_{⋅ ⋅ ⋅ (𝑓}

𝑚(𝑡))𝛼−1𝑑𝑡)

1/(1+𝑚(𝛼−1))

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studied recently by Breaz et al. [13].

(3) For𝑟 = 1, 𝑠 = 0, 𝑎₁ = 𝑞, 𝛾_𝑘 = 1/𝛼_𝑘, and𝛿 = 1, we obtain the integral operator

𝐹_𝛼(𝑧) = ∫𝑧

0 (

𝑓1(𝑡)

𝑡 )

𝛼1

⋅ ⋅ ⋅ (𝑓𝑚_𝑡(𝑡))𝛼𝑚𝑑𝑡 (14)

introduced and studied by D. Breaz and N. Breaz [14]. (4) For𝑟 = 1, 𝑠 = 0, 𝑎₁ = 𝑞2, 𝛾_𝑘 = 1/(𝛼 − 1), and𝛿 =

1 + 𝑚(𝛼 − 1), we obtain the integral operator

𝐺_𝛼(𝑧) = (1 + 𝑚 (𝛼 − 1)

× ∫𝑧

0 𝑡

𝑚(𝛼−1)_(𝑓󸀠 1(𝑡))

𝛼−1

⋅ ⋅ ⋅ (𝑓_𝑚󸀠 (𝑡))𝛼−1𝑑𝑡)1/(1+𝑚(𝛼−1))

(15)

introduced by Selvaraj and Karthikeyan [12].

(5) For𝑟 = 1, 𝑠 = 0, 𝑎₁ = 𝑞2, 𝛾_𝑘 = 1/𝛼, and𝛿 = 1, we obtain the integral operator

𝐺_𝛼(𝑧) = ∫𝑧

0 (𝑓 󸀠 1(𝑡))

𝛼

⋅ ⋅ ⋅ (𝑓_𝑚󸀠 (𝑡))𝛼𝑑𝑡, (16)

recently introduced and studied by Breaz and G¨uney [15]. (6) For𝑟 = 1, 𝑠 = 0, 𝑎₁= 𝑞, 𝑓₁= ⋅ ⋅ ⋅ = 𝑓_𝑚 = 𝑓 ∈A, 𝛾_𝑘=

1/(𝛼 − 1), and𝛿 = 𝛼, where𝛼 ∈ CandR(𝛼) > 0, we obtain the integral operator

𝐺_𝛼(𝑧) = (𝛼 ∫𝑧

0 (𝑓 (𝑡))

𝛼−1₎1/𝛼_𝑑𝑡, ₍₁₇₎

introduced and studied by Pescar [16].

(3)

Lemma 4 (see [17,18]). Let𝛿 ∈CwithRe(𝛿) > 0. If𝑓 ∈A satisfies

1 − |𝑧|2Re(𝛿)

Re(𝛿) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

𝑧𝑓󸀠󸀠(𝑧)

𝑓󸀠_(𝑧) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ 1, (18)

for all𝑧 ∈U, then the integral operator

𝐹_𝛿(𝑧) = {𝛿 ∫𝑧

0 𝑡

𝛿−1_𝑓󸀠_{(𝑡) 𝑑𝑡}}1/𝛿 ₍₁₉₎

is in the classS.

Lemma 5 (see [16]). Let𝛿 ∈ CwithRe(𝛿) > 0, 𝑐 ∈ C, with

|𝑐| ≤ 1, 𝑐 ̸= − 1. If𝑓 ∈Asatisfies

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨𝑐|𝑧|

2𝛿_{+ (1 − |𝑧|}2𝛿₎𝑧𝑓󸀠󸀠(𝑧)

𝛿𝑓󸀠_(𝑧)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ 1, (20)

for all𝑧 ∈Uthen the integral operator

𝐹_𝛿(𝑧) = {𝛿 ∫𝑧

0 𝑡

𝛿−1_𝑓󸀠_{(𝑡) 𝑑𝑡}}1/𝛿 ₍₂₁₎

is in the classS.

Lemma 6 (Generalized Schwarz Lemma, see [19]).

(General-ized Schwarz Lemma) Let the function𝑓be analytic in the disk

U𝑅 = {𝑧 : |𝑧| < 𝑅}, with|𝑓(𝑧)| < 𝑀for fixed𝑀. If𝑓(𝑧)has

one zero with multiplicity order bigger that𝑚for𝑧 = 0, then

󵄨󵄨󵄨󵄨𝑓(𝑧)󵄨󵄨󵄨󵄨 ≤ 𝑀_𝑅𝑚|𝑧|𝑚, (𝑧 ∈U𝑅) . (22)

Equality can hold only if

𝑓 (𝑧) = 𝑒𝑖𝜃(_𝑅𝑀_𝑚) 𝑧𝑚, (23)

where𝜃is constant.

2. Univalence Conditions for

𝐼

_{𝛾𝑘,𝛿}

(𝑎

_𝑖

,𝑏

_𝑗

;𝑞;𝑧)

Theorem 7. Let𝑓_𝑘 ∈ Afor all𝑘 = 1, . . . , 𝑚, 𝛾_𝑘 ∈ C, and

𝑀 ≥ 1with

1

Re(𝛿)

𝑚

∑

𝑘=1

[(2 − 𝜇_𝑘) 𝑀 + 1]

󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨 ≤ 1. (24)

If for all𝑘 = 1, . . . , 𝑚, 𝑓_𝑘 ∈B𝑟_𝑠(𝑎_𝑖, 𝑏_𝑗, 𝑞, 𝜇_𝑘), 0 ≤ 𝜇_𝑘< 1, and

󵄨󵄨󵄨󵄨

󵄨M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)󵄨󵄨󵄨󵄨_{󵄨 ≤ 𝑀, (𝑧 ∈}U) (25)

then the integral operator 𝐼_𝛾_𝑘_,𝛿(𝑎_𝑖, 𝑏_𝑗; 𝑞; 𝑧) defined by (11) is

analytic and univalent inU.

Proof. From the definition of the operatorM(𝑎_𝑖, 𝑏_𝑗; 𝑞)𝑓(𝑧)𝑓

it can be observed that

M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧)

𝑧 ̸= 0, (𝑧 ∈U) , (26)

and for𝑧 = 0, we have

(M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓_𝑧 1(𝑧))

1/𝛾1

⋅ ⋅ ⋅ (M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓_𝑧 𝑚(𝑧))

1/𝛾𝑚

= 1.

(27)

We define the functionℎ(𝑧)by the form

ℎ (𝑧) = ∫𝑧

0 𝑚

∏

𝑘=1

(M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑡)

𝑡 )

1/𝛾𝑘

𝑑𝑡. (28)

Therefore

ℎ󸀠(𝑧) =∏𝑚

𝑘=1

(M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓_𝑧 𝑘(𝑧))

1/𝛾𝑘

. (29)

Differentiating logarithmically and multiplying by𝑧on both sides of (29)

𝑧ℎ󸀠󸀠_(𝑧)

ℎ󸀠_(𝑧) = 𝑚

∑

𝑘=1

1 𝛾_𝑘(

𝑧(M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)) 󸀠

M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)

− 1) .

(30)

Thus we have

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨

𝑧ℎ󸀠󸀠_(𝑧)

ℎ󸀠_(𝑧) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ 𝑚

∑

𝑘=1

1 󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑧(M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧))󸀠

− 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨.

(31)

So

1 − |𝑧|2Re(𝛿)

Re(𝛿) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

𝑧ℎ󸀠󸀠_(𝑧)

ℎ󸀠_(𝑧) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

≤ 1 − |𝑧|2Re(𝛿)

Re(𝛿)

× [ [

𝑚

∑

𝑘=1

1 󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨 (

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑧(M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧))󸀠

M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨+ 1)]] ≤ 1 − |𝑧|2Re(𝛿)

Re(𝛿)

× [ [

𝑚

∑

𝑘=1

1 󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨 (

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑧2₍_M_(𝑎

𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧))󸀠

[M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)]2

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

×󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨

𝑧

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨+ 1)]]

.

(4)

Since|M(𝑎_𝑖, 𝑏_𝑗; 𝑞)𝑓(𝑧)𝑓_𝑘(𝑧)| ≤ 𝑀, (𝑧 ∈ U, 𝑘 = 1, . . . , 𝑚), and𝑓_𝑘 ∈ B𝑟_𝑠(𝑎_𝑖, 𝑏_𝑗, 𝑞, 𝜇_𝑘)for all𝑘 = 1, . . . , 𝑚, then from the Schwarz Lemma and (10), we obtain

1 − |𝑧|2Re(𝛿)

Re(𝛿) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

𝑧ℎ󸀠󸀠(𝑧) ℎ󸀠_(𝑧) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

≤ 1 − |𝑧|2Re(𝛿)

Re(𝛿)

× [(∑𝑚

𝑘=1

1 󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

𝑧2₍_M_(𝑎

𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧))󸀠

[M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)]2

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑀

+𝑀 + 1)] ]

≤ 1

Re(𝛿)

𝑚

∑

𝑘=1

1

󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨 [(2 − 𝜇𝑘) 𝑀 + 1] , (𝑧 ∈U)

(33)

which, in the light of the hypothesis (24), yields

1 − |𝑧|2Re(𝛿)

Re(𝛿) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

𝑧ℎ󸀠󸀠(𝑧)

ℎ󸀠_(𝑧) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ 1, (𝑧 ∈U) . (34)

Applying Lemma (1) for the function ℎ(𝑧) we obtain that

𝐼_𝛾_𝑘_,𝛿(𝑎_𝑖, 𝑏_𝑗; 𝑞; 𝑧)is univalent.

Taking 𝜇_𝑘 = 0 (for all𝑘 = 1, . . . , 𝑚), 𝑀 = 1, 𝑎_𝑖 =

𝑞𝛼𝑖_{, 𝑏}

𝑗= 𝑞𝛽𝑗, 𝑞 → 1, and𝛾𝑘= 1/(𝛼 − 1), 𝛿 = 1 + 𝑚(𝛼 − 1)in

Theorem7, we have the following.

Corollary 8 (see [12]). Let𝑓_𝑘 ∈ Afor all𝑘 = 1, . . . , 𝑚and

𝛼 ∈Cwith

|𝛼 − 1| ≤ Re(𝛼)

3𝑚 . (35)

If

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨

𝑧2_(𝐻𝑟

𝑠(𝛼1, 𝛽1) 𝑓𝑘(𝑧))󸀠

(𝐻𝑟

𝑠(𝛼1, 𝛽1) 𝑓𝑘(𝑧))2

− 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨󵄨< 1, (𝑧 ∈U) (36)

and for all𝑘 = 1, . . . , 𝑚, then the integral operator𝐹_𝛼(𝛼₁, 𝛽₁; 𝑧)

defined by(12)is analytic and univalent inU.

Taking𝜇_𝑘 = 0 (for all𝑘 = 1, . . . , 𝑚), 𝑀 = 1, 𝑟 = 1, 𝑠 =

0, 𝑎1= 𝑞, and𝛾𝑘 = 1/(𝛼 − 1), 𝛿 = 1 + 𝑚(𝛼 − 1)in Theorem7,

we have the following.

Corollary 9. Let𝑓_𝑘∈Afor all𝑘 = 1, . . . , 𝑚and𝛼 ∈Cwith

|𝛼 − 1| ≤ Re_3𝑚(𝛼). (37)

If

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨󵄨

𝑧2_𝑓󸀠 𝑘(𝑧)

(𝑓_𝑘(𝑧))2 − 1 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨

󵄨󵄨< 1, (𝑧 ∈U) (38)

and for all𝑘 = 1, . . . , 𝑚, then the integral operator 𝐹_𝑚,𝛼(𝑧)

defined by(13)is analytic and univalent inU.

Theorem 10. Let𝑓_𝑘 ∈Afor all𝑘 = 1, . . . , 𝑚, 𝛿, 𝛾_𝑘 ∈C, and

𝑀 ≥ 1with

|𝑐| ≤ 1 −1_𝛿∑𝑚

𝑘=1

[(2 − 𝜇_𝑘) 𝑀 + 1]

󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨 , 𝑐 ∈C. (39)

If for all𝑘 = 1, . . . , 𝑚, 𝑓_𝑘∈B𝑟_𝑠(𝑎_𝑖, 𝑏_𝑗, 𝑞, 𝜇_𝑘), 0 ≤ 𝜇_𝑘< 1, and

󵄨󵄨󵄨󵄨

󵄨M(𝑎𝑖, 𝑏𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧)󵄨󵄨󵄨󵄨_{󵄨 ≤ 𝑀, (𝑧 ∈}U) , (40)

then the integral operator 𝐼_𝛾_𝑘_,𝛿(𝑎_𝑖, 𝑏_𝑗; 𝑞) defined by (11) is

analytic and univalent inU.

Proof. From the proof of Theorem7, we have

𝑧ℎ󸀠󸀠_(𝑧)

ℎ󸀠_(𝑧) = 𝑚

∑

𝑘=1

1 𝛾𝑘(

𝑧(M(𝑎_𝑖, 𝑏_𝑗; 𝑞) 𝑓 (𝑧) 𝑓𝑘(𝑧))󸀠

− 1) .

(41)

Thus we have

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨𝑐|𝑧|

2𝛿_{+ (1 − |𝑧|}2𝛿₎𝑧ℎ󸀠󸀠(𝑧)

𝛿ℎ󸀠_(𝑧)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ |𝑐| +󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨(1 − |𝑧|2𝛿)

𝑧ℎ󸀠󸀠_(𝑧)

𝛿ℎ󸀠_(𝑧)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨.

(42)

From this result and using the proof of Theorem7we obtain

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨𝑐|𝑧|

2𝛿_{+ (1 − |𝑧|}2𝛿₎𝑧ℎ󸀠󸀠(𝑧)

𝛿ℎ󸀠_(𝑧)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨

≤ |𝑐| +1_𝛿∑𝑚

𝑘=1

1

󵄨󵄨󵄨󵄨𝛾𝑘󵄨󵄨󵄨󵄨 [(2 − 𝜇𝑘) 𝑀 + 1] .

(43)

Since|𝑐| ≤ 1 − (1/𝛿) ∑𝑚_𝑘=1(1/𝛾_𝑘)[(2 − 𝜇_𝑘)𝑀 + 1], then we have

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨𝑐|𝑧|

2𝛿_{+ (1 − |𝑧|}2𝛿₎𝑧ℎ󸀠󸀠(𝑧)

𝛿ℎ󸀠_(𝑧)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨≤ 1, (𝑧 ∈U) . (44)

Applying Lemma (4) for the functionℎ(𝑧) we obtain that

𝐼_𝛾_𝑘_,𝛿(𝑎_𝑖, 𝑏_𝑗; 𝑞; 𝑧)is univalent.

Taking𝜇_𝑘 = 0 (for all𝑘 = 1, . . . , 𝑚), 𝑟 = 1, 𝑠 = 0, 𝑎₁ = 𝑞, and𝛾_𝑘 = 1/(𝛼 − 1), 𝛿 = 1 + 𝑚(𝛼 − 1)(𝛼 ∈R)in Theorem10, we have the following.

Corollary 11. Let𝑓_𝑘 ∈Afor all𝑘 = 1, . . . , 𝑚;𝑐 ∈C, 𝛼 ∈R,

and𝑀 ≥ 1with

|𝑐| ≤ 1 + ( 1 − 𝛼

1 + 𝑚 (𝛼 − 1)) (2𝑀 + 1) 𝑚,

𝛼 ∈ [1,2𝑀𝑚 + 1_2𝑀𝑚 ] .

(45)

If for all𝑘 = 1, . . . , 𝑚

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨

𝑧2_𝑓󸀠 𝑘(𝑧)

𝑓2 𝑘(𝑧)

− 1󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨}

󵄨< 1, (𝑧 ∈U) , 󵄨󵄨󵄨󵄨𝑓𝑘(𝑧)󵄨󵄨󵄨󵄨 ≤ 𝑀, (𝑧 ∈U; 𝑘 = 1, . . . , 𝑚) ,

(46)

then the integral operator𝐹_𝑚,𝛼(𝑧)defined by(13)is analytic

(5)

Letting𝑚 = 1, 𝑀 = 1, and𝑓₁ = 𝑓in Corollary11, we have the following.

Corollary 12. Let𝑓 ∈A,𝑐 ∈Cand𝛼 ∈Rwith

|𝑐| ≤ 3 − 2𝛼_𝛼 , (𝑐 ̸= − 1) ,

𝛼 ∈ [1,3 2] .

(47)

If

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨

𝑧2_𝑓󸀠_(𝑧)

𝑓2_(𝑧) − 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨< 1, (𝑧 ∈U) ,

󵄨󵄨󵄨󵄨𝑓(𝑧)󵄨󵄨󵄨󵄨 ≤ 1, (𝑧 ∈U) ,

(48)

then the integral operator𝐺_𝛼(𝑧)defined by(17)is analytic and

univalent inU.

Remark 13. Many other interesting corollaries and results can

be obtained by specializing the parameters in Theorem10; for example, see [13,20,21].

Acknowledgments

The work presented here was partially supported by GUP-2012-023 and UKM-DLP-2011-050.

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(6)

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