J256979

International Journal of Engineering Science Invention ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org Volume 2 Issue 5 Ç May. 2013 Ç PP.69-79

INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS CHENA RAM AND SAROJ SOLANKI Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur, Rajasthan, India.

ABSTRACT: In this paper, we study a new class of functions defined by Wright generalized hypergeometric function. Characterization property, the result on modified Hadamard product and integral transform are obtained. Distortion theorem and radii of starlikeness and convexity are also obtained.

KEY WORDS: Analytic function, Uniformly convex, Wright generalized hypergeometric function, Linear operator , Hadamard product.

I.

INTRODUCTION

Let đ??´ denote the class of functions of the form ∞

�� � � ,

� � =�+

�� ≼ 0

1.1

đ?‘›=2

which are analytic and univalent in the unit disk đ?&#x2018;&#x2C6; = đ?&#x2018;§: đ?&#x2018;§ â&#x2C6;&#x2C6; â&#x201E;&#x201A; and đ?&#x2018;§ < 1 .

A function đ?&#x2018;&#x201C; â&#x2C6;&#x2C6; đ??´ is said to be in the class of uniformly convex functions of order đ?&#x203A;ž, denoted by đ?&#x2018;&#x2C6;đ??śđ?&#x2018;&#x2030; đ?&#x203A;ž cf. [5] if đ?&#x2018;§đ?&#x2018;&#x201C; â&#x20AC;˛â&#x20AC;˛ (đ?&#x2018;§)

Re

đ?&#x2018;&#x201C; â&#x20AC;˛(đ?&#x2018;§)

đ?&#x2018;§đ?&#x2018;&#x201C; â&#x20AC;˛â&#x20AC;˛ đ?&#x2018;§

+1â&#x2C6;&#x2019;đ?&#x203A;ž â&#x2030;Ľ đ?&#x153;&#x201A;

đ?&#x2018;&#x201C;â&#x20AC;˛ đ?&#x2018;§

â&#x2C6;&#x2019;1 ,

(1.2)

and is said to be in a corresponding subclass of đ?&#x2018;&#x2C6;đ??śđ?&#x2018;&#x2030; đ?&#x203A;ž denoted by đ?&#x2018;&#x2020;đ?&#x2018;? đ?&#x203A;ž if Re

đ?&#x2018;§đ?&#x2018;&#x201C; â&#x20AC;˛ (đ?&#x2018;§) đ?&#x2018;&#x201C;(đ?&#x2018;§ )

đ?&#x2018;§đ?&#x2018;&#x201C; â&#x20AC;˛ đ?&#x2018;§

â&#x2C6;&#x2019;đ?&#x203A;ž â&#x2030;Ľđ?&#x153;&#x201A;

đ?&#x2018;&#x201C; đ?&#x2018;§

â&#x2C6;&#x2019;1 ,

(1.3)

where â&#x2C6;&#x2019;1 â&#x2030;¤ đ?&#x203A;ž â&#x2030;¤ 1 and đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2C6;.

The class of uniformly convex and uniformly starlike function has been studied by Goodman and Minda [11].

If đ?&#x2018;&#x201C;(đ?&#x2018;§) of the form (1.1) in class đ??´ and function đ?&#x2018;&#x201D;(đ?&#x2018;§) â&#x2C6;&#x2C6; đ??´ defined as

1], [2

and Ma

â&#x2C6;&#x17E;

đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;§đ?&#x2018;&#x203A; ,

đ?&#x2018;&#x201D; đ?&#x2018;§ =đ?&#x2018;§+

(1.4)

đ?&#x2018;&#x203A;=2

then the Hadamard product of đ?&#x2018;&#x201C; đ?&#x2018;§ and đ?&#x2018;&#x201D; đ?&#x2018;§ is given by â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;§ đ?&#x2018;&#x203A; .

đ?&#x2018;&#x201C;â&#x2C6;&#x2014;đ?&#x2018;&#x201D; đ?&#x2018;§ = đ?&#x2018;§+

(1.5)

đ?&#x2018;&#x203A; =2

Let đ?&#x2018;&#x2021; denote the subclass of đ??´ consisting of functions of the form cf. [7] â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;§ đ?&#x2018;&#x203A; ,

đ?&#x2018;&#x201C; đ?&#x2018;§ =đ?&#x2018;§â&#x2C6;&#x2019;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; â&#x2030;Ľ 0 .

(1.6)

đ?&#x2018;&#x203A;=2

A function đ?&#x153;&#x201C;đ?&#x2018;? đ?&#x153;&#x201C;đ?&#x2018;?

đ?&#x203A;źđ?&#x2018;&#x2014; đ??´đ?&#x2018;&#x2014; đ?&#x203A;źđ?&#x2018;&#x2014; đ??´đ?&#x2018;&#x2014;

1,đ?&#x2018;&#x17E; 1,đ?&#x2018;&#x17E;

; đ?&#x203A;˝đ?&#x2018;&#x2014; đ??ľđ?&#x2018;&#x2014;

; đ?&#x203A;˝đ?&#x2018;&#x2014; đ??ľđ?&#x2018;&#x2014;

1,đ?&#x2018; 1,đ?&#x2018;

; đ?&#x2018;§ is studied by R.K. Raina [9] as

; đ?&#x2018;§ = đ?&#x153;&#x201D;đ?&#x2018;§ đ?&#x2018;&#x17E; đ?&#x153;&#x201C;đ?&#x2018;

đ?&#x203A;źđ?&#x2018;&#x2014; đ??´đ?&#x2018;&#x2014;

1,đ?&#x2018;&#x17E;

; đ?&#x203A;˝đ?&#x2018;&#x2014; đ??ľđ?&#x2018;&#x2014; ,

(1.7)

where s

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014;

jď&#x20AC;˝1

đ?&#x153;&#x201D;=

, đ?&#x2018;&#x17E;, đ?&#x2018; â&#x2C6;&#x2C6; đ?&#x2018; 0

q

ď&#x192;&#x2022;

(1.8)

Î&#x201C; đ?&#x203A;źđ?&#x2018;&#x2014;

jď&#x20AC;˝1

and đ?&#x2018;&#x17E; đ?&#x153;&#x201C;đ?&#x2018;

đ?&#x203A;źđ?&#x2018;&#x2014; đ??´đ?&#x2018;&#x2014;

1,đ?&#x2018;&#x17E;

; đ?&#x203A;˝đ?&#x2018;&#x2014; đ??ľđ?&#x2018;&#x2014;

1,đ?&#x2018;

; đ?&#x2018;§ is the Wright generalized hypergeometric function introduced by Wright [3] as

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69 | P a g e

INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT q

𝑞 𝜓𝑠

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑠

∞



𝑛 =0

s

Γ 𝛼𝑗 + 𝐴𝑗 𝑛 𝑧 𝑛

j1

;𝑧 =

,



Γ 𝛽𝑗 +𝐵𝑗 𝑛

𝑧∈𝑈

(1.9)

𝑛 !

j1

Ξ𝑝

by making use of the Hadamard product, Raina [9] defined a linear operator

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧 ∶

𝑇 → 𝑇 as Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

𝑓 𝑧 = 𝜓𝑝

1,𝑠

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗 ∞

1,𝑠

; 𝑧 ∗ 𝑓(𝑧)

𝜎𝑛 𝑎𝑛 𝑧 𝑛 ,

=𝑧−

(1.10)

𝑛=2

where q

𝜔



Γ(𝛼𝑗 + 𝐴𝑗 (𝑛 − 1))

j1

𝜎𝑛 =

(1.11)

s



Γ 𝛽𝑗 +𝐵𝑗 𝑛 − 1)

𝑛−1 !

j1

and 𝜔 is defined by equation (1.8).

Definition 1. A function 𝑓 ∈ 𝐴 for −1 ≤ 𝛾 < 1 is said to be in the class 𝑇 ∗ (𝛾) if and only if ′

𝑧 Ξ𝑝

𝛼 𝑗 𝐴𝑗

Re Ξ𝑝

𝛼 𝑗 𝐴𝑗

1,𝑞 1,𝑞

; 𝛽 𝑗 𝐵𝑗

; 𝛽 𝑗 𝐵𝑗

′

𝑓 𝑧

1,𝑠

𝑧 Ξ𝑝

𝛼 𝑗 𝐴𝑗

Ξ𝑝

𝛼 𝑗 𝐴𝑗

−𝛾 ≥ 𝑓 𝑧

1,𝑠

1,𝑞 1,𝑞

; 𝛽 𝑗 𝐵𝑗

; 𝛽 𝑗 𝐵𝑗

1,𝑠 1,𝑠

𝑓 𝑧

−1 , 𝑧 ∈ 𝑈. (1.12) 𝑓 𝑧

Definition 2. A function 𝑓 ∈ 𝐴 for −1 ≤ 𝛾 < 1 is said to be in the class 𝑈𝐶𝑉𝑇 ∗ (𝛾) if and only if ′′

𝑧 Ξ𝑝

𝛼 𝑗 𝐴𝑗

Re 1 +

1,𝑞

; 𝛽 𝑗 𝐵𝑗

1,𝑠

′

Ξ𝑝

𝛼 𝑗 𝐴𝑗

′′

𝑓 𝑧

𝛼 𝑗 𝐴𝑗

1,𝑞

Ξ𝑝

𝛼 𝑗 𝐴𝑗

; 𝛽 𝑗 𝐵𝑗 𝑓 𝑧 1,𝑞 1,𝑠

−𝛾 ≥

; 𝛽 𝑗 𝐵𝑗 𝑓 𝑧 1,𝑞 1,𝑠

II.

𝑧 Ξ𝑝

; 𝛽 𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧 ′

−1 , 𝑧 ∈ 𝑈.

(1.13)

CHARACTERIZATION PROPERTY

Theorem 2.1. A function 𝑓 𝑧 defined by (1.6) is in the class 𝑇 ∗ (𝛾) if and only if ∞

2𝑛 − 1 − 𝛾 𝜎𝑛 𝑎𝑛 ≤ 1 − 𝛾, −1 ≤ 𝛾 < 1 .

(2.1)

𝑛 =2

The result is sharp for the function 1−𝛾 𝑓 𝑧 =𝑧− 𝑧𝑛 , 2𝑛 − 1 − 𝛾 𝜎𝑛 s



q

Γ 𝛽𝑗

j1

where 𝜎𝑛 =



Γ(𝛼𝑗 + 𝐴𝑗 (𝑛 − 1))

j1

. s

q

 j1

Γ 𝛼𝑗



Γ 𝛽𝑗 +𝐵𝑗 𝑛 − 1)

𝑛−1 !

j1

Proof. It is sufficient to show that

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70 | P a g e

INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT ′

𝑧 Ξ𝑝

𝛼𝑗 𝐴𝑗

Ξ𝑝

𝛼𝑗 𝐴𝑗

𝑧 Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

1,𝑞

; 𝛽𝑗 𝐵𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑠

1,𝑠

′

𝑧 Ξ𝑝

𝑓 𝑧

𝛼𝑗 𝐴𝑗

− 1 ≤ Re Ξ𝑝

𝛼𝑗 𝐴𝑗

𝑧 Ξ𝑝

𝛼𝑗 𝐴𝑗

Ξ𝑝

𝛼𝑗 𝐴𝑗

𝑓 𝑧

1,𝑞

1,𝑞

; 𝛽𝑗 𝐵𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑠

1,𝑠

𝑓 𝑧 −𝛾 , 𝑓 𝑧

′

Ξ𝑝

1,𝑞

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑠

1,𝑠

′

𝑓 𝑧 − 1 − Re 𝑓 𝑧

1,𝑞

1,𝑞

; 𝛽𝑗 𝐵𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑠

1,𝑠

𝑓 𝑧 −1 ≤ 1−𝛾 𝑓 𝑧

we have ′

𝑧 Ξ𝑝 Ξ𝑝

𝛼𝑗 𝐴𝑗 𝛼𝑗 𝐴𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑞

1,𝑞

𝑧 Ξ𝑝

𝑓 𝑧

1,𝑠

; 𝛽𝑗 𝐵𝑗

′

𝛼𝑗 𝐴𝑗

− 1 − Re Ξ𝑝

𝑓 𝑧

1,𝑠

𝛼𝑗 𝐴𝑗

1,𝑞

1,𝑞

; 𝛽𝑗 𝐵𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑠

1,𝑠

𝑓 𝑧 −1 𝑓 𝑧

′

𝑧 Ξ𝑝

𝛼𝑗 𝐴𝑗

Ξ𝑝

𝛼𝑗 𝐴𝑗

≤2

1,𝑞

1,𝑞

; 𝛽𝑗 𝐵𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧 −1 𝑓 𝑧

1,𝑠



1−

 n 2

≤2

𝑛𝜎𝑛 𝑎𝑛 𝑧 𝑛−1 −1





1−

n 2

𝜎𝑛 𝑎𝑛 𝑧 𝑛−1

or, 

 ≤2

(𝑛 − 1)𝜎𝑛 𝑎𝑛 𝑧 𝑛−1

n 2

≤ 1 − 𝛾.



1−

 n 2

𝜎𝑛 𝑎𝑛 𝑧 𝑛−1

The above inequality must hold for all 𝑧 in 𝑈. Letting 𝑧 → 1− , we have 

2 ≤

 n 2

(𝑛 − 1)𝜎𝑛 𝑎𝑛 ≤ 1−𝛾 .



1−

 n 2

𝜎𝑛 𝑎𝑛

Therefore, ∞

2𝑛 − 1 − 𝛾 𝜎𝑛 𝑎𝑛 ≤ 1 − 𝛾 . 𝑛=2

Conversely, if 𝑓 ∈ 𝑇 ∗ (𝛾) and 𝑧 is real then (2.1) gives

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71 | P a g e

INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT 



1−

n 2



𝑛𝜎𝑛 𝑎𝑛 𝑧



,



𝜎𝑛 𝑎𝑛 𝑧 𝑛−1

n 2

𝑛 − 1 𝜎𝑛 𝑎𝑛 𝑧𝑛−1

n 2

−𝛾≥



1−



𝑛−1

1−



𝜎𝑛 𝑎𝑛 𝑧 𝑛−1

n 2

by letting 𝑧 → 1− along the real axis, we get 



1−

n 2





≥ 𝛾,



𝜎𝑛 𝑎𝑛

n 2

𝑛 − 1 𝜎𝑛 𝑎𝑛

n 2

−



1−



𝑛𝜎𝑛 𝑎𝑛

1−



𝜎𝑛 𝑎𝑛

n 2

∞

∞

1−

(2𝑛 − 1) 𝜎𝑛 𝑎𝑛 ≥ 1 − 𝛾

1−

𝑛=2

𝜎𝑛 𝑎𝑛 , 𝑛 =2

which yields the required result, where 𝜎𝑛 is given by (1.11).

Corollary 2.1. Let the function 𝑓(𝑧) defined by (1.6) be in the class 𝑇 ∗ (𝛾), then 1−𝛾 𝑎𝑛 ≤ , 𝑛 ≥ 2, 2𝑛 − 1 − 𝛾 𝜎𝑛 where s

q





Γ 𝛽𝑗

j1

𝜎𝑛 =

Γ(𝛼𝑗 + 𝐴𝑗 (𝑛 − 1))

j1

, q



𝑛 ≥ 2.

s

Γ 𝛼𝑗

j1



Γ 𝛽𝑗 +𝐵𝑗 𝑛 − 1)

𝑛−1 !

j1

III.

Growth and Distortion theorems

Theorem II.1. Let the function 𝑓 𝑧 defined by (1.6) be in the class 𝑇 ∗ 𝛾 , then 𝑧 −

1−𝛾 3−𝛾

𝑧

2

≤

Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧

≤ 𝑧 +

1−𝛾 3−𝛾

2

(3.1)

𝑧.

(3.2)

𝑧

and ′

1−2

1−𝛾 3−𝛾

𝑧 ≤

Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧

≤ 1+2

1−𝛾 3−𝛾

Equality holds for the function 1 − 𝛾 𝑧2 𝑓 𝑧 =𝑧− , 3 − 𝛾 𝜎2 s

 where 𝜎2 =

q

Γ 𝛽𝑗

j1

j1

Γ(𝛼 𝑗 +𝐴𝑗 )

j1

, 𝑛 ≥ 2.

s

q





Γ 𝛼𝑗



3.3

Γ 𝛽 𝑗 +𝐵 𝑗

j1

Proof. Let 𝑓(𝑧) ∈ 𝑇 ∗ 𝛾 . By using(1.10), we get

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72 | P a g e

INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT ∞

Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧

𝜎𝑛 𝑎𝑛 𝑧 𝑛 ,

≤ 𝑧 + 𝑛=2

by using (2.1) i.e. ∞

1−𝛾 1−𝛾 = , 𝑛 ≥ 2, 2𝑛 − 1 − 𝛾 3−𝛾

𝜎𝑛 𝑎𝑛 ≤ 𝑛=2

then, we obtain

∞

Ξ𝑝

𝛼𝑗 𝐴𝑗

; 𝛽𝑗 𝐵𝑗

1,𝑞

𝑓 𝑧

1,𝑠

≤ 𝑧 + 𝑧

2

𝑎𝑛 𝜎𝑛 , 𝑛=2

and

Ξ𝑝

Ξ𝑝

𝛼𝑗 𝐴𝑗

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

𝑓 𝑧

1,𝑠

𝑓 𝑧

1,𝑠

1−𝛾

≤ 𝑧 1+

≥ 𝑧 1−

1−𝛾

3−𝛾

𝑧 ,

(3.4)

𝑧 .

3−𝛾

(3.5)

′

Similarly

Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

𝑓 𝑧

1,𝑠

1−𝛾

≥ 1−2

3−𝛾

𝑧,

(3.6)

𝑧.

(3.7)

′

And

Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

𝑓 𝑧

1,𝑠

By using (3.4) and (3.5), we obtain 1−𝛾 𝑧 − 𝑧 2 ≤ Ξ𝑝 3−𝛾 and by using (3.6) and (3.7), we get

1−𝛾

≤ 1+2

𝛼𝑗 𝐴𝑗

1,𝑞

3−𝛾

; 𝛽𝑗 𝐵𝑗

1,𝑠

𝑓 𝑧

≤ 𝑧 +

1−𝛾 𝑧 2, 3−𝛾

′

1−2

1−𝛾 3−𝛾

𝑧 ≤

Ξ𝑝

𝛼𝑗 𝐴𝑗

1,𝑞

; 𝛽𝑗 𝐵𝑗

1,𝑠

;𝑧 𝑓 𝑧

≤ 1+2

1−𝛾 3−𝛾

𝑧.

The bounds (3.1) and (3.2) are attained for functions 𝑓 𝑧 given by 1−𝛾 𝑓 𝑧 =𝑧− 𝑧2 , 3 − 𝛾 𝜎2 where 𝜎2 is given by (3.3).

Theorem III.2 Let the function 𝑓 𝑧 defined by (1.6) and ∞

𝑏𝑛 𝑧𝑛

𝑔 𝑧 =𝑧−

(3.8)

𝑛=2

be in the class 𝑇 ∗ (𝛾). Then the function 𝑕(𝑧) defined by ∞

𝑞𝑛 𝑧 𝑛 ,

𝑕 𝑧 = 1 − 𝜆 𝑓 𝑧 + 𝜆𝑔 𝑧 = 𝑧 −

3.9

𝑛=2

where 𝑞𝑛 = 1 − 𝜆 𝑎𝑛 + 𝜆𝑏𝑛 , (0 ≤ 𝜆 ≤ 1) is also in class 𝑇 ∗ 𝛾 Proof. By using (2.1) for 𝑓 𝑧 and 𝑔(𝑧), we have ∞

2𝑛 − 1 − 𝛾 𝜎𝑛 𝑎𝑛 ≤ 1 − 𝛾

(3.10)

2𝑛 − 1 − 𝛾 𝜎𝑛 𝑏𝑛 ≤ 1 − 𝛾 .

(3.11)

𝑛 =2

And

∞

𝑛 =2

On using (3.10) and (3.11) in (3.9), we get ∞

𝑕 𝑧 = 1−𝜆 ∞

𝑧−

𝑎𝑛 𝑧

𝑛

∞

𝑏𝑛 𝑧𝑛 ,

+𝜆 𝑧−

𝑛 =2

𝑛=2

[ 1 − 𝜆 𝑎𝑛 + 𝜆𝑏𝑛 ]𝑧 𝑛 ,

=𝑧− 𝑛 =2

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INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT then â&#x2C6;&#x17E;

â&#x2C6;&#x17E;

â&#x2C6;&#x17E;

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D;đ?&#x2018;&#x203A; [ 1 â&#x2C6;&#x2019; đ?&#x153;&#x2020; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; + đ?&#x153;&#x2020;đ?&#x2018;?đ?&#x2018;&#x203A; ] = đ?&#x2018;&#x203A;=2

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 1 â&#x2C6;&#x2019; đ?&#x153;&#x2020; đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; + đ?&#x153;&#x2020; đ?&#x2018;&#x203A; =2

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;?đ?&#x2018;&#x203A; , đ?&#x2018;&#x203A;=2

â&#x2030;¤ 1â&#x2C6;&#x2019;đ?&#x153;&#x2020; 1â&#x2C6;&#x2019;đ?&#x203A;ž +đ?&#x153;&#x2020; 1â&#x2C6;&#x2019;đ?&#x203A;ž , â&#x2030;¤ 1â&#x2C6;&#x2019;đ?&#x203A;ž .

So đ?&#x2018;&#x2022;(đ?&#x2018;§) is also in the class đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž).

IV Theorem 4.1. Let the functions

CLOSURE THEOREM â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014; đ?&#x2018;§ đ?&#x2018;&#x203A; , đ?&#x2018;&#x2014; = 1,2, â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś đ?&#x2018;&#x161;

đ?&#x2018;&#x201C;đ?&#x2018;&#x2014; đ?&#x2018;§ = đ?&#x2018;§ â&#x2C6;&#x2019;

(4.1)

đ?&#x2018;&#x203A; =2

be in the classes đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;žđ?&#x2018;&#x2014;

đ?&#x2018;&#x2014; = 1,2, â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś đ?&#x2018;&#x161; respectively. Then the function đ?&#x2018;&#x2022;(đ?&#x2018;§) defined by

1 đ?&#x2018;&#x2022; đ?&#x2018;§ =đ?&#x2018;§â&#x2C6;&#x2019; đ?&#x2018;&#x161;

â&#x2C6;&#x17E;

đ?&#x2018;&#x161;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014; đ?&#x2018;§ đ?&#x2018;&#x203A; đ?&#x2018;&#x203A;=2

(4.2)

đ?&#x2018;&#x2014; =1

is in the class đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž), where đ?&#x203A;ž = min1â&#x2030;¤đ?&#x2018;&#x2014; â&#x2030;¤đ?&#x2018;&#x161; đ?&#x203A;žđ?&#x2018;&#x2014; , with 0 â&#x2030;¤ đ?&#x203A;žđ?&#x2018;&#x2014; < 1.

(4.3)

Proof. Since đ?&#x2018;&#x201C;đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;žđ?&#x2018;&#x2014; , (đ?&#x2018;&#x2014; = 1,2, â&#x20AC;Ś â&#x20AC;Ś â&#x20AC;Ś đ?&#x2018;&#x161;). By using (2.1) in (4.2), we get â&#x2C6;&#x17E;

1 đ?&#x2018;&#x161;

đ?&#x153;&#x17D;đ?&#x2018;&#x203A; 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x2018;&#x203A; =2

1 â&#x2030;¤ đ?&#x2018;&#x161;

đ?&#x2018;&#x161;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014; đ?&#x2018;&#x2014; =1

1 = đ?&#x2018;&#x161;

đ?&#x2018;&#x161;

â&#x2C6;&#x17E;

đ?&#x2018;&#x2014; =1

đ?&#x2018;&#x203A;=2

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014;

đ?&#x2018;&#x161;

1 â&#x2C6;&#x2019; đ?&#x203A;žđ?&#x2018;&#x2014; â&#x2030;¤ 1 â&#x2C6;&#x2019; đ?&#x203A;ž by using 4.3 . đ?&#x2018;&#x2014; =1

Hence đ?&#x2018;&#x2022; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž so the proof is completed.

V

Results involving modified Hadamard product

Let đ?&#x2018;&#x201C; đ?&#x2018;§ and đ?&#x2018;&#x201D;(đ?&#x2018;§) defined by (1.6) and (3.8) respectively. Then â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;§ đ?&#x2018;&#x203A;

đ?&#x2018;&#x201C; â&#x2C6;&#x2014; đ?&#x2018;&#x201D;)(đ?&#x2018;§ = đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;&#x203A; =2

be the modified Hadamard product of function đ?&#x2018;&#x201C;(đ?&#x2018;§) and đ?&#x2018;&#x201D;(đ?&#x2018;§).

Theorem 5.1. For functions â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014; đ?&#x2018;§ đ?&#x2018;&#x203A; , đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;,đ?&#x2018;&#x2014; â&#x2030;Ľ 0; đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2C6;; đ?&#x2018;&#x2014; = 1,2 ,

đ?&#x2018;&#x201C;đ?&#x2018;&#x2014; đ?&#x2018;§ = đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;&#x203A;=2

let đ?&#x2018;&#x201C;1 (đ?&#x2018;§) â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž) and đ?&#x2018;&#x201C;2 đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x153;&#x201A; . Then (đ?&#x2018;&#x201C;1 â&#x2C6;&#x2014; đ?&#x2018;&#x201C;2 ) đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x153;&#x2030; , where đ?&#x153;&#x2030; = 1 â&#x2C6;&#x2019;

2 1â&#x2C6;&#x2019;đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;˝

s

ď&#x192;&#x2022;

(5.1)

3â&#x2C6;&#x2019;đ?&#x203A;ž 3â&#x2C6;&#x2019;đ?&#x203A;˝ đ?&#x153;&#x17D;2 â&#x2C6;&#x2019; 1â&#x2C6;&#x2019;đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;˝

q

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014;

jď&#x20AC;˝1

ď&#x192;&#x2022;

Î&#x201C;(đ?&#x203A;źđ?&#x2018;&#x2014; + đ??´đ?&#x2018;&#x2014; )

jď&#x20AC;˝1

and đ?&#x153;&#x17D;2 =

.

ď&#x192;&#x2022;

5.2

s

q

Î&#x201C; đ?&#x203A;źđ?&#x2018;&#x2014;

jď&#x20AC;˝1

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014; +đ??ľđ?&#x2018;&#x2014;

jď&#x20AC;˝1

The result is best possible for the functions đ?&#x2018;&#x201C;1 = đ?&#x2018;§ â&#x2C6;&#x2019;

1â&#x2C6;&#x2019;đ?&#x203A;ž 3â&#x2C6;&#x2019;đ?&#x203A;ž đ?&#x153;&#x17D;2

đ?&#x2018;§2 ,

(5.3)

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INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT 𝑓2 = 𝑧 −

1−𝜂 3−𝜂 𝜎2

𝑧2 .

(5.4)

Proof . By using (2.1), it is sufficient to prove that ∞

𝑛 =2

2𝑛 − 1 − 𝜉 𝜎𝑛 𝑎𝑛,1 𝑎𝑛,2 ≤ 1, 1−𝜉

(5.5)

where ξ is defined by (5.1) under the hypothesis, it follows from (2.1) ∞

2𝑛 − 1 − 𝛾 𝜎𝑛 𝑎𝑛,1 ≤ 1 1−𝛾

(5.6)

2𝑛 − 1 − 𝜂 𝜎𝑛 𝑎𝑛,2 ≤ 1.

1−𝜂

(5.7)

𝑛 =2

and ∞

𝑛=2

By using Cauchy-Schwarz inequality ∞

2𝑛 − 1 − 𝛾

1

2𝑛 − 1 − 𝜂

2

1

2

1−𝛾 1−𝜂

𝑛 =2

𝜎𝑛

𝑎𝑛,1 𝑎𝑛,2 ≤ 1,

(5.8)

thus, to find largest 𝜉 such that ∞

𝑛 =2

2𝑛 − 1 − 𝜉 𝜎𝑛 𝑎𝑛,1 𝑎𝑛,2 ≤ 1−𝜉

𝑎𝑛,1 𝑎𝑛,2 ≤

2𝑛−1−𝛾

1

∞

2𝑛 − 1 − 𝛾

1−𝛾 1−𝜂

2

2𝑛 − 1 − 𝜂

1−𝛾 1−𝜂

𝑛=2

2 2𝑛−1−𝜂

1

1

1

2

𝜎𝑛

𝑎𝑛,1 𝑎𝑛,2 ≤ 1,

2 (1−𝜉 )

2𝑛−1−𝜉

, for 𝑛 ≥ 2

(5.9)

, for 𝑛 ≥ 2

(5.10)

then (5.8) reduces to, 𝑎𝑛,1 𝑎𝑛,2 ≤

1−𝛾 1−𝜂 2𝑛−1−𝛾

1

2 2𝑛−1−𝜂

1

2 𝜎𝑛

it is sufficient to find the largest 𝜎 such that 1−𝛾 1−𝜂 1

1

≤

2𝑛 − 1 − 𝛾

1

2

2𝑛 − 1 − 𝜂

1−𝛾 1−𝜂 2𝑛 − 1 − 𝛾 2𝑛 − 1 − 𝜂 𝜎𝑛 2𝑛 − 1 − 𝜉 2𝑛 − 1 − 𝛾 2𝑛 − 1 − 𝜂 𝜎𝑛 ≤ , 1−𝜉 1−𝛾 1−𝜂 2

2

1

2

(1 − 𝜉)

2𝑛 − 1 − 𝜉

, for 𝑛 ≥ 2

(5.11)

𝜉 2𝑛 − 1 − 𝛾 2𝑛 − 1 − 𝜂 𝜎𝑛 − 1 − 𝛾 1 − 𝜂 ≤ 2𝑛 − 1 − 𝛾 2𝑛 − 1 − 𝜂 𝜎𝑛 − 2𝑛 − 2 + 1 1 − 𝛾 1 − 𝜂 , ≤

2𝑛 − 1 − 𝛾 2𝑛 − 1 − 𝜂 𝜎𝑛 − 1 − 𝛾 1 − 𝜂 − 2 𝑛 − 1 1 − 𝛾 1 − 𝜂 ,

𝜉 ≤1−

2(𝑛 − 1) 1 − 𝛾 1 − 𝜂 , 2𝑛 − 1 − 𝛾 2𝑛 − 1 − 𝜂 𝜎𝑛 − 1 − 𝛾 1 − 𝜂

where s



q

Γ 𝛽𝑗

j1

𝜎𝑛 =



Γ(𝛼𝑗 + 𝐴𝑗 (𝑛 − 1))

j1

for 𝑛 ≥ 2. s

q

 j1

Γ 𝛼𝑗



Γ 𝛽𝑗 +𝐵𝑗 𝑛 − 1)

𝑛−1 !

j1

𝜎𝑛 is decreasing function of 𝑛(𝑛 ≥ 2), we get

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INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT s

q

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014;

jď&#x20AC;˝1

Î&#x201C;(đ?&#x203A;źđ?&#x2018;&#x2014; + đ??´đ?&#x2018;&#x2014; )

jď&#x20AC;˝1

0 â&#x2030;¤ đ?&#x153;&#x17D;đ?&#x2018;&#x203A; â&#x2030;¤ đ?&#x153;&#x17D;2 =

.

s

q

ď&#x192;&#x2022;

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;źđ?&#x2018;&#x2014;

jď&#x20AC;˝1

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014; +đ??ľđ?&#x2018;&#x2014;

jď&#x20AC;˝1

2 1â&#x2C6;&#x2019;đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x153;&#x201A; , 3 â&#x2C6;&#x2019; đ?&#x203A;ž 3 â&#x2C6;&#x2019; đ?&#x153;&#x201A; đ?&#x153;&#x17D;2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 1 â&#x2C6;&#x2019; đ?&#x153;&#x201A;

đ?&#x153;&#x2030; = 1â&#x2C6;&#x2019;

where đ?&#x153;&#x17D;2 is given by (5.2).This completes the proof of Theorem.

Theorem V.2. Let the functions â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014; đ?&#x2018;§ đ?&#x2018;&#x203A; , đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;,đ?&#x2018;&#x2014; â&#x2030;Ľ 0; đ??˝ = 1,2

đ?&#x2018;&#x201C;đ?&#x2018;&#x2014; đ?&#x2018;§ = đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;&#x203A; =2

be in the class đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž . Then đ?&#x2018;&#x201C;1 â&#x2C6;&#x2014; đ?&#x2018;&#x201C;2 đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x153;&#x152; , where 2 1â&#x2C6;&#x2019;đ?&#x203A;ž 2 đ?&#x153;&#x152; = 1â&#x2C6;&#x2019; 3 â&#x2C6;&#x2019; đ?&#x203A;ž 2 đ?&#x153;&#x17D;2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 2 and đ?&#x153;&#x17D;2 is given by (5.2).

Proof. if we set đ?&#x203A;ž = đ?&#x153;&#x201A; in the above Theorem, the results follows.

Theorem V.3. Let the function đ?&#x2018;&#x201C; đ?&#x2018;§ defined by (1.6) be in the class đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž) and also let â&#x2C6;&#x17E;

đ?&#x2018;?đ?&#x2018;&#x203A; đ?&#x2018;§đ?&#x2018;&#x203A; , for đ?&#x2018;?đ?&#x2018;&#x203A; â&#x2030;¤ 1 .

đ?&#x2018;&#x201D; đ?&#x2018;§ =đ?&#x2018;§â&#x2C6;&#x2019;

(5.12)

đ?&#x2018;&#x203A;=2

Then đ?&#x2018;&#x201C; â&#x2C6;&#x2014; đ?&#x2018;&#x201D; đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž .

Proof. By using (2.1), we get â&#x2C6;&#x17E;

â&#x2C6;&#x17E;

đ?&#x153;&#x17D; đ?&#x2018;&#x203A; 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; đ?&#x2018;? đ?&#x2018;&#x203A; = đ?&#x2018;&#x203A; =2

đ?&#x153;&#x17D; đ?&#x2018;&#x203A; 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; đ?&#x2018;? đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; =2

â&#x2C6;&#x17E;

đ?&#x153;&#x17D; đ?&#x2018;&#x203A; 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; ,

â&#x2030;¤ đ?&#x2018;&#x203A; =2

â&#x2030;¤ 1â&#x2C6;&#x2019;đ?&#x203A;ž , where đ?&#x153;&#x17D; đ?&#x2018;&#x203A; is given by (5.2). Hence it follows that đ?&#x2018;&#x201C; â&#x2C6;&#x2014; đ?&#x2018;&#x201D; đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž Theorem 5.4. Let the functions â&#x2C6;&#x17E;

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; ,đ?&#x2018;&#x2014; đ?&#x2018;§ đ?&#x2018;&#x203A; , đ?&#x2018;&#x2014; = 1,2

đ?&#x2018;&#x201C;đ?&#x2018;&#x2014; đ?&#x2018;§ =đ?&#x2018;§ â&#x2C6;&#x2019;

(5.13)

đ?&#x2018;&#x203A; =2

be in the class đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž ). Then the function đ?&#x2018;&#x2022; đ?&#x2018;§ defined by â&#x2C6;&#x17E;

(đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,1 + đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,2 )đ?&#x2018;§

đ?&#x2018;&#x2022; đ?&#x2018;§ =đ?&#x2018;§ â&#x2C6;&#x2019;

đ?&#x2018;&#x203A;

(5.14)

đ?&#x2018;&#x203A; =2

is in the class đ?&#x2018;&#x2021; â&#x2C6;&#x2014; Î&#x201D; , where 4(1â&#x2C6;&#x2019;đ?&#x203A;ž )2

Î&#x201D; =1â&#x2C6;&#x2019;

(5.15)

3â&#x2C6;&#x2019;đ?&#x203A;ž 2 đ?&#x153;&#x17D; 2â&#x2C6;&#x2019;2(1â&#x2C6;&#x2019;đ?&#x203A;ž )2

and đ?&#x153;&#x17D; 2 is given by (5.2).

Proof. By using theorem 2.1, it is sufficient to prove that â&#x2C6;&#x17E;

đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; =2

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; Î&#x201D; (đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,1 + đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,2 ) â&#x2030;¤ 1, 1â&#x2C6;&#x2019;Î&#x201D;

where đ?&#x2018;&#x201C; đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž đ?&#x2018;&#x2014; â&#x2C6;&#x17E;

đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; =2

(5.16)

đ?&#x2018;&#x2014; = 1,2 . Then

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; ,1 â&#x2030;¤ 1, 1â&#x2C6;&#x2019;đ?&#x203A;ž

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INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT â&#x2C6;&#x17E;

đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; =2

And

â&#x2C6;&#x17E;

đ?&#x2018;&#x203A; =2 đ?&#x153;&#x17D; đ?&#x2018;&#x203A;

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1â&#x2C6;&#x2019;đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;ž â&#x2C6;&#x17E;

đ?&#x2018;&#x203A; =2

(5.17)

đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,2 â&#x2030;¤ 1,

(5.18)

2

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;ž

đ?&#x2018;&#x203A; =2 â&#x2C6;&#x17E;

đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,1 â&#x2030;¤ 1,

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; ,2 â&#x2030;¤ 1, đ?&#x153;&#x17D;đ?&#x2018;&#x203A;

then

2

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;ž

1 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D;đ?&#x2018;&#x203A; 2 1â&#x2C6;&#x2019;đ?&#x203A;ž

2

đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,1 + đ?&#x2018;&#x17D; 2đ?&#x2018;&#x203A; ,2 â&#x2030;¤ 1.

(5.19)

By comparing (5.16) and (5.19), we obtain 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; Î&#x201D; 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž = đ?&#x153;&#x17D;đ?&#x2018;&#x203A; 1â&#x2C6;&#x2019;Î&#x201D; 1â&#x2C6;&#x2019;đ?&#x203A;ž

2

2

,

therefore, Î&#x201D; 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž Î&#x201D; =1â&#x2C6;&#x2019;

2

2

đ?&#x153;&#x17D;đ?&#x2018;&#x203A; â&#x2C6;&#x2019;2 1â&#x2C6;&#x2019;đ?&#x203A;ž

= 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž

4(đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1) 1 â&#x2C6;&#x2019; đ?&#x203A;ž 2 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 2 đ?&#x153;&#x17D; đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 2 1 â&#x2C6;&#x2019; đ?&#x203A;ž

4 1â&#x2C6;&#x2019;đ?&#x203A;ž 2 3 â&#x2C6;&#x2019; đ?&#x203A;ž 2đ?&#x153;&#x17D; 2 â&#x2C6;&#x2019; 2 1 â&#x2C6;&#x2019; đ?&#x203A;ž which proves the Theorem. Î&#x201D; =1â&#x2C6;&#x2019;

2

2

2

đ?&#x153;&#x17D; đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 2 2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 1 â&#x2C6;&#x2019; đ?&#x203A;ž

2

,

.

, đ?&#x2018;&#x203A; â&#x2030;Ľ 2,

VI. INTEGRAL TRANSFORM OF THE CLASS đ?&#x2018;ť â&#x2C6;&#x2014; (đ?&#x153;¸ ) Let the function đ?&#x2018;&#x201C; đ?&#x2018;§ â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž ). Then The integral transform 1 đ?&#x2018;&#x201C; đ?&#x2018;Ąđ?&#x2018;§ đ?&#x2018;&#x2030;đ?&#x153;&#x2020; đ?&#x2018;&#x201C; đ?&#x2018;§ = đ?&#x153;&#x2020; đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą , đ?&#x2018;Ą 0 where đ?&#x153;&#x2020; is a real valued, non negative weight function normalized so that

6.1

1

đ?&#x153;&#x2020; đ?&#x2018;Ą đ?&#x2018;&#x2018;đ?&#x2018;Ą = 1.

6.2

0

Since special cases of đ?&#x153;&#x2020; (đ?&#x2018;Ą ) are particularly interesting such as đ?&#x153;&#x2020; đ?&#x2018;Ą = 1 + đ?&#x2018;? đ?&#x2018;Ą đ?&#x2018;? , đ?&#x2018;? > â&#x2C6;&#x2019;1 , for which đ?&#x2018;&#x2030; đ?&#x153;&#x2020; is known as Bernadi operator and 1+đ?&#x2018;? đ?&#x203A;ż đ?&#x2018;?

1 đ?&#x203A;ż â&#x2C6;&#x2019;1 đ?&#x153;&#x2020; đ?&#x2018;Ą = đ?&#x2018;Ą log , đ?&#x2018;? > â&#x2C6;&#x2019;1, đ?&#x203A;ż â&#x2030;Ľ 0 Î&#x201C;đ?&#x203A;ż đ?&#x2018;Ą this gives the Komatu operator cf. 12 .

ThoremVI.1. Let the function đ?&#x2018;&#x201C; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž , then đ?&#x2018;&#x2030; đ?&#x153;&#x2020; (đ?&#x2018;&#x201C; ) â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; (đ?&#x203A;ž ).

Proof. By using (6.1) and (6.4), we have

đ?&#x2018;&#x2030;đ?&#x153;&#x2020;

đ?&#x203A;ż

đ?&#x2018;? +1 đ?&#x2018;&#x201C; = Î&#x201C; đ?&#x203A;ż

=

đ?&#x2018;? +1

đ?&#x203A;ż â&#x2C6;&#x2019;1 đ?&#x2018;Ą đ?&#x2018;?

(log đ?&#x2018;Ą

)đ?&#x203A;ż â&#x2C6;&#x2019;1

đ?&#x2018;§ â&#x2C6;&#x2019;

0

â&#x2C6;&#x2019;1

đ?&#x2018;&#x2030;đ?&#x153;&#x2020; đ?&#x2018;&#x201C; = đ?&#x2018;§ â&#x2C6;&#x2019; đ?&#x2018;&#x203A; =2

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;§ đ?&#x2018;&#x203A;đ?&#x2018;Ą

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A;đ?&#x2018;§ đ?&#x2018;&#x203A;đ?&#x2018;Ą

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

đ?&#x2018;&#x2018;đ?&#x2018;Ą ,

đ?&#x2018;&#x203A; =2 đ?&#x203A;ż â&#x2C6;&#x2019;1

1

â&#x2C6;&#x17E;

đ?&#x2018;?

đ?&#x2018;Ą (log đ?&#x2018;Ą

Î&#x201C;đ?&#x203A;ż â&#x2C6;&#x17E;

6.4

â&#x2C6;&#x17E;

1

â&#x2C6;&#x2019;1

đ?&#x203A;ż

6.3

)đ?&#x203A;ż â&#x2C6;&#x2019;1

đ?&#x2018;§ â&#x2C6;&#x2019;

0

đ?&#x2018;? +1 đ?&#x2018;? +đ?&#x2018;&#x203A;

,

đ?&#x2018;&#x203A; =2

đ?&#x203A;ż

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;§ đ?&#x2018;&#x203A; ,

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(6.5)

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INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT by (2.1), đ?&#x2018;&#x201C; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž , if and only if s

â&#x2C6;&#x17E;

đ?&#x2018;&#x203A; =2

q

ď&#x192;&#x2022;

Î&#x201C;(đ?&#x203A;ź đ?&#x2018;&#x2014; + đ??´ đ?&#x2018;&#x2014; (đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1))

jď&#x20AC;˝1

s

q

ď&#x192;&#x2022;

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;źđ?&#x2018;&#x2014;

Î&#x201C; đ?&#x203A;˝ đ?&#x2018;&#x2014; +đ??ľ đ?&#x2018;&#x2014; đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1)

đ?&#x2018;? +1 đ?&#x2018;? +đ?&#x2018;&#x203A;

đ?&#x203A;ż

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; â&#x2030;¤ 1,

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1 !

jď&#x20AC;˝1

jď&#x20AC;˝1

since

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014;

jď&#x20AC;˝1

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;ž

đ?&#x2018;? +1 < 1, therefore, đ?&#x2018;? +đ?&#x2018;&#x203A; s

ď&#x192;&#x2022;

â&#x2C6;&#x17E;

đ?&#x2018;&#x203A; =2

q

Î&#x201C; đ?&#x203A;˝đ?&#x2018;&#x2014;

jď&#x20AC;˝1

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž 1â&#x2C6;&#x2019;đ?&#x203A;ž

ď&#x192;&#x2022; jď&#x20AC;˝1

Î&#x201C;(đ?&#x203A;ź đ?&#x2018;&#x2014; + đ??´ đ?&#x2018;&#x2014; (đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1))

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; â&#x2030;¤ 1.

s

q

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;źđ?&#x2018;&#x2014;

ď&#x192;&#x2022;

Î&#x201C; đ?&#x203A;˝ đ?&#x2018;&#x2014; +đ??ľ đ?&#x2018;&#x2014; đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1)

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1 !

jď&#x20AC;˝1

jď&#x20AC;˝1

Proof is completes.

Theorem 6.2. Let the function đ?&#x2018;&#x201C; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž . Then đ?&#x2018;&#x2030; đ?&#x153;&#x2020; (đ?&#x2018;&#x201C; ) is starlike of order đ?&#x153; 0 â&#x2030;¤ đ?&#x153; < 1 in đ?&#x2018;§ < đ?&#x2018;&#x2026; 1 , where

đ?&#x2018;? +đ?&#x2018;&#x203A; đ?&#x2018;&#x2026; 1 = inf đ?&#x2018;&#x203A; đ?&#x2018;? +1

đ?&#x203A;ż

1â&#x2C6;&#x2019;đ?&#x153;

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D; đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; â&#x2C6;&#x2019;đ?&#x153; 1â&#x2C6;&#x2019;đ?&#x203A;ž

1

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

(6.6)

and đ?&#x153;&#x17D; đ?&#x2018;&#x203A; is given by (1.11).

Proof. It suffices to prove that đ?&#x2018;§ đ?&#x2018;&#x2030; đ?&#x153;&#x2020; đ?&#x2018;&#x201C; (đ?&#x2018;§ ) â&#x20AC;˛ â&#x2C6;&#x2019;1 <1â&#x2C6;&#x2019;đ?&#x153; , đ?&#x2018;&#x2030; đ?&#x153;&#x2020; đ?&#x2018;&#x201C; (đ?&#x2018;§ ) ď&#x201A;Ľ

â&#x2C6;&#x2019;

ď&#x192;Ľ

đ?&#x2018;? +1 đ?&#x2018;? +đ?&#x2018;&#x203A;

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

n ď&#x20AC;˝2

therefore

ď&#x192;Ľ

đ?&#x2018;? +1 đ?&#x2018;? +đ?&#x2018;&#x203A;

n ď&#x20AC;˝2

â&#x2C6;&#x17E;

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1 đ?&#x2018;&#x203A; =2

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

đ?&#x203A;ż

â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;§

đ?&#x2018;&#x203A; =2

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

â&#x2030;¤1â&#x2C6;&#x2019;đ?&#x153; â&#x2C6;&#x2019; 1â&#x2C6;&#x2019;đ?&#x153; đ?&#x2018;&#x203A; =2

đ?&#x2018;? +đ?&#x2018;&#x203A; đ?&#x2018;? +1

â&#x2030;¤

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; đ?&#x2018;§ đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

đ?&#x203A;ż

đ?&#x2018;? +1 đ?&#x2018;? +đ?&#x2018;&#x203A;

â&#x2C6;&#x17E;

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;§

đ?&#x2018;&#x17D; đ?&#x2018;&#x203A; đ?&#x2018;§ đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1 < 1â&#x2C6;&#x2019;đ?&#x153; ,

ď&#x201A;Ľ

1â&#x2C6;&#x2019;

đ?&#x203A;ż

(6.7)

đ?&#x203A;ż

1â&#x2C6;&#x2019;đ?&#x153; , đ?&#x2018;&#x203A; â&#x2C6;&#x2019;đ?&#x153;

đ?&#x2018;? +1 đ?&#x2018;? +đ?&#x2018;&#x203A;

đ?&#x203A;ż

đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;§

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1 ,

6.8

(6.8) is true if

đ?&#x2018;§ â&#x2030;¤

đ?&#x2018;? +đ?&#x2018;&#x203A; đ?&#x2018;? +1

đ?&#x203A;ż

1â&#x2C6;&#x2019;đ?&#x153;

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D; đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; â&#x2C6;&#x2019;đ?&#x153; 1â&#x2C6;&#x2019;đ?&#x203A;ž

1

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1 , đ?&#x2018;&#x203A;

â&#x2030;Ľ 2,

6.9

where đ?&#x153;&#x17D; đ?&#x2018;&#x203A; is defined by (1.11).

Proof of the Theorem is completed.

Theorem 6.3. Let the function đ?&#x2018;&#x201C; â&#x2C6;&#x2C6; đ?&#x2018;&#x2021; â&#x2C6;&#x2014; đ?&#x203A;ž . Then đ?&#x2018;&#x2030; đ?&#x153;&#x2020; (đ?&#x2018;&#x201C; ) is convex of order đ?&#x153; 0 â&#x2030;¤ đ?&#x153; < 1 in đ?&#x2018;§ < đ?&#x2018;&#x2026; 2 , where

đ?&#x2018;? +đ?&#x2018;&#x203A; đ?&#x2018;&#x2026; 2 = inf đ?&#x2018;&#x203A; đ?&#x2018;? +1

đ?&#x203A;ż

1â&#x2C6;&#x2019;đ?&#x153;

2đ?&#x2018;&#x203A; â&#x2C6;&#x2019; 1 â&#x2C6;&#x2019; đ?&#x203A;ž đ?&#x153;&#x17D;đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; đ?&#x2018;&#x203A; â&#x2C6;&#x2019;đ?&#x153; 1â&#x2C6;&#x2019;đ?&#x203A;ž

1

đ?&#x2018;&#x203A; â&#x2C6;&#x2019;1

,

6.10

and đ?&#x153;&#x17D; đ?&#x2018;&#x203A; is defined by (1.11).

Proof. The proof of the Theorem can be obtained as Theorem 6.2.

REFERENCES [1].

[2].

[3].

A.W. Goodman (1991): On uniformly convex functions, Ann. Polon. Math., 56, 87-92.

A.W. Goodman (1991): On uniformly starlike functions, J. Math. Anal. and Appl., 155, 364-370.

E. M. Wright (1935): The asymptotic expansion of the hyper-geometric functions I, J. London, Math. Soc., 10, 286-293

www.ijesi.org

78 | P a g e

INTEGRAL TRANSFORM OF CERTAIN SUBCLASSES OF UNIVALENT [4].

F. Ronning (1993): Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118, 189-196.

[5].

F. Ronning (1995): Integral representations for bounded starlike functions, Anal. Prolon. Math., 60, 289-297.

[6].

G. Murugusundaramoorthy, T. Rosy and M. Darus (2005): A subclass of uniformly convex functions associated with certain fractional calculus operators, 6(3), Art 86. I 1-21.

[7].

H. Silverman (1975): Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51, 109-116.

[8].

R. Bharati, R. Parvathan and A. Swaminathan (1997): On subclass of uniformly convex functions and corresponding class of starlike functions. Tamkang J. of Math., 28(1), 17-33.

[9].

R.K. Raina (2006): Certain subclasses of analytic functions with fixed argument of coefficients involving the wrightâ&#x20AC;&#x2122;s function, Tamsui Oxford J. of Math. Sci., 22(1).

[10].

Schild and H. Silverman (1975): Convolution of univalent functions with negative

[11].

coefficients, Ann. Univ. Marie curie-sklodowska sect. A, 29, 99-107.

[12].

W. Ma and D. Minda, (1992): Uniformly convex functions, Ann. Polon. Math., 57,165-175.

[13].

Y.C. Kim and F. Ronning (2001): Integral transform of certain subclasses of analytic

[14].

functions, J. Math. Anal. Appl., 258, 466-489.

www.ijesi.org

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