• No results found

10. Certain subclasses of strongly starlike and strongly convex functions defined by convolution

N/A
N/A
Protected

Academic year: 2020

Share "10. Certain subclasses of strongly starlike and strongly convex functions defined by convolution"

Copied!
8
0
0

Loading.... (view fulltext now)

Full text

(1)

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 3(2010), Pages 85-92.

CERTAIN SUBCLASSES OF STRONGLY STARLIKE AND STRONGLY CONVEX FUNCTIONS DEFINED BY

CONVOLUTION

ALKA RAO , VATSALA MATHUR

Abstract. In the present paper certain subclasses of strongly starlike and strongly convex functions defined by convolution with the generalized Hur-witz -Lerch Zeta function are investigated. Some inclusion relations are also mentioned as special cases of our main results.

1. Introduction And Preliminaries

Let𝒜 denote the class of analytic functions𝑓(𝑧) defined in the open unit disk

𝐷={𝑧∈𝐶; ∣𝑧∣<1}by

𝑓(𝑧) =𝑧+

𝑘=2

𝑎𝑘𝑧𝑘. (1.1)

If𝑔∈ 𝒜is given by

𝑔(𝑧) =𝑧+

𝑘=2 𝑏𝑘𝑧𝑘,

then the Hadamard product (or convolution)𝑓 ∗𝑔 is defined by

(𝑓∗𝑔) (𝑧) =𝑧+

𝑘=2

𝑎𝑘𝑏𝑘𝑧𝑘.

We say that a function𝑓 ∈ 𝒜is starlike of order𝛼and belongs to the class𝑆∗(𝛼),if

it satisfies the inequality:

Re (

𝑧𝑓′(𝑧)

𝑓(𝑧) )

> 𝛼 (𝑧∈𝐷; 0≤𝛼 <1). (1.2)

A function𝑓 ∈ 𝒜is called strongly starlike of order𝛼and𝑓 ∈𝑆𝑠∗(𝛽), if it satisfies

the inequality:

arg

(𝑧𝑓(𝑧)

𝑓(𝑧) )

<𝜋

2𝛽 (𝑧∈𝐷; 0< 𝛽≤1).

2000Mathematics Subject Classification. 26A33, 30C45, 33C20.

Key words and phrases. Strongly starlike and strongly convex functions, generalized Hurwitz -Lerch Zeta function, hypergeometric functions.

c

⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted March 2, 2010. Published August 4, 2010.

(2)

The class𝒦 of convex functions of order𝛼, is a subclass of𝒜where the functions

𝑓 ∈ 𝒜satisfy the inequality:

Re (

1 +𝑧𝑓

′′(𝑧)

𝑓′(𝑧)

)

> 𝛼 (𝑧∈𝐷; 0≤𝛼 <1). (1.3)

We denote by𝒦𝑐(𝛽) a class of strongly convex functions of order𝛽, if the following

inequality holds:

arg (

1 + 𝑧𝑓

′′(𝑧)

𝑓′(𝑧)

)

< 𝜋

2𝛽 (𝑧∈𝐷; 0≤𝛼 <1).

A function 𝑓(𝑧) ∈ 𝒜 is called strongly starlike of order 𝛽 and type 𝛼 (say 𝑓 ∈

𝑆𝑠∗(𝛼, 𝛽)), if it satisfies the inequality:

arg

(𝑧𝑓(𝑧) 𝑓(𝑧) −𝛼

)

< 𝜋

2𝛽 (𝑧∈𝐷; 0≤𝛼 <1,0< 𝛽≤1). (1.4)

Also, that a function𝑓(𝑧)∈ 𝒜is in the class of strongly convex functions of order

𝛽 and type𝛼(denoted by𝑓 ∈ 𝒦𝑐(𝛼, 𝛽)), if it satisfies the following inequality:

arg (

1 + 𝑧𝑓

′′(𝑧)

𝑓′(𝑧) −𝛼

)

< 𝜋

2𝛽 (𝑧∈𝐷; 0≤𝛼 <1,0< 𝛽≤1). (1.5)

It readily follows that

𝑓(𝑧)∈ 𝒦𝑐(𝛼, 𝛽) ⇐⇒ 𝑧𝑓′(𝑧)∈𝑆∗𝑠(𝛼, 𝛽)

and we note that

𝑆𝑠∗(0, 𝛽) =𝑆𝑠∗(𝛽) ;𝒦𝑐(0, 𝛽) =𝒦𝑐(𝛽)

and

𝑆𝑠∗(𝛼,1) =𝑆𝑠∗(𝛼) ;𝒦𝑐(𝛼,1) =𝒦𝑐(𝛼).

Srivastava and Attiya [13] introduced and investigated following family of linear operator which was further studied by Li [6] and Prajapat and Goyal [11]. This operator is defined in terms of the Hadamard product of two analytic functions by

𝐽𝜆,𝜇𝑓 =𝐻𝜆,𝜇∗𝑓(𝑧) (𝑧∈𝐷;𝜆∈𝐶, 𝜇∈𝐶∖𝑍0;𝑓 ∈𝐴), (1.6)

where

𝐻𝜆,𝜇= (1 +𝜇) 𝜆

[𝜑(𝑧, 𝜆, 𝜇)−𝜇𝜆] (𝑧∈𝐷) (1.7)

and𝜙is the generalized Hurwitz-Lerch Zeta function [14] defined by

𝜙(𝑧, 𝜆, 𝜇) =

𝑘=0 𝑧𝑘

(𝜇+𝑘)𝜆 (1.8)

(𝜆∈𝐶, 𝜇∈𝐶/𝑍0 𝑤ℎ𝑒𝑛 ∣𝑧∣<1,Re(𝜆)>1𝑤ℎ𝑒𝑛 ∣𝑧∣= 1).

The function𝐽𝜆,𝜇𝑓(𝑧) is also in the class𝒜, since by using (1.1), we can write

𝐽𝜆,𝜇𝑓(𝑧) =𝑧+ ∞

𝑘=2

(1 +𝜇

𝑘+𝜇

)𝜆

𝑎𝑘𝑧𝑘. (1.9)

Recently, a further extension of the Srivastava Attiya operator𝐽𝜆,𝜇was introduced

and investigated by Darus and Al-Shaqsi [3] (see also Xiang et al.[15]) which is defined by

𝐽𝜆,𝜇𝑐,𝑑𝑓(𝑧) =𝑧+

𝐾=2

(1 +𝜇

𝑘+𝜇

)𝜆 𝑐! (𝑘+𝑑2) !

(𝑑−2) ! (𝑘+𝑐−1) !𝑎𝑘𝑧

(3)

(𝑧∈𝐷, 𝜆∈𝐶, 𝜇∈𝐶∖𝑍0; 𝑓 ∈𝐴; 𝑐 >−1 𝑎𝑛𝑑 𝑑 >0).

It may be noted here that the operator 𝐽𝜆,𝜇𝑐,𝑑 contains the known Choi-Saigo-Srivastava operator [2], the Choi-Saigo-Srivastava-Attiya operator [13], the Owa and Choi-Saigo-Srivastava integral operator [9], the generalized Benardi-Libera-Livingston integral operator, the operator, closely related to the multiplier transformation studied by Flett [4] and Li [6], fractional differintegral operator studied by Patel and Mishra [10] and several other operators ( see also [1], [5] and [7] ). We now observe some special cases of the operator (1.10) which are given below.

𝐽01,𝜇,2𝑓(𝑧) =𝑓(𝑧), (1.11)

𝐽11,,02𝑓(𝑧) =𝑧+

𝑘=2

1

𝑘𝑎𝑘𝑧 𝑘 =

0 𝑓(𝑡)

𝑡 𝑑𝑡, (1.12)

𝐽11,𝑏,2𝑓(𝑧) = 𝑧+

𝑘=2

(

1+𝑏 𝑘+𝑏 )

𝑎𝑘𝑧𝑘

= 1+𝑧𝑏𝑏

𝑧 ∫

0

𝑡𝑏−1𝑓(𝑡)𝑑𝑡=𝐹(𝑓)(𝑧), 𝑏 >−1,

(1.13)

𝐽𝜆,1,21𝑓(𝑧) =𝑧+

𝑛=2

( 2

𝑘+ 1 )𝜆

𝑎k𝑧𝑘 =𝐼𝜆𝑓(𝑧), (1.14)

where (1.12) and (1.13) are the well known Libera, generalized Benardi- Libera-Livingston integral operators and (1.14) represents the operator closely related to the multiplier transformation studied by Flett [4].

Using (1.10), it is easy to show that

𝑧(𝐽𝜆,𝜇𝑐+1,𝑑𝑓)′(𝑧) = (𝑐+ 1)𝐽𝜆,𝜇𝑐,𝑑𝑓(𝑧)−𝑐𝐽𝜆,𝜇𝑐+1,𝑑𝑓(𝑧), (1.15)

𝑧(𝐽𝜆𝑐,𝑑+1,𝜇𝑓)′(𝑧) = (𝜇+ 1)𝐽𝜆,𝜇𝑐,𝑑𝑓(𝑧) −𝜇𝐽𝜆𝑐,𝑑+1,𝜇𝑓(𝑧), (1.16)

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧) =𝑑𝐽𝜆,𝜇𝑐,𝑑+1𝑓(𝑧)−(𝑑−1)𝐽𝜆,𝜇𝑐,𝑑𝑓(𝑧). (1.17)

Definition 1. We define a subclass of strongly starlike functions𝑆𝑠∗(𝛼, 𝛽) by

𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽) = {

𝑓 :𝑓 ∈𝐴, 𝐽𝜆,𝜇𝑐,𝑑𝑓(𝑧)∈𝑆𝑠∗(𝛼, 𝛽)𝑎𝑛𝑑

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧)

(𝐽𝜆,𝜇𝑐,𝑑𝑓)(𝑧) ∕=𝛼;𝑧∈𝐷 }

.

(1.18)

Definition 2. We define a subclass of strongly convex functions𝒦𝑐(𝛼, 𝛽) by

𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽) = {

𝑓 :𝑓 ∈𝐴, 𝐽𝜆,𝜇𝑐,𝑑𝑓(𝑧)∈ 𝒦𝑐(𝛼, 𝛽)𝑎𝑛𝑑

(𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′)′(𝑧)

(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧) ∕=𝛼; 𝑧∈𝐷

}

.

(1.19) From the above two definitions, the following relation holds:

𝑓(𝑧)∈ 𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⇐⇒𝑧𝑓′(𝑧)∈𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽). (1.20)

(4)

To establish our main results, we shall apply the following lemma:

Lemma 1[8]Let a function 𝑝(𝑧) be analytic in D with 𝑝(0) = 1 , 𝑝′(0) = 0 and

𝑝(𝑧)∕= 0 (𝑧∈𝐷). If there exists a point𝑧0∈𝐷 such that

∣arg (𝑝(𝑧))∣<𝜋

2𝛽 (∣𝑧∣<∣𝑧0∣) 𝑎𝑛𝑑 ∣arg (𝑝(𝑧0))∣=

𝜋

2𝛽 (0< 𝛽 ≤1), then

𝑧0𝑝′(𝑧0)

𝑝(𝑧0) =𝑖𝑘𝛽, where

𝑘≥1 2(𝑎+

1

𝑎)≥1 𝑤ℎ𝑒𝑛 arg (𝑝(𝑧0)) = 𝜋

2𝛽,

𝑘≤ −1 2(𝑎+

1

𝑎)≤ −1 𝑤ℎ𝑒𝑛 arg (𝑝(𝑧0)) =− 𝜋

2𝛽, and

(𝑝(𝑧0))1/𝛽=±𝑖𝑎 (𝑎 >0).

2. Main Results Our first main result is given as follows:

Theorem 1. Let 0≤𝛼 <1, 0 < 𝛽 ≤1, 𝜆 ∈𝐶 and min{𝜇+𝛼, 𝑐+ 1, 𝑑}>0, then

𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⊂𝑆𝑠∗(𝑐, 𝑑;𝜆+ 1.𝜇;𝛼, 𝛽). (2.1)

Proof. Following [12], let us assume that the function𝑓 belongs the class𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)

and define a function𝑝(𝑧) by

𝑝(𝑧) = 1 1 𝛼

(

𝑧(𝐽𝜆𝑐,𝑑+1,𝜇𝑓)′(𝑧) (𝐽𝜆𝑐,𝑑+1,𝜇𝑓)(𝑧) −𝛼

)

(𝑧∈𝐷). (2.2)

The function (2.2) is analytic in the unit disk𝐷 and𝑝(0) = 1. Differentiation and using (1.16), we find that

1 1𝛼

(

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧)

(𝐽𝜆,𝜇𝑐,𝑑𝑓)(𝑧) −𝛼 )

=𝑝(𝑧) + 𝑧𝑝

(𝑧)

𝜇+𝛼+ (1−𝛼)𝑝(𝑧). (2.3)

Hence, the relations (2.2) and (2.3) imply that𝑝(𝑧)∕= 0 and∣arg(𝑝(𝑧))∣< 𝜋2𝛽for 0< 𝛽≤1 in𝐷. Otherwise, there exists a𝑧0 in𝐷 where the function 𝑝(𝑧) satisfies the conditions of Lemma 1 and in the case when arg(𝑝(𝑧0)) =𝜋2𝛽 and (𝑝(𝑧0))1/𝛽= 𝑖𝑎, we get

arg (

1 (1𝛼)

(

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧

0)

(𝐽𝑐,𝑑𝜆,𝜇𝑓)(𝑧0)

−𝛼

))

= arg(𝑝(𝑧0)) + arg

(

1 + 𝑧𝑝′(𝑧0)/𝑝(𝑧0)

𝜇+𝛼+(1𝛼)𝑝(𝑧0)

)

= 𝜋

2𝛽+ tan

1( 𝑘𝛽(𝜇+𝛼+(1𝛼)𝑎𝛽cos𝜋𝛽2 )

(𝜇+𝛼)2+(1𝛼)2𝑎2𝛽+2(𝜇+𝛼)(1𝛼)𝑎𝛽cos𝜋𝛽

2 +𝑘𝛽(1𝛼)𝑎𝛽sin 𝜋𝛽

2

)

≥ 𝜋 2𝛽

(

𝑘≥ 1 2(𝑎+

1

𝑎)≥1 𝑎𝑛𝑑 0< 𝛽≤1 )

.

(2.4)

But this leads to a contradiction, as𝑓 ∈𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽).

On the same lines, we can show that when arg(𝑝(𝑧0)) =−𝜋

2𝛽 and (𝑝(𝑧0))

1/𝛽=𝑖𝑎,

arg (

1 (1𝛼)

(

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧0)

(𝐽𝜆,𝜇𝑐,𝑑𝑓)(𝑧0) −𝛼 ))

≤ −𝜋 2𝛽

(

𝑘≤ −1 2(𝑎+

1

𝑎)≤ −1 𝑎𝑛𝑑 0< 𝛽≤1

)

(5)

This is again a contradiction to the fact that 𝑓 ∈ 𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽). Therefore,

∣arg(𝑝(𝑧))∣< 𝜋2𝛽 in 𝐷 and 𝑓 ∈𝑆∗𝑠(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽),which completes the proof

of Theorem 1.

Theorem 2. Let0≤𝛼 <1, 0< 𝛽≤1𝜆∈𝐶 andmin{𝜇+𝛼, 𝑐+ 1, 𝑑}>0, then 𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⊂ 𝒦𝑐(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽).

Proof. In view of Theorem 1 and relation (1.20), we find that

𝑓(𝑧)∈ 𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽) =⇒𝑧𝑓′(𝑧)∈𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)

=⇒𝑧𝑓′(𝑧)∈𝑆𝑠∗(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽)

=⇒𝑓(𝑧)∈ 𝒦𝑐(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽).

This completes the proof.

Remark 1. Above two theorems imply the following inclusions:

𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⊂𝑆𝑠∗(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽)...⊂𝑆𝑠∗(𝑐, 𝑑;𝜆+𝑛, 𝜇;𝛼, 𝛽),

𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⊂ 𝒦𝑐(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽)...⊂ 𝒦𝑐(𝑐, 𝑑;𝜆+𝑛, 𝜇;𝛼, 𝛽),

for𝑛∈𝑁.

Our next result follows by taking into account the relation (1.15) and is given by:

Theorem 3. Let 0 ≤ 𝛼 <1,0 < 𝛽 ≤1, 𝜆 ∈ 𝐶, 𝜇∈ 𝐶∖𝑍0− andmin{𝑐+𝛼, 𝑐+ 1, 𝑑}>0, then

𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⊂𝑆𝑠∗(𝑐+ 1, 𝑑;𝜆, 𝜇;𝛼, 𝛽).

Proof. The above inclusion can easily be proved by applying the relation (1.15) and the method followed in Theorem 1.

Theorem 4. Let 0 ≤ 𝛼 <1,0 < 𝛽 ≤1, 𝜆 ∈ 𝐶, 𝜇∈ 𝐶∖𝑍0− andmin{𝑐+𝛼, 𝑐+ 1, 𝑑}>0, then

𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽)⊂ 𝒦𝑐(𝑐+ 1, 𝑑;𝜆, 𝜇;𝛼, 𝛽).

Proof. With the help of (1.15) and Theorem 3, the result given by Theorem 4 can easily be proved by following the proof of Theorem 2.

Next, we prove the following result.

Theorem 5. Let 0≤𝛼 <1,0< 𝛽 ≤1, 𝜆∈𝐶, 𝜇∈𝐶∖𝑍0− andmin{𝑐+ 1, 𝑑, 𝑑+

𝛼−1}>0, then

𝑆𝑠∗(𝑐, 𝑑+ 1;𝜆, 𝜇;𝛼, 𝛽)⊂𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽).

Proof. First we set

𝑝(𝑧) = 1 1 𝛼

(

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝑓)′(𝑧)

(𝐽𝜆,𝜇𝑐,𝑑𝑓)(𝑧) −𝛼 )

(𝑧∈𝐷),

for𝑓 ∈𝑆𝑠∗(𝑐, 𝑑+ 1;𝜆, 𝜇;𝛼, 𝛽). Now using (1.17), we get

𝑑+𝛼−1 + (1−𝛼)𝑝(𝑧) =𝑑

(

𝐽𝜆,𝜇𝑐,𝑑+1𝑓)(𝑧) (

𝐽𝜆,𝜇𝑐,𝑑𝑓)(𝑧)

(6)

Following the steps similar to that of Theorem 1, we can show that𝑓 ∈𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽),

which establishes the desired inclusion relation.

The following results Theorems 6-8 can be established by following the methods given in [11] and [12]. Their proof-details can well be omitted here.

Theorem 6. Let 0≤𝛼 <1,0< 𝛽 ≤1, 𝜆∈𝐶, 𝜇∈𝐶∖𝑍0− andmin{𝑐+ 1, 𝑑, 𝑑+

𝛼−1}>0, then

𝒦𝑐(𝑐, 𝑑+ 1;𝜆, 𝜇;𝛼, 𝛽)⊂ 𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽).

Theorem 7. Let 𝑓 ∈𝐴, 0 ≤𝛼 <1,0 < 𝛽 ≤1, 𝜆 ∈𝐶, 𝜇∈𝐶∖𝑍0− and min{𝑏+ 1, 𝑏+𝛼, 𝑐+ 1, 𝑑}>0, and also let

(

𝑧(𝐽𝜆,𝜇𝑐,𝑑𝐹(𝑓))′(𝑧) (𝐽𝜆,𝜇𝑐,𝑑𝐹(𝑓))(𝑧)

)

∕=𝛼 (𝑧∈𝐷), then

𝑓(𝑧)∈𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽) =⇒𝐹(𝑓(𝑧))∈𝑆𝑠∗(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽),

where the operator 𝐹 is defined by (1.13).

Theorem 8. Let 𝑓 ∈ 𝐴 and let (

(𝑧(𝐽𝜆,𝜇𝑐,𝑑𝐹(𝑓))′)′(𝑧) (𝐽𝜆,𝜇𝑐,𝑑𝐹(𝑓))′(𝑧)

)

∕= 𝛼 (𝑧 ∈ 𝐷) under the

restrictions to the parameters given in Theorem 7, then

𝑓(𝑧)∈ 𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽) =⇒𝐹(𝑓(𝑧))∈ 𝒦𝑐(𝑐, 𝑑;𝜆, 𝜇;𝛼, 𝛽).

3. Special cases of main results

First we note the following consequences of Theorems 1 and 2 when𝜇= 1.

Corollary 1. Let 𝑓 ∈𝐴and𝑧(𝐽𝜆𝑐,𝑑+1,𝜇𝐼𝛾(𝑓(𝑧)))

∕=𝛼(𝐽𝜆𝑐,𝑑+1,𝜇𝐼𝛾(𝑓))(𝑧), 𝑧∈𝐷. If

arg ⎛

⎜ ⎝

𝑧(𝐽𝜆𝑐,𝑑+𝛾,𝜇𝐼𝛾𝑓(𝑧))

𝐽𝜆𝑐,𝑑+𝛾,𝜇𝐼𝛾𝑓(𝑧) −𝛼 ⎞

⎟ ⎠

< 𝜋

2𝛽, (0≤𝛼 <1, 0< 𝛽≤1, 𝛾 >0)

then𝐼𝛾𝑓(𝑧)𝑆

𝑠(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽), where𝐼𝛾 is the operator (1.14).

Corollary 2. Let𝑓 ∈𝐴and (

𝑧(𝐽𝜆𝑐,𝑑+1,𝜇𝐼𝛾(𝑓(𝑧)))′ )′

∕=𝛼(𝐽𝜆𝑐,𝑑+1,𝜇𝐼𝛾(𝑓))′(𝑧), (𝑧

𝐷). If

arg ⎛

⎜ ⎜ ⎜ ⎝

(

𝑧(𝐽𝜆𝑐,𝑑+𝛾,𝜇𝐼𝛾𝑓(𝑧))′ )′

(

𝐽𝜆𝑐,𝑑+𝛾,𝜇𝐼𝛾𝑓(𝑧))′

− 𝛼

⎟ ⎟ ⎟ ⎠

< 𝜋

2𝛽, (0≤𝛼 <1, 0< 𝛽 ≤1, 𝛾 >0)

then𝐼𝛾𝑓(𝑧)∈ 𝒦𝑐(𝑐, 𝑑;𝜆+ 1, 𝜇;𝛼, 𝛽),where 𝐼𝛾 is the operator (1.14).

If we put 𝜆 = 𝛾 = 1 and 𝑐 = 1, 𝑑 = 2 in the above Corollaries 1 and 2, we obtain Corollaries 2 and 4 of [11] .

(7)

Corollary 3. Let 𝑓 ∈ 𝐴 and 𝑧𝑓(𝑧) ∕= (𝛼+ 1)

𝑧 ∫

0

𝑓(𝑡)𝑑𝑡; (𝑧 ∈ 𝐷). If 𝑓(𝑧)

satis-fies following conditionRe (𝑧𝑓(𝑧))∕= (𝛼+ 1) Re(

𝑧 ∫

0

𝑓(𝑡)𝑑𝑡) (0≤𝛼 <1), then

4

𝑧 𝑧 ∫

0

1

𝑢 𝑢 ∫

0

𝑓(𝑡)𝑑𝑡 𝑑𝑢∈𝑆∗(𝛼), (𝑧, 𝑢∈𝐷).

Corollary 4. Let𝑓 ∈𝐴and𝑧2𝑓(𝑧)∕= (𝛼+ 1)

(

𝑧𝑓(𝑧)−

𝑧 ∫

0 𝑓(𝑡)𝑑𝑡

)

;𝑧∈𝐷. If𝑓(𝑧)

satisfies the condition,

Re(𝑧2𝑓′(𝑧))∕= (𝛼+ 1) Re ⎛

⎝(𝛼+ 1) ⎛

⎝𝑧𝑓(𝑧)− 𝑧 ∫

0 𝑓(𝑡)𝑑𝑡

⎠ ⎞

⎠ (0≤𝛼 <1),

then

4

𝑧 𝑧 ∫

0

1

𝑢 𝑢 ∫

0

𝑓(𝑡)𝑑𝑡 𝑑𝑢∈ 𝒦(𝛼), (𝑧, 𝑢∈𝐷).

Remark 2. If we put 𝑐 = 1, 𝑑 = 2 Theorems 1, 2 , 7 and 8 would reduce to the corresponding results due to Prajapat and Goyal [11]. Further by setting the parameter𝜇= 1 , the results due to Liu [7] follow for𝜆 >0.

If𝜆= 0 in our results the corresponding results for Choi-Saigo-Srivastava operator [2] can be easily obtained.

Acknowledgments

Authors are thankful to the anonymous referee for helpful comments. Thanks are also due to Prof. R.K. Raina, India and J. K. Prajapat, Central University, Rajasthan, for their useful suggestions. The first author is thankful to UGC India for financial assistance.

References

[1] S. D. BERNARDI, Convex and starlike univalent functions, Trans. Amer. Math. Soc.,

135(1969), 429-446.

[2] J. H. CHOI, M. SAIGO AND H. M. SRIVASTAVA,Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl.,276(2002), 432-445.

[3] M. DARUS AND K. AL- SHAQSI,On subordinations for certain analytic functions associ-ated with generalized integral operator,Lobachevskii J. Math.,29(2008), 90-97.

[4] T. M. FLETT,The dual of an inequality of Hardy and Littlewood and some related inequal-ities, J. Math. Anal. Appl.,38(1972), 746-765.

[5] I. B. JUNG, Y. C. KIM and H. M. SRIVASTAVA, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl.,

176(1993), 138–147.

[6] J. L. LI,Some properties of two integral operators, Soochow J. Math.,25(1999), 91–96. [7] J. L. LIU,A linear operator and strongly starlike functions, J. Math. Soc. Japon,544(2002),

975-981.

[8] M. NUNOKAWA,On properties of non-Carath𝑒´odory functions, Proc. Japan Acad. Ser. A Math. Sci.,68(1992),152-153.

(8)

[10] J. PATEL and A. K. MISHRA,On certain subclasses of multivalent function associated with an extended fractional differintegral operator, J. Math. Anal. Appl.,332(2007), 109–122. [11] J. K. PRAJAPAT AND S.P. GOYAL, Applications of Srivastava-Attiya operator to the

classes of strongly starlike and strongly convex functions, J. Math. Ineq.,3(2009), 129-137. [12] J. K. PRAJAPAT, R.K. RAINA AND H.M. SRIVASTAVA,Some inclusion properties for

certain subclasses of strongly starlike and strongly convex functions involving a family of fractional integral operators, Integral Transform. Spec. Funct.,18(9) (2007),639-651. [13] H. M. SRIVASTAVA AND A. A. ATTIYA, An integral operator associated with the

Hurwitz-Lerch zeta function and differential subordination,Integral Transforms and Special Func-tions,18(3) (2007),639-651.

[14] H. M. SRIVASTAVA AND J. CHOI,Series Associated with the Zeta and Related Functions-Order by: relevance—pagesrelevance—pages- PreviousNext -View all, Kluwer Academic Pub-lishers, Dordrecht, Boston and London, 2001

[15] RI-GUANG XIANG, ZHI-GANG WANG, AND M. DARUS,A family of integral operators preserving subordination and superordination, Bull. Malaysian Math. Soc.,33(1)(2010), 121– 131.

Alka Rao

Department of mathematics, Maharana Pratap Government P.G. College, Chittorgarh, Rajasthan, INDIA-312001

E-mail address:: alkarao.ar@rediffmail.com

Vatsala Mathur

Department of mathematics, Malaviya National Institute of Technology, Jaipur, Rajasthan, INDIA

References

Related documents

Efficacy and safety of short duration azithromycin eye drops versus azithromycin single oral dose for the treatment of trachoma in children: a randomised,

FIGURE 6: TEM IMAGES OBTAINED FROM POLYMERIZED TUBULIN IN THE PRESENCE OF DIFFERENT CONCENTRATIONS OF BANEH EXTRACT AND CONTROL CHEMICALS (COLCHICINE AND

Stimulation of marma points present in head region can prove to be of great prophylactic assist in managing migraine pain.. There are various aspects of this

Six blends were prepared and 3 batches of tablets of each formulation code blend were formulated and evaluated for pre-compression parameters like

Diabetes induced neuropathic pain is recognized as one of the most difficult type of pain to treat and conventional analgesics are well known to be partially effective or

Both the proposed model and the analysis of the EE in this study take into account (i) the path losses, fading, and shadowing that affect the received signal at the UE within the

In this paper we carried out an exploratory study with object-oriented systems that applies design patterns to (i) investigate if design patterns avoid the bad smells emergence

This piece was constructed from seven short musical passages that explore Antoni Gaudí’s exterior design and his morphological ideas that can be found in Casa Batlló. This