Applications of differential subordinations for certain classes of p-valent functions associated with generalized Srivastava-Attiya operator

The object of the present paper is to investigate some inclusion relations and other interesting properties for certain classes of p-valent functions involving generalized Srivastava-Attiya operator by using the principle of differential subordination.

MSC: 30C45.

Let be the class of functions which are analytic and p-valent in the unit disc of the form

Let also . For , given by , the Hadamard product (or convolution) of and is defined by

Next, in the usual notation, let denote the Hurwitz-Lerch Zeta function defined as follows:

For further interesting properties and characteristics of the Hurwitz-Lerch Zeta function see [2,5,8,9,11], and [21].

Recently, Srivastava and Attiya [20] have introduced the linear operator , defined in terms of the Hadamard product by

where

The Srivastava-Attiya operator contains, among its special cases, the integral operators introduced and investigated by Alexander [1], Libera [7] and Jung et al. [6].

Analogous to , Liu [10] defined the operator by

where

and

It is easy to observe from (1.6) and (1.7) that

We note that

(i) ;

(ii) (), where the operator L was introduced by Alexander [1];

(iii) (), where the operator was introduced by Srivastava-Attiya [20];

(iv) (), where the operator was introduced by Choi et al. [3];

(v) (), where the operator was introduced by Shams et al. [18];

(vi) (), where the operator was introduced by El-Ashwah and Aouf [4];

(vii) (), where the operator was introduced by El-Ashwah and Aouf [4].

It follows from (1.8) that

For two analytic functions , we say that f is subordinate to g, written if there exists a Schwarz function , which (by definition) is analytic in U with and for all , such that . Furthermore, if the function is univalent in U, then we have the following equivalence (see [14]):

Definition 1 For fixed parameters A and B, with , we say that is in the class if it satisfies the following subordination condition:

In view of the definition of subordination (1.10) is equivalent to the following condition:

For convenience, we write , where denotes the class of functions in satisfying the inequality

In the present paper, we investigate some inclusion relations and other interesting properties for certain classes of p-valent functions involving an integral operator.

To establish our main results, we need the following lemmas.

Lemma 1 ([13,14])

Lethbe analytic and convex (univalent) inUwith. Suppose also that the functionφgiven by

is analytic inU, wheremis a positive integer. If

then

andis the best dominant of (2.2).

We denote by the class of functions given by

which are analytic in U and satisfy the following inequality:

Lemma 2 ([17])

Let the function, wheregiven by (2.4). Then

Lemma 3 ([22])

For, ,

The result is best possible.

Lemma 4 ([24])

Letμbe a positive measure on the unit interval. Letbe a complex valued function defined onsuch thatis analytic inUfor eachand such thatisμintegrable onfor all. In addition, suppose that, is real and

IfGis defined by

then

Lemma 5 ([19])

Let the functiongbe analytic inUwithand (). Then, for any functionFanalytic inU, is contained in the convex hull of.

Lemma 6 ([16])

Letφbe analytic inUwithandforand letwith, .

(i) Letandsatisfy eitheror. Ifφsatisfies

then

and this is best dominant.

(ii) Letandbe such that. Ifφsatisfies (2.5), then

and this is the best dominant.

For real or complex numbers an and c () and , the Gaussian hypergeometric function defined by

where and . We note that the series defined by (2.6) converges absolutely for , and hence, represents an analytic function in U (see, for details, [23], Ch.14]).

Lemma 7 ([23])

For real or complex numbersa, nandc ()

and

Unless otherwise mentioned, we assume throughout this paper that , , , , , m is a positive integer and the powers are understood as principle values.

Theorem 1Letfgiven by (1.1) satisfy the following subordination condition:

Then

where

is the best dominant of (3.2). Furthermore,

where

The estimate (3.4) is best possible.

Proof Let

where θ is of the form (2.1) and is analytic in U. Differentiating (3.6) with respect to z, we get

Applying Lemma 1 for and Lemma 7, we have

This proves the assertion (3.2) of Theorem 1. Next, in order to prove the assertion (3.4) of Theorem 1, it suffices to show that

Indeed, we have

Setting

which is a positive measure on the closed interval , we get

Then

Letting in the above inequality, we obtain the assertion (3.4). Finally, the estimate (3.4) is best possible as Ψ is the best dominant of (3.2). This completes the proof of Theorem 1. □

Theorem 2If (), then

where

The result is best possible.

Proof Let , then we write

where u is of the form (2.1), is analytic in U and has a positive real part in U. Differentiating (3.8) with respect to z, we have

Applying the following well-known estimate [12]:

in (3.9), we have

such that the right-hand side of (3.10) is positive, if , where R is given by (3.7).

In order to show that the bound R is best possible, we consider the function defined by

Note that

for . This completes the proof of Theorem 2. □

For a function , the generalized Bernardi-Libera-Livingston integral operator is defined by

From (1.8) and (3.11), we have

and

Theorem 3Letandbe defined by (3.11). Then

where

is the best dominant of (3.13). Furthermore,

where

The result is best possible.

Proof Let

where K is of the form (2.1) and is analytic in U. Using (3.12) in (3.15) and differentiating the resulting equation with respect to z, we have

The remaining part of the proof is similar to that of Theorem 1, and so we omit it. □

We note that

Putting () and in Theorem 3 and using (3.16), we obtain the following corollary.

Corollary 1Ifsatisfies the following inequality:

then

The result is best possible.

Theorem 4Letsatisfy the following inequality:

If

then

where

Proof Let

where is analytic in U with and . Then, by applying the familiar Schwartz Lemma [15], we have , where is analytic in U and . Therefore (3.18) leads to

Differentiating (3.19) logarithmically with respect to z, we have

Letting

where ω is in the form (2.1), is analytic in U and

then we have

Using the following known estimates [12] (see also [15]):

in (3.21), we have

which is certainly positive, provided that , where is given by (3.17). This completes the proof of Theorem 4. □

Theorem 5Let () and. If each of the functionssatisfies the following subordination condition:

then

where

and

The result is best possible when.

Proof Suppose that the functions () satisfy the condition (3.22). Then by setting

we have

And

from (3.24), (3.26) and (3.27), we have

For convenience,

Since (), it follows from Lemma 3 that

By using (3.30) in (3.29) and applying Lemmas 2 and 3, we have

When , we consider () satisfy the condition (3.22) and are defined by

By using (3.29) and applying Lemma 3, we have

This completes the proof of Theorem 5. □

Remark 1 Putting () and () in Theorem 5, we obtain the result obtained by Liu [10], Theorem 5].

Putting (), () and in Theorem 5, we obtain the following corollary.

Corollary 2Letandsatisfy the following inequality:

then

where

The result is best possible.

Theorem 6Letandsatisfy the following inequality:

then

Proof We have

where satisfies (3.31) and is convex (univalent) in U. By using (1.10) and applying Lemma 5, we complete the proof of Theorem 6. □

Theorem 7Letandsatisfy the following subordination condition:

Then

where

The result is best possible.

Proof Let

where M is of the form (2.1) and is analytic in U. Differentiating (3.33) with respect to z, we have

Now, by following steps similar to the proof of Theorem 1 and using the elementary inequality

we obtain the result asserted by Theorem 7. □

Theorem 8Letandwithand. Suppose that

Ifwithfor all, then

implies

where

is the best dominant.

Proof Let us put

Then φ is analytic in U, and for all . Taking the logarithmic derivatives in both sides of (3.34) and using the identity (1.9), we have

Now the assertions of Theorem 8 follow by using Lemma 6 for . □

Putting and , , in Theorem 8, we obtain the following corollary.

Corollary 3Assume thatsatisfies eitheror. Ifwithfor, then

implies

andis the best dominant.

Remark 2 Specializing the parameters s and b in the above results of this paper, we obtain the results for the corresponding operators , , and which are defined in the introduction.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

The authors would like to thank the referees of the paper for their helpful suggestions.

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