#### Abstract

We introduce an integrodifferential operator () which plays an important role in the Geometric Function Theory. Some theorems in differential subordination for () are used. Applications in Analytic Number Theory are also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function.

#### 1. Introduction

Let denote the class of functions normalized by which are analytic in the open unit disc .

Also, let denote the class of analytic functions in the form

We begin by recalling that a general Hurwitz-Lerch Zeta function defined by (cf., e.g., [1, P. 121 et seq.]) , when , when

which contains important functions of Analytic Number Theory, as the Polylogarithmic function:

Several properties of can be found in the recent papers, for example Choi et al. , Ferreira and López , Gupta et al. , and Luo and Srivastava . See, also .

Recently, Srivastava and Attiya  introduced the operator which makes a connection between Geometric Function Theory and Analytic Number Theory, defined by where and denotes the Hadamard product (or convolution).

Furthermore, Srivastava and Attiya  showed that As special cases of , Srivastava and Attiya  introduced the following identities: where, the operators and are the integral operators introduced earlier by Alexander  and Libera , respectively, is the generalized Bernardi operator, introduced by Bernardi , and is the Jung-Kim-Srivastava integral operator introduced by Jung et al. .

Moreover, in , Srivastava and Attiya defined the operator for , by using the following relationship:

Some applications of the operator to certain classes in Geometric Function Theory can be found in [21, 22].

In our investigations we need the following definitions and lemma.

Definition 1.1. Let and be analytic functions. The function is said to be subordinate to , written , if there exists a function analytic in , with and , and such that . If is univalent, then if and only if and .

Definition 1.2. Let be analytic in domain , and let be univalent in . If is analytic in with when , then we say that satisfies a first order differential subordination if The univalent function is called dominant of the differential subordination (1.10), if for all satisfying (1.10), if for all dominant of (1.10), then we say that is the best dominant of (1.10).

Lemma 1.3 (see ). If and , then

The purpose of the present paper is to extend the use of as integrodifferential operator, and some theorems in differential subordination for are used. Applications in Analytic Number Theory are also obtained which give new results for Hurwitz-Lerch Zeta function and Polylogarithmic function.

#### 2. Making Use of as a Differential Operator

From the definition of in (1.5) and using (1.7), we obtain the following identities.

For and , we have where is the Sălăgean differential operator which introduced by Sălăgean , is the generalized of operator, introduced by Al-Oboudi , was studied by Cho and Srivastava  and by Cho and Kim , and the operator was studied by Uralegaddi and Somanatha .

Also, we note that where is the Polylogarithmic function defined by (1.4).

Now, we prove the following lemma.

Lemma 2.1. If and , then where to -times, and denotes the composition .

Proof. Putting in (1.11), we have therefore, Noting that the relation (2.5) is a recurrence relation, by using mathematical induction, we get (2.3), which completes the proof of the lemma.

Putting in Lemma 2.1, we obtain the following properties for both Hurwitz-Lerch Zeta function and Polylogarithmic function .

Corollary 2.2. Let and be the Hurwitz-Lerch Zeta function and Polylogarithmic function defined by (1.3) and (1.4), respectively, then we have where and .

Example 2.3. Using Corollary 2.2, we have the following well known results for .(i). (ii). (iii). (iv). (v). (vi).

#### 3. Applications of Differential Subordination for

To prove our results, we need the following lemmas due to Hallenbeck and Ruscheweyh  and Miller and Mocanu , respectively, see also Miller and Mocanu .

Lemma 3.1. Let be convex univalent in , with and If and then where
The function is convex univalent and is the best dominant.

Lemma 3.2. Let , and let be the root of the equation as follows:
In addition, let , for .
If and then

Now, we define the function as the following:

Theorem 3.3. Let the function defined by (3.7) and for some . If then

The constant is the best estimate.

Proof. Defining the function , then we have .
If we take , and the convex univalent function defined by then, we have Using Lemma 1.3 and (3.7), therefore (3.11) can be written as then, where is defined by (3.10) satisfying .
Applying Lemma 3.1, we obtain that , where the convex univalent function defined by Since and , we have .
This implies that Hence, the constant cannot be replace by any larger one.
This completes the proof of Theorem 3.3.

Theorem 3.4. Let the function with ; real, defined by (3.7), and let satisfy the following equation:
If then

Proof. Defining the function , then we have Using Lemma 1.3 and (3.7), therefore (3.11) can be written as This completes the proof of Theorem 3.4 after applying Lemma 3.2

#### 4. Applications in Analytic Number Theory

Putting in Theorem 3.3, then we have the following property of Hurwitz-Lerch Zeta function.

Corollary 4.1. Let the function defined by (1.6). If then where and .

The constant is the best estimate.

Putting in Theorem 3.4, then we have another property of Hurwitz-Lerch Zeta function.

Corollary 4.2. Let the function defined by (1.6), and let satisfy the following equation:
If then where and ; real.

Putting and in Theorem 3.3, then we have the following property of Polylogarithmic function.

Corollary 4.3. Let the function defined by
If then where and .
The constant is the best estimate.

Putting and in Theorem 3.4, then we have the following property of Polylogarithmic function.

Corollary 4.4. Let the functions and defined by (1.6) and (4.6), respectively, and let satisfy the following:
If then where and ; real.

Setting , and in Theorem 3.3, then we have the following property of Polylogarithmic function.

Corollary 4.5. Let the function defined by (4.6).
If then where and .
The constant is the best estimate.

Taking , and in Theorem 3.4, then we have the following property of polylogarithmic function.

Corollary 4.6. Let the function defined by (4.6).
If then where and .

Corollary 4.7. Let the function defined by (4.6) as follows:
If then where and .

Proof. Let satisfy the condition (4.16). Also, putting , , and in Theorem 3.4.
Using (4.16), then we have therefore Corollary 4.5, gives Applied (4.11) again and to -times, which gives (4.17). This completes the proof of Corollary 4.7.

Finally, we can put Corollary 4.7 in the following form.

Corollary 4.8. Let the function defined by (4.6).
If then where and .

#### Acknowledgments

This research was funded by the Deanship of Scientific Research (DSR), King Abdul-Aziz University, Jeddah, Saudi Arabia, under Grant no. 103-130-D1432. The authors, therefore, acknowledge with thanks DSR technical and financial support.