Journal of Complex Analysis
Volume 2018 (2018), Article ID 5915864, 7 pages
https://doi.org/10.1155/2018/5915864

Research Article

Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function

Xiao-Yuan Wang,¹ Lei Shi,² and Zhi-Ren Wang¹

¹College of Science, Yanshan University, Hebei, Qinhuangdao 066004, China
²School of Mathematics and Statistics, Anyang Normal University, Henan, Anyang 455000, China

Correspondence should be addressed to Xiao-Yuan Wang; mewangxiaoyuan@163.com

Received 16 August 2017; Accepted 6 March 2018; Published 8 April 2018

Academic Editor: Pranay Goswami

Copyright © 2018 Xiao-Yuan Wang et al. 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.

Abstract.

The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator involving the Hurwitz–Lerch Zeta function. We make some applications of the operator for meromorphic functions.

1. Introduction

Denote by the class of functions analytic in the unite diskof the form and let

For two functions and to be analytic in , is said to be subordinate to in and written by if there exists a Schwarz function , which is analytic in , withsuch that It is generally known that Furthermore, if the function is univalent in , then

Denote by the set of functions that are analytic and univalent on , where are such that for . Furthermore, let

Denote by the class of functions of the formwhich are analytic in the punctured unit disk

We recall the general Hurwitz–Lerch Zeta function (see, e.g., [1, p. 121] and [2, p. 194]) defined bywhere

In recent years, the general Hurwitz–Lerch Zeta function was investigated by many researchers. A huge amount of interesting properties and consequences can be found in, for example, Choi and Srivastava [3], Garg et al. [4], Lin and Srivastava [5], and Srivastava et al. [6].

In 2007, by involving the general Hurwitz–Lerch Zeta function , Srivastava and Attiya [7] (also see [8–11]) introduced the integral operator

Analogous to abovementioned operator , Wang and Shi [12] introduced a new integral operatordefined bywhereand “” denotes the Hadamard product.

From (10), (12), (16), and (17), we easily find thatIt is true that , the integral operator defined as

We can deduce that

We also see thatFurthermore, by (18), we observe thatOperator (23) was introduced and studied by Alhindi and Darus [13]; operators (24) and (25) were introduced by Lashin [14].

The main purpose of this paper is to derive some third-order differential subordination, differential superordination properties, and sandwich-type theorems of the integral operator .

2. Preliminary Results

We will investigate our main results by using following definitions and lemmas.

Definition 1 (see [15, p. 440, Definition  1]). Suppose that , , and are univalent in . If is analytic in and satisfies the third-order differential subordinationthen is called a solution of the differential subordination. is called a dominant of the solutions of the differential subordination or more simply a dominant if for all satisfying (28). A dominant that satisfies for all dominants of (28), is called the best dominant of (28).

As the second-order differential superordinations were introduced and investigated by Miller and Mocanu [16], Tang et al. [17] introduced the following third-order differential superordinations.

Definition 2 (see [17, p. 3, Definition 5]). Suppose that and the function is analytic in . If the functions and are univalent in and satisfy the third-order differential superordinationthen is called a solution of the differential superordination. An analytic function is called a subordinant of the solutions of the differential superordination or more simply a subordinant if satisfies (31) for satisfying (31). A univalent subordinant that satisfies for all superordinants of (31) is said to be the best superordinant.

Lemma 3 (see [18, p. 132], [19, p. 190]). Suppose that is univalent in the open unit disk and and are analytic in a domain containing with when . Set and . Suppose that(1) is star-like in ;(2) If for some with andthen and is the best dominant.

Lemma 4 (see [20, p. 332]). Suppose that is univalent in the open unit disk and and are analytic in a domain containing . Set . Suppose that (1) is star-like in ;(2) If , with , is univalent in , and then and is the best dominant.

Lemma 5 (see [16, p. 822]). Suppose that is univalent complex in the open unit disk and , with . If , is univalent in , and then and is the best dominant.

3. Main Results

In this section, we state several third-order differential subordination and differential superordination results associated with the operator .

Theorem 6. Suppose that the function is nonzero univalent in with andLet and . If satisfies thenand is the best dominant in (39). When the left hand side expressions in (39) are interpreted as 1.

Proof. Suppose thatThen is analytic in . Logarithmically differentiating both sides of (40) with respect to , we haveTo apply Lemma 3, we set By means of (36) we see that is univalent star-like in . Since , we furthermore get thatBy a routine calculation using (40) and (41) we find that Therefore, hypothesis (38) is equivalently written as We know that condition (33) is also satisfied. From an application of Lemma 3, we have Thus, we get the assertions in (39). Thus, the proof of Theorem 6 is completed.

Theorem 7. Suppose that the function is a univalent mapping of into the right half plane with andLet and , satisfy Ifwherethenand is the best dominant in (51). When , the left hand side expression of (51) is interpreted as 1.

Proof. Suppose that the function is defined by (40). If set we easily get By virtue of (41), hypothesis (49) can be rewritten as Therefore, by making use of Lemma 3, we derive that Thus, the assertion in (49) follows. The proof of Theorem 7 is completed.

Theorem 8. Suppose that the function is a univalent mapping of into the right half plane with and satisfies conditionLet , , and satisfy Let function be univalent in , where is defined by (50). Ifthenand is the best subordinant in (59). When , the left hand side expressions of (59) are interpreted as 1.

Proof. By putting obviously, is star-like in and Suppose that function is defined by (40). By simple calculation, from (41), we know that Hence, condition (58) can be equivalently written as Therefore, by Lemma 4, we have and is the best subordinant. The proof of Theorem 8 is completed.

Theorem 9. Suppose that , , the function is univalent in , andLet satisfy Denote byIfthenand is the best dominant in (69). When , the left side hand expressions of (69) are interpreted as 1.

Proof. Suppose that function is defined by (40). Making using of (41), we haveTherefore, by putting obviously, is star-like in and Furthermore, by substituting the expression for from (40) and (70), respectively, we get where is given by (67). Hypothesis (68) can be equivalently written as From Lemma 3, we get Thus, we get assertion (69) of Theorem 9.

Theorem 10. Suppose that ; function is univalent in with . Let function satisfy If defined by (67) is univalent and satisfiesthenand is the best subordinant in (78). When , the left hand side expressions of (78) are interpreted as 1.

Proof. Suppose that function is defined by (40). From (41), we get Hypothesis (77) can be rewritten as Then, combining Lemma 5 with , we have (78). Theorem 10 follows immediately.

Following that, we display some sandwich-type theorems associated with the operator .

Theorem 11. Suppose that functions are univalent mapping of into the right half plane and satisfy conditions Let , , and satisfy If function is given by (50) and satisfies thenwhere and are, respectively, the best subordinant and the best dominant in (84).

Combining Theorems 9 and 10, we get the following result.

Corollary 12. Suppose that , , and with Functions and are univalent convex in with Let satisfy If function is given by (67) and satisfies thenwhere and are, respectively, the best subordinant and the best dominant in (87).

4. Conclusions

In the present paper, making use of the integral operator involving the Hurwitz–Lerch Zeta function, we have derived several third-order differential subordination and differential superordination consequences of meromorphic functions in the punctured unit disk. Furthermore, the sandwich-type theorems are considered. These subordinate relationships have shown the upper and lower bounds of the operator in the punctured unit disk.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of the paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant no. 11301008, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Foundation of Educational Committee of Henan Province under Grant no. 17A110014.

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