Research Article | Open Access

# Strong Differential Subordinations Obtained with New Integral Operator Defined by Polylogarithm Function

**Academic Editor:**Heinrich Begehr

#### Abstract

By using the polylogarithm function, a new integral operator is introduced. Strong differential subordination and superordination properties are determined for some families of univalent functions in the open unit disk which are associated with new integral operator by investigating appropriate classes of admissible functions. New strong differential sandwich-type results are also obtained.

#### 1. Introduction

Let denote the class of analytic function in the open unit disk . For a positive integer and , let and let . We also denote by the subclass of , with the usual normalization .

Let and be formal Maclaurin series. Then, the* Hadamard product* or* convolution* of and is defined by the power series .

Let the functions and in ; then we say that is subordinate to in , and write , if there exists a Schwarz function in with and such that in . Furthermore, if the function is univalent in , then and (cf [1–3]).

Let denote the well-known generalization of the Riemann zeta and polylogarithm functions, or simply the th order polylogarithm function, given by where any term with is excluded; see Lerch [4] and also [5, Sections 1.10 and 1.12]. Using the definition of the Gamma function [5, page 27], a simply transformation produces the integral formula

Note that is Koebe function. For more details about polylogarithms in theory of univalent functions, see Ponnusamy and Sabapathy [6] and Ponnusamy [7].

Now, for , we defined the following integral operator: where , and .

We also note that the operator defined by (4) can be expressed by the series expansion as follows: Obviously, we have, for , Moreover, from (5), it follows that

We note that,(i)for and ( is any integer), the multiplier transformation was studied by Flett [8] and Sălăgean [9];(ii)for and (), the differential operator was studied by Sălăgean [9];(iii)for and ( is any integer), the operator was studied by Uralegaddi and Somanatha [10];(iv)for , the multiplier transformation was studied by Jung et al. [11];(v)for , the integral operator was studied by Komatu [12].To prove our results, we need the following definition and theorems considered by Antonino and Romaguera [13], Antonino [14], G. I. Oros and G. Oros [15], and Oros [16].

*Definition 1 (see [13] cf [14, 15]). *Let be analytic in and let be analytic and univalent in . Then, the function is said to be strongly subordinate to , or is said to be strongly superordinate to , written as , if, for , as the function of is subordinate to . We note that if and only if and .

*Definition 2 ([15] cf [1]). *Let and let be univalent in . If is analytic in and satisfies the (second-order) differential subordination
then is called a solution of the strong differential subordination. The univalent function is called a dominant of the solution of the strong differential subordination, or more simply a dominant, if for all satisfying (8). A dominant that satisfies for all dominants of (8) is said to be best dominant.

Recently, Oros [16] introduced the following strong differential superordinations as the dual concept of strong differential subordination.

*Definition 3 (see [16] cf [17]). *Let and let be analytic in . If and are univalent in for and satisfy the (second-order) strong differential superordination
then is called a solution of the strong differential superordination. An analytic function is called a subordinant of the solution of the strong differential superordination, or more simply a subordinant, if for all satisfying (9). A univalent subordinant that satisfies for all subordinantes of (9) is said to be best subordinant.

Denote by the class of function that are analytic and injective on , where and such that for . Further, let the subclass of for which be denoted by and .

*Definition 4 (see [15]). *Let be a set in and let be a positive integer. The class of admissible functions consists of those function that satisfy the admissibility condition
whenever , and
for , , , and . We write as .

*Definition 5 (see [16]). *Let be a set in and with . The class of admissible functions consists of those function that satisfy the admissibility condition
whenever , for and
for , , , and . We write as .

For the above two classes of admissible function, Oros and Oros proved the following theorems.

Theorem 6 (see [15]). *Let with . If satisfies
**
then .*

Theorem 7 (see [16]). *Let with . If and
**
is univalent in for , then
**
implies that .*

In the present paper, making use of polylogarithm function, we introduce a new integral operator. By using the differential subordination and superordination results given by G. I. Oros and G. Oros [15] and Oros [16], we determine certain classes of admissible functions and obtain some subordination and superordination implications of multivalent functions associated with the new integral operator defined by (4). New differential sandwich-type theorems are also obtained. We remark that we use the same technique given by Cho [18].

#### 2. Subordination Results

Firstly, we begin by proving the subordination theorem involving the integral operator defined by (4). For this purpose, we need the following class of admissible functions.

*Definition 8. *Let be a set in , , and . The class of admissible functions consists of those functions that satisfy the admissibility condition
whenever
for , , , and .

Theorem 9. *Let . If satisfies
**
then
*

*Proof. *Define the function in by

From (22) with the relation (7), we get
Further computations show that
Define the transformation from to by
Let
Using (22), (23), and (24), from (26), we obtain
Hence, (20) becomes
Note that
and so the admissibility condition for is equivalent to the admissibility condition for . Therefore, by Theorem 6, or
which evidently completes the proof of Theorem 9.

If is a simply connected domain, then for some conformal mapping of onto . In this case, the class is written as . The following result is an immediate consequence of Theorem 9.

Theorem 10. *Let . If satisfies
**
then
*

Our next result is an extension of Theorem 9 to the case where the behavior of on is not known.

Corollary 11. *Let and let be univalent in with . Let for some where . If satisfies
**
then
*

*Proof. *Theorem 9 yields . The result is now deduced from .

Theorem 12. *Let and be univalent in with and set and . Let satisfy one of the following conditions: *(1)*, for some ,*(2)*there exists such that for all .** If satisfies (31), then
*

*Proof. *Using the same technique given in [3, Theorem 2.3d].*Case 1*. By applying Theorem 9, we obtain . Since , we deduce that .*Case 2*. If we let , then
By using Theorem 9 and the comment associated with (20) with , we obtain , for . By letting , we obtain .

The next theorem yields the best dominant of the differential subordination.

Theorem 13. *Let be univalent in and let . Suppose that the differential equation
**
has a solution with and satisfies one of the following conditions: *(1)* and ,*(2)* is univalent in and , for some ,*(3)* is univalent in and there exists such that for all .**If satisfies (31), and
**
is analytic in , then
**
and is the best dominant.*

*Proof. *Using the same technique given in [3, Theorem 2.3e].

We deduce that is a dominant from Theorems 10 and 12. Since satisfies (37), it is also a solution of (31) and therefore will be dominated by all dominants. Hence, is the best dominant.

In the particular case , , and, in view of Definition 8, the class of admissible function , denoted by , is described below.

*Definition 14. *Let be a set in , and . The class of admissible function consists of function , such that
whenever , , and , and .

Corollary 15. *Let . If satisfies
**
then
*

In the special case , the class is simply denoted by .

Corollary 16. *Let . If satisfies
**
then
*

Corollary 17. *Let , and let be an anlaytic function in with for . If satisfies
**
then
*

*Proof. *This follows from Corollary 15 by taking and , where . To use Corollary 15, we need to show that ; that is, the admissible condition (40) is satisfied. This follows since
for , , and , and . Hence, by Corollary 15, we deduce the required results.

#### 3. Superordination and Sandwich-Type Results

The dual problem of differential subordination, that is, differential superordination of the new integral operator defined by (4), is investigated in this section. For this purpose, the class of admissible functions is given in the following definition.

*Definition 18. *Let be a set in with and . The class of admissible functions consists of those functions that satisfy the admissibility condition
whenever
for , , , and .

Theorem 19. *Let . If , and
**
is univalent in , then
**
implies that
*

*Proof. *From (27) and (51), we have
From (25), we see that the admissibility condition for is equivalent to the admissibility condition for as given in Definition 2. Hence, and, by Theorem 9, or
which evidently completes the proof of Theorem 19.

If is a simply connected domain, then for some conformal mapping of onto . In this case, the class is written as . Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 19.

Theorem 20. *Let and let be analytic in and let . If , and
**
is univalent in , then
**
implies that
*

Theorems 19 and 20 can only be used to obtain subordinantes of differential superordination of the form (51) or (56). The following theorem proves the existence of the best subordinant of (56) for certain .

Theorem 21. *Let be univalent in and let . Suppose that the differential equation
**
has a solution . If , , , and
**
is univalent in , then
**
implies that
**
and is the best subordinant.*

*Proof. *The proof is similar to that of Theorem 13 and so it is omitted.

Combining Theorems 10 and 20, we obtain the following sandwich-type theorem.

Theorem 22. *Let and be analytic functions in and let be analytic function in with and . If , and
**
is univalent in , then
**
implies that
*

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The work presented here was supported by Ministry of Manpower, Sultanate of Oman.

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#### Copyright

Copyright © 2014 K. AL-Shaqsi. 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.