Strong Differential Subordinations Obtained with New Integral Operator Defined by Polylogarithm Function
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.
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  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
Now, for , we defined the following integral operator: where , and .
We note that,(i)for and ( is any integer), the multiplier transformation was studied by Flett  and Sălăgean ;(ii)for and (), the differential operator was studied by Sălăgean ;(iii)for and ( is any integer), the operator was studied by Uralegaddi and Somanatha ;(iv)for , the multiplier transformation was studied by Jung et al. ;(v)for , the integral operator was studied by Komatu .To prove our results, we need the following definition and theorems considered by Antonino and Romaguera , Antonino , G. I. Oros and G. Oros , and Oros .
Definition 1 (see  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 ( cf ). 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  introduced the following strong differential superordinations as the dual concept of strong differential subordination.
Definition 3 (see  cf ). 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 ). 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 ). 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 ). Let with . If satisfies then .
Theorem 7 (see ). 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  and Oros , 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 .
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.
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.
The work presented here was supported by Ministry of Manpower, Sultanate of Oman.
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