Research Article | Open Access

Volume 2014 |Article ID 260198 | https://doi.org/10.1155/2014/260198

K. AL-Shaqsi, "Strong Differential Subordinations Obtained with New Integral Operator Defined by Polylogarithm Function", International Journal of Mathematics and Mathematical Sciences, vol. 2014, Article ID 260198, 6 pages, 2014. https://doi.org/10.1155/2014/260198

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

Accepted20 Feb 2014
Published06 Apr 2014

#### 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 ).

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

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

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  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.

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.

1. S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981. View at: Google Scholar | MathSciNet
2. S. S. Miller and P. T. Mocanu, “On some classes of first-order differential subordinations,” The Michigan Mathematical Journal, vol. 32, no. 2, pp. 185–195, 1985. View at: Publisher Site | Google Scholar | MathSciNet
3. S. S. Miller and P. T. Mocanu, Differential Subordination, Theory and Application, vol. 225, Marcel Dekker, New York, NY, USA, 2000. View at: MathSciNet
4. M. Lerch, “Note sur la fonction $K\left(w,x,s\right)={\sum }_{k=0}^{\infty }{e}^{2k\pi ix}$,” Acta Mathematica, vol. 11, no. 1–4, pp. 19–24, 1887. View at: Publisher Site | Google Scholar | MathSciNet
5. H. Bateman, Higher Transcendental Functions, vol. 1 of Edited by: A. Erdelyi, W. Mangnus, F. Oberhettinger, F. G. Tricomi, McGraw-Hill, New York, NY, USA, 1953.
6. S. Ponnusamy and S. Sabapathy, “Polylogarithms in the theory of univalent functions,” Results in Mathematics, vol. 30, no. 1-2, pp. 136–150, 1996. View at: Publisher Site | Google Scholar | MathSciNet
7. S. Ponnusamy, “Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane,” The Rocky Mountain Journal of Mathematics, vol. 28, no. 2, pp. 695–733, 1998. View at: Publisher Site | Google Scholar | MathSciNet
8. T. M. Flett, “The dual of an inequality of Hardy and Littlewood and some related inequalities,” Journal of Mathematical Analysis and Applications, vol. 38, pp. 746–765, 1972. View at: Google Scholar | MathSciNet
9. G. S. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983. View at: Publisher Site | Google Scholar | MathSciNet
10. B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in Analytic Function Theory, H. M. Srivastava and S. Own, Eds., pp. 371–374, World Scientific, Singapore, 1992. View at: Google Scholar | MathSciNet
11. I. B. Jung, Y. C. Kim, and H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 138–147, 1993. View at: Publisher Site | Google Scholar | MathSciNet
12. Y. Komatu, “On analytic prolongation of a family of operators,” Mathematica, vol. 32, no. 2, pp. 141–145, 1990. View at: Google Scholar | MathSciNet
13. J. A. Antonino and S. Romaguera, “Strong differential subordination to Briot-Bouquet differential equations,” Journal of Differential Equations, vol. 114, no. 1, pp. 101–105, 1994. View at: Publisher Site | Google Scholar | MathSciNet
14. J. A. Antonino, “Strong differential subordination and applications to univalency conditions,” Journal of the Korean Mathematical Society, vol. 43, no. 2, pp. 311–322, 2006. View at: Publisher Site | Google Scholar | MathSciNet
15. G. I. Oros and G. Oros, “Strong differential subordination,” Turkish Journal of Mathematics, vol. 33, no. 3, pp. 249–257, 2009. View at: Google Scholar | MathSciNet
16. G. Oros, “Strong differential superordination,” Acta Universitatis Apulensis, vol. 19, pp. 101–106, 2009. View at: Google Scholar
17. S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations,” Complex Variables. Theory and Application, vol. 48, no. 10, pp. 815–826, 2003. View at: Publisher Site | Google Scholar | MathSciNet
18. N. E. Cho, “Strong differential subordination properties for analytic functions involving the Komatu integral operator,” Boundary Value Problems, vol. 2013, article 44, 2013. View at: Publisher Site | Google Scholar | MathSciNet