Open Access This article is
- freely available
Axioms 2018, 7(2), 27; doi:10.3390/axioms7020027
Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator
Department of Mathematics, Faculty of Science, Menofia University, Shebin Elkom 32511, Egypt
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Author to whom correspondence should be addressed.
Received: 19 January 2018 / Accepted: 17 April 2018 / Published: 24 April 2018
Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved.
Keywords:differential subordination; p-valent functions; generalized fractional differintegral operator
JEL Classification:30C45; 30C50
1. Introduction and Definitions
Denote by the class of analytic and p-valent functions of the form:
For functions analytic in , f is subordinate to g, written if there exists a function w, analytic in with and such that If g is univalent in then (see [1,2]):
If is analytic in and satisfies:then is a solution of (2). The univalent function q is called dominant, if for all . A dominant is called the best dominant, if for all dominants q.
Let be the well-known (Gaussian) hypergeometric function defined by:where:
We will recall some definitions that will be used in our paper.
For and in terms of the generalized fractional integral and generalized fractional derivative operators defined by Srivastava et al.  (see also ) as:where is an analytic function in a simply-connected region of the complex plane containing the origin with the order when , and the multiplicity of is removed by requiring to be real when .
We note that:where and are the fractional integral and fractional derivative operators studied by Owa .
For , we have:where “∗” stands for convolution of two power series, and is the well-known generalized hypergeometric function.
It is easy to verify that:and:
By using the operator , we introduce the following class.
For , is in the class ifwhich is equivalent to:
For convenience, we write , which satisfies the inequality:
In this paper, we investigate some subordination and convolution properties for classes of multivalent functions, which are defined by a generalized fractional differintegral operator. The theory of subordination received great attention, particularly in many subclasses of univalent and multivalent functions (see, for example, [13,15,16,17]).
To prove our main results, we shall need the following lemmas.
Denote by the class of functions given by:which are analytic in and satisfy the following inequality:
Using the well-known growth theorem for the Carathéodory functions (cf., e.g., ), we may easily deduce the following result:
. If . Then
. ForThe result is the best possible.
 Let ϕ be such that and and with
If or and satisfies:then:and this is the best dominant.
If and and if satisfies:then:and this is the best dominant.
. Let , and satisfy for all and all . If is analytic in and if:then in .
. Let be analytic in with and for all If there exist two points such that:for some and and for all then:where:
3. Properties Involving
Unless otherwise mentioned, we assume throughout this paper that , and the powers are considered principal ones.
The estimate in (13) is sharp.
Then, φ is analytic in . After some computations, we get:
Now, by using Lemma 1, we deduce that:or, equivalently,and so:
Now, for defined by (18), we have:
Letting we obtain:which ends our proof. ☐
Putting and using Lemma 1 for Equation (15) in Theorem 1, we obtain the following example.
Let the function Then, following containment property holds,
The result is sharp.
Putting in Theorem 2, we obtain the following example.
Let the function Then, following inclusion property holds
For a function the generalized Bernardi–Libera–Livingston integeral operator is defined by (see ):
If prove that:
Now, the first part of this lemma follows. Furthermore,
If we replace by and using the first part of this lemma, we get (21). ☐
Suppose that and defined by (20). Then:
The result is sharp.
Then, φ is analytic in . After some calculations, we have:
Employing the same technique that was used in proving Theorem 1, the remaining part of the theorem can be proven. ☐
Let If each of the functions satisfies:then:where:and:
The result is possible when
Suppose that satisfy the condition (25). Setting:we have:
Thus, by making use of the identity (3) in (29), we get:which, in view of F given by (27) and (30), yields:where:
Since it follows from Lemma 3 that:
When we consider the functions , which satisfy (25), are defined by:
Thus, it follows from (32) that:which evidently ends the proof. ☐
Let , and let with and Suppose that:
If with for all then:implies:where:is the best dominant.
Then, is analytic in , and for all Taking the logarithmic derivatives on both sides of (34) and using (3), we have:
Now, the assertions of Theorem 5 follow by Lemma 4. ☐
Let If satisfies:then:where is the positive root of the equation:
Then, φ is analytic in , and for all Taking the logarithmic derivatives on both sides of (37) and using the identity (3), we have:and so:
Then, for we have:where is the positive root of Equation (36). Suppose that:
For and for This proves that Thus, for , , and so, we obtain the required result by an application of Lemma 5. ☐
Suppose that If:then:where:
Then, from Theorem 1, we have:
Let be the function that maps onto the domain:with then:
The authors would like to thank all referees for their valuable comments which led to the improvement of this paper.
All the authors read and approved the final manuscript as a consequence of the authors meetings.
Conflicts of Interest
The authors declare no conflict of interest.
- Bulboacă, T. Differential Subordinations and Superordinations, Recent Results; House of Scientific Book Publ.: Cluj-Napoca, Romania, 2005. [Google Scholar]
- Miller, S.S.; Mocanu, P.T. Differential Subordination: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics; Marcel Dekker Inc.: New York, NY, USA, 2000; Volume 225. [Google Scholar]
- Owa, S. On the distortion theorems I. Kyungpook Math. J. 1978, 18, 53–59. [Google Scholar]
- Owa, S.; Srivastava, H.M. Univalent and starlike generalized hypergeometric functions. Can. J. Math. 1987, 39, 1057–1077. [Google Scholar] [CrossRef]
- Srivastava, H.M.; Saigo, M.; Owa, S. A class of distortion theorems involving certain operators of fractional calculus. J. Math. Anal. Appl. 1988, 131, 412–420. [Google Scholar] [CrossRef]
- Prajapat, J.K.; Raina, R.K̇.; Srivastava, H.M. Some inclusion properties for certain subclasses of strongly starlike and strongly convex functions involving a family of fractional integral operators. Integral Transform. Spec. Funct. 2007, 18, 639–651. [Google Scholar] [CrossRef]
- Goyal, S.P.; Prajapat, J.K. A new class of analytic p-valent functions with negative coefficients and fractional calculus operators. Tamsui Oxf. J. Math. Sci. 2004, 20, 175–186. [Google Scholar]
- Prajapat, J.K.; Aouf, M.K. Majorization problem for certain class of p-valently analytic function defined by generalized fractional differintegral operator. Comput. Math. Appl. 2012, 63, 42–47. [Google Scholar] [CrossRef]
- Tang, H.; Deng, G.; Li, S.; Aouf, M.K. Inclusion results for certain subclasses of spiral-like multivalent functions involving a generalized fractional differintegral operator. Integral Transform. Spec. Funct. 2013, 24, 873–883. [Google Scholar] [CrossRef]
- Aouf, M.K.; Mostafa, A.O.; Zayed, H.M. Some characterizations of integral operators associated with certain classes of p-valent functions defined by the Srivastava-Saigo-Owa fractional differintegral operator. Complex Anal. Oper. Theory 2016, 10, 1267–1275. [Google Scholar] [CrossRef]
- Aouf, M.K.; Mostafa, A.O.; Zayed, H.M. Subordination and superordination properties of p-valent functions defined by a generalized fractional differintegral operator. Quaest. Math. 2016, 39, 545–560. [Google Scholar] [CrossRef]
- Aouf, M.K.; Mostafa, A.O.; Zayed, H.M. On certain subclasses of multivalent functions defined by a generalized fractional differintegral operator. Afr. Mat. 2017, 28, 99–107. [Google Scholar] [CrossRef]
- Mostafa, A.O.; Aouf, M.K.; Zayed, H.M.; Bulboacă, T. Multivalent functions associated with Srivastava-Saigo-Owa fractional differintegral operator. RACSAM 2017. [Google Scholar] [CrossRef]
- Mostafa, A.O.; Aouf, M.K.; Zayed, H.M. Inclusion relations for subclasses of multivalent functions defined by Srivastava–Saigo–Owa fractional differintegral operator. Afr. Mat. 2018. [Google Scholar] [CrossRef]
- Mostafa, A.O.; Aouf, M.K.; Zayed, H.M. Subordinating results for p-valent functions associated with the Srivastava–Saigo–Owa fractional differintegral operator. Afr. Mat. 2018. [Google Scholar] [CrossRef]
- Mostafa, A.O.; Aouf, M.K. Some applications of differential subordination of p-valent functions associated with Cho-Kwon-Srivastava operator. Acta Math. Sin. (Engl. Ser.) 2009, 25, 1483–1496. [Google Scholar] [CrossRef]
- Wang, Z.; Shi, L. Some properties of certain extended fractional differintegral operator. RACSAM 2017, 1–11. [Google Scholar] [CrossRef]
- Hallenbeck, D.Z.; Ruscheweyh, S. Subordination by convex functions. Proc. Am. Math. Soc. 1975, 52, 191–195. [Google Scholar] [CrossRef]
- Pommerenke, C. Univalent Functions; Vandenhoeck & Ruprecht: Göttingen, Germany, 1975. [Google Scholar]
- Stankiewicz, J.; Stankiewicz, Z. Some applications of Hadamard convolution in the theory of functions. Ann. Univ. Mariae Curie-Sklodowska Sect. A 1986, 40, 251–265. [Google Scholar]
- Obradović, M.; Owa, S. On certain properties for some classes of starlike functions. J. Math. Anal. Appl. 1990, 145, 357–364. [Google Scholar] [CrossRef]
- Takahashi, N.; Nunokawa, M. A certain connection between starlike and convex functions. Appl. Math. Lett. 2003, 16, 653–655. [Google Scholar] [CrossRef]
- Choi, J.H.; Saigo, M.; Srivastava, H.M. Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 2002, 276, 432–445. [Google Scholar] [CrossRef]
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).