Generalized convolution properties based on the modified Mittag-Leffler function

Volume 10, Issue 8, pp 4284--4294 http://dx.doi.org/10.22436/jnsa.010.08.23

Publication Date: August 22, 2017 Submission Date: February 24, 2017

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Authors

H. M. Srivastava - Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada. - Center for General Education (Department of Science and Technology), China Medical University, Taichung 40402, Taiwan, Republic of China. Adem Kılıçman - Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia. Zainab E. Abdulnaby - Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia. - Department of Mathematics, College of Science, Al-Mustansiriyah University, Baghdad, Iraq. Rabha W. Ibrahim - Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia.

Abstract

Studies of convolution play an important role in Geometric Function Theory (GFT). Such studies attracted a large number of researchers in recent years. By making use of the Hadamard product (or convolution), several new and interesting subclasses of analytic and univalent functions have been introduced and investigated in the direction of well-known concepts such as the subordination and superordination inequalities, integral mean and partial sums, and so on. In this article, we apply the Hadamard product (or convolution) by utilizing some special functions. Our contribution in this paper includes defining a new linear operator in the form of the generalized Mittag-Leffler function in terms of the extensively-investigated Fox-Wright \(\:_p\Psi_q\)-function in the right-half of the open unit disk where where \(\Re(z)>0.\) We then show that the new linear convolution operator is bounded in some spaces. In particular, several boundedness properties of this linear convolution operator under mappings from a weighted Bloch space into a weighted-log Bloch space are also investigated. For uniformity and convenience, the Fox-Wright \(\:_p\Psi_q\)-notation is used in our results.

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Keywords

• Fractional calculus
• analytic functions
• fractional calculus operator
• univalent functions
• convex functions
• Mittag-Leffler function
• Fox-Wright \(\:_p\Psi_q\)-function
• weighted \(\mu\)-Bloch space
• weighted-log Bloch space.

MSC

•  33E12
•  26A33
•  47B38

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