Second-order differential superordination for analytic functions in the upper half-plane

Volume 10, Issue 10, pp 5271--5280 http://dx.doi.org/10.22436/jnsa.010.10.13

Publication Date: October 19, 2017 Submission Date: July 27, 2017

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Authors

Huo Tang - School of Mathematics and Statistics, Chifeng University, Chifeng 024000, Inner Mongolia, People's Republic of China. H. M. Srivastava - Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada. - Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China. Guan-Tie Deng - School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China. Shu-Hai Li - School of Mathematics and Statistics, Chifeng University, Chifeng 024000, Inner Mongolia, People's Republic of China.

Abstract

Let \(\Omega\) be a set in the complex plane \(\mathbb{C}\). Also let \(p(z)\) be analytic in the upper half-plane \(\Delta=\{z:z\in\mathbb{C}\ \text{and}\ \Im(z)>0\}\) and suppose that \(\psi: \mathbb{C}^3\times\Delta\rightarrow\mathbb{C}\). In this paper, we investigate the problem of determining properties of functions \(p(z)\) that satisfy the following second-order differential superordination: \[\Omega\subset\left\{\psi\left(p(z),p'(z),p''(z);z\right): z\in\Delta\right\}.\] Applications of these results to second-order differential superordination for analytic functions in \(\Delta\) are also presented.

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Keywords

• Differential subordination
• differential superordination
• analytic functions
• admissible functions
• upper half-plane

MSC

•  30C45
•  30C80

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