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Third-Order Differential Superordination Involving the Generalized Bessel Functions

  • Huo TangEmail author
  • H. M. Srivastava
  • Erhan Deniz
  • Shu-Hai Li
Article

Abstract

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let \(\Omega \) be a set in the complex plane \(\mathbb {C}\), let \(\mathfrak {p}(z)\) be analytic in the unit disk \(\mathbb {U}=\{z:z\in \mathbb {C}\quad \text {and} \quad |z|<1\}\), and let \(\psi : \mathbb {C}^4\times \mathbb {U}\rightarrow \mathbb {C}\). In this paper, we investigate the problem of determining properties of functions \(\mathfrak {p}(z)\) that satisfy the following third-order differential superordination:
$$\begin{aligned} \Omega \subset \left\{ \psi (\mathfrak {p}(z),z\mathfrak {p}'(z),z^2\mathfrak {p}''(z), z^3\mathfrak {p}'''(z);z): z\in \mathbb {U}\right\} . \end{aligned}$$
As applications, we derive some third-order differential superordination results for analytic functions in \(\mathbb {U}\), which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained.

Keywords

Differential subordination Differential superordination  Analytic functions Univalent functions  Hadamard product (or convolution)  Admissible functions  Generalized Bessel functions Bessel and modified Bessel functions Spherical Bessel function 

Mathematics Subject Classification

Primary 30C45 33C10 Secondary 30C80 

Notes

Acknowledgments

The research of the first-named author (Huo Tang) was partly supported by the Natural Science Foundation of the People’s Republic of China under Grant 11271045, the Higher School Doctoral Foundation of the People’s Republic of China under Grant 20100003110004, the Natural Science Foundation of Inner Mongolia of the People’s Republic of China under Grants 2010MS0117, 2014MS0101 and the Higher School Foundation of Inner Mongolia of the People’s Republic of China under Grant NJZY13298. The research of the third-named author (Erhan Deniz) was supported by the Commission for the Scientific Research Projects of Kafkas Univertsity (Project Number 2012-FEF-30). The authors would like to thank Professors Rosihan M. Ali and V. Ravichandran for their valuable suggestions and the referees for their careful reading of the paper and for their helpful comments to improve our paper.

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Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2014

Authors and Affiliations

  • Huo Tang
    • 1
    • 2
    Email author
  • H. M. Srivastava
    • 3
  • Erhan Deniz
    • 4
  • Shu-Hai Li
    • 1
  1. 1.School of Mathematics and StatisticsChifeng UniversityChifengPeople’s Republic of China
  2. 2.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China
  3. 3.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  4. 4.Department of Mathematics, Faculty of Science and LettersKafkas UniversityKarsTurkey

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