Third-Order Differential Superordination Involving the Generalized Bessel Functions

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.