An application of generalized Bessel functions on subclasses of uniformly spirallike functions

TL;DR

This paper establishes necessary and sufficient conditions for generalized Bessel functions and related operators to belong to subclasses of uniformly spirallike functions, expanding understanding of their geometric properties.

Contribution

It provides new criteria for generalized Bessel functions and integral operators to be in specific subclasses of uniformly spirallike functions, linking special functions with geometric function theory.

Findings

Conditions for $zu_{p}(z)$ to be in $ ext{SP}_{p}( ext{α,β})$ and $ ext{UCSP}( ext{α,β})$

Criteria for $z(2-u_{p}(z))$ to belong to the same classes

Necessary and sufficient conditions for the operator $ ext{I}( ext{κ,c})f$ and the integral operator $ ext{G}( ext{κ,c,z})$ to be in $ ext{UCSPT}( ext{α,β})$

Abstract

The main object of this paper is to find necessary and sufficient conditions for generalized Bessel functions of first kind $zu_{p}(z)$ to be in the classes $\mathcal{SP}_{p}(\alpha ,\beta )$ and $\mathcal{UCSP}(\alpha ,\beta )$ of uniformly spirallike functions and also give necessary and sufficient conditions for $z(2-u_{p}(z))$ to be in the above classes. Furthermore, we give necessary and sufficient conditions for $\mathcal{I}(\kappa ,c)f$ \ to be in $\mathcal{UCSPT}(\alpha ,\beta )$ provided that the function $f$ is in the class $\mathcal{R}^{\tau }(A,B)$. Finally, we give conditions for the integral operator $\mathcal{G(}\kappa ,c,z\mathcal{)=}% \int_{0}^{z}(2-u_{p}(t))dt$ to be in the class $\mathcal{UCSPT}(\alpha ,\beta ).$ Several corollaries and consequences of the main results are also considered.