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

Abstract

The main object of this paper is to find necessary and sufficient conditions for generalized Bessel functions of first kind $z{u}_p(z)$ to be in the classes ${\mathcal{S}\mathcal{P}}_p(\alpha ,\beta )$ and $\mathcal{U}\mathcal{CS}\mathcal{P}(\alpha ,\beta )$ 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 $\mathcal{U}\mathcal{CS}\mathcal{P}\mathcal{T}(\alpha ,\beta )$ provided that the function $f$ is in the class ${\mathcal{R}}^{\tau }(A,B)$. Finally, we give conditions for the integral operator $\mathcal{G}(\kappa ,c,z)=$ to be in the class $\mathcal{U}\mathcal{CS}\mathcal{P}\mathcal{T}(\alpha ,\beta ).$ Several corollaries and consequences of the main results are also considered.