Sufficient conditions for strong starlikeness

TL;DR

This paper establishes sufficient conditions for certain analytic functions to be subordinate to a fractional power of the Möbius transformation, with applications to hypergeometric and Bessel functions, advancing the theory of strong starlikeness.

Contribution

It provides new sufficient conditions for subordination of analytic functions to specific fractional transformations, including applications to hypergeometric and Bessel functions, and explores related differential inequalities.

Findings

Derived conditions for subordination involving hypergeometric functions.

Established criteria for Bessel functions to be strongly starlike.

Analyzed subordination under differential inequalities with parameters.

Abstract

Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$ with $p(0)=1$ and $0< \alpha \leq 1$. The conditions on complex valued functions $C$, $D$ and $E$ are obtained for $p$ to be subordinate to $((1+z)/(1-z))^{\alpha}$ when $C(z) z^{2}p''(z)+D(z)zp'(z) + E(z)p(z)=0$. Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to $((1+z)/(1-z))^{\alpha}$ are obtained as applications. The conditions on $\alpha$ and $\beta$ are derived for $p$ to be subordinate to $((1+z)/(1-z))^{\alpha}$ when $1+\beta zp'(z)/p^{n}(z)$ with $n=1,2$ is subordinate to $1+4z/3+2z^{2}/3=:\varphi_{CAR}(z)$. Similar problems were investigated for $\RE p(z)>0$ when the functions $p(z)+\beta zp'(z)/p^{n}(z)$ with $n=0,2$ is subordinate to $\varphi_{CAR}(z)$. The condition on $\beta$ is determined for $p$ to be subordinate to $((1+z)/(1-z))^{\alpha}$ when $p(z)+\beta zp'(z)/p^{n}(z)$ with $n=0,1,2$ is subordinate to $((1+z)/(1-z))^{\alpha}$.