On a certain subclass of strongly starlike functions
R. Kargar, J. Sokół, H. Mahzoon
TL;DR
The paper investigates the two-parameter subclass $\mathcal{S}^*(\alpha_1,\alpha_2)$ of strongly starlike functions, defined by a sector bound on $\arg\{\frac{zf'(z)}{f(z)}\}$ and equivalently by subordination to a convex map $G$. It derives general and sharp coefficient bounds, including $|a_2|\le\lambda$ and a bound for $|a_n|$ $(n\ge3)$, with $\lambda=(\alpha_1+\alpha_2)\cos(\pi\theta/2)$ and $\theta=(\alpha_2-\alpha_1)/(\alpha_2+\alpha_1)$, and identifies extremal functions via Schwarz mappings. The authors also obtain sharp logarithmic-coefficient bounds $|\gamma_n|\le\lambda/(2n)$ and establish convexity-based representations for $\log\{f(z)/z\}$, yielding the sharp case $|\gamma_n|\le\beta/n$ when $\alpha_1=\alpha_2=\beta$. Furthermore, they provide explicit lower and upper bounds for $\Re\{\frac{zf'(z)}{f(z)}\}$ on disks, expressed in terms of $\alpha_1,\alpha_2$, and the disc radius, thereby enriching the coefficient-estimation theory for strongly starlike subclasses with concrete, potentially sharp, geometric bounds.
Abstract
Let $\mathcal{S}^*(α_1,α_2)$, where $ α_1, α_2 \in (0,1]$, represent the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$, normalized by $f(0) = f'(0) - 1=0$, and satisfying the following double-sided inequality: \begin{equation*} -\frac{πα_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\} <\frac{πα_2}{2}, \quad (z\in\mathbb{D}). \end{equation*} In this manuscript, we estimate the coefficients and logarithmic coefficients associated with functions that belong to the class $\mathcal{S}^*(α_1,α_2)$. As a result, we provide a general bound for the coefficients of a strongly starlike function, which has been an open question until now. Finally, we derive upper and lower bounds for the expression ${\rm Re}\{zf'(z)/f(z)\}$, where $f\in \mathcal{S}^*(α_1,α_2)$.