On a certain class of starlike functions
Md Firoz Ali, Md Nurezzaman
TL;DR
The paper addresses sharp coefficient and Hankel-determinant problems for the starlike subclass $\mathcal{S}_u^*$, defined by $\left|\frac{z f'(z)}{f(z)}-1\right|<1$. It deploys the subordination framework $\frac{z f'(z)}{f(z)}-1=\frac{p(z)-1}{p(z)+1}$ with $p\in\mathcal{P}$, leverages rotational invariance, and uses classical coefficient relations to derive exact bounds and extremals. The main contributions are sharp bounds for $|H_{2,2}(f)|$, $|H_{3,1}(f)|$, $|H_{2,1}(\mathrm{F}_f/2)|$, and $|H_{2,2}(f^{-1})|$, along with interrelations among Hankel invariants (e.g., $|4H_{2,1}(\mathrm{F}_f/2)-H_{2,2}(f)|=\frac{1}{12}|a_2|^4$) and precise bounds for the successive coefficients, with explicit extremal functions such as $f_2(z)=z e^{z}$ and $f_n(z)=z e^{z^{n-1}/(n-1)}$. These results advance Hankel-determinant and coefficient problems within the $\mathcal{S}_u^*$ class and provide concrete extremals linking logarithmic and inverse-coefficient data.
Abstract
Let $\mathcal{S}_u^*$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, normalized by $f(0)=f'(0)-1=0$ that satisfies the inequality $\left|zf'(z)/f(z)-1\right|<1$ in $\mathbb{D}$. In the present article, we obtain the sharp estimate of Hankel determinants whose entries are coefficients of $f\in\mathcal{S}_u^*$, logarithmic coefficients of $f\in\mathcal{S}_u^*$ and coefficients of inverse of $f\in\mathcal{S}_u^*$, respectively. We also obtain, the sharp estimate of the successive coefficients for functions in the class $\mathcal{S}_u^*$.