Sharp estimates of generalized Zalcman functional of early coefficients for Ma-Minda type functions

Abstract

Let $\varphi$ be an analytic function in the unit disk $\mathbb{D}:=\{ z\in\mathbb{C}:|z|<1 \}$ which has the form $\varphi(z)=1+p_1 z+p_2 z^2 +p_3 z^3+\cdots$ with $p_1>0$, $p_2$, $p_3 \in\mathbb{R}$. For given such $\varphi$, let ${\mathcal S}^{*}(\varphi)$, ${\mathcal K}(\varphi)$ and ${\mathcal R}(\varphi)$ denote the classes of standardly normalized analytic functions $f$ in $\mathbb{D}$ which satisfy

\begin{equation*}

\frac{zf'(z)}{f(z)} \prec \varphi(z), \quad 1+\frac{zf''(z)}{f'(z)} \prec \varphi(z) \quad f'(z) \prec \varphi(z),\quad z\in \mathbb{D},

\end{equation*}

respectively, where $\prec$ means the usual subordination.

In this paper, we find the sharp bounds of $|a_2 a_3 - a_4|$, where $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ over classes ${\mathcal S}^{*}(\varphi)$, ${\mathcal K}(\varphi)$ and ${\mathcal R}(\varphi)$.