Schwarzian Norm Estimates for Analytic Functions Associated with Convex Functions

TL;DR

This paper establishes sharp bounds on the Schwarzian and pre-Schwarzian derivatives for functions in certain subclasses of analytic functions related to convex functions, based on their second derivative at zero.

Contribution

It introduces new bounds for Schwarzian and pre-Schwarzian norms for subclasses of analytic functions defined by a real part condition, extending existing results.

Findings

Sharp bounds for Schwarzian derivatives are derived.

Bounds depend on the second derivative at zero.

Results include distortion and growth theorems for the classes.

Abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disc $\mathbb{D}=\{z\in\mathbb{C}:\;|z|<1\}$ normalized by $f(0)=0$ and $f^{\prime}(0)=1$. In the present article, we consider and $\mathcal{F}(c)$ the subclasses of $\mathcal{A}$ are defined by \begin{align*} \mathcal{F}(c)=\bigg\{f\in\mathcal{A}:\;{\rm Re}\;\bigg(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg)>1-\frac{c}{2},\;\;\mbox{for some}\;c\in(0,3]\bigg\}, \end{align*} and derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in and $\mathcal{F}(c)$ expressed in terms of their value $f^{\prime\prime}(0)$, in particular, when the quantity is equal to zero. Moreover, we obtain sharp bounds for distortion and growth theorems for functions in the class $\mathcal{F}(c)$.