The Schwarzian norm estimates for Janowski convex functions

TL;DR

This paper establishes the precise bounds for the Schwarzian norm of Janowski convex functions, using Dieudonné's lemma to identify extremal functions and improve understanding of their geometric properties.

Contribution

It provides the first sharp estimates of the Schwarzian norm for Janowski convex functions, introducing a new method for constructing extremal functions.

Findings

Sharp Schwarzian norm bounds for Janowski convex functions.

A novel approach to extremal function construction.

Enhanced understanding of geometric function theory.

Abstract

For $-1\leq B<A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class $\mathcal{C}(A,B)$. The Dieudonn\'{e}'s lemma which gives the exact region of variability for derivatives at a point of bounded functions, plays the key role in this study, and we also use this lemma to construct the extremal functions for the sharpness by a new method.