Pre-Schwarzian and Schwarzian norm Estimates for class of Ozaki Close-to-Convex functions

TL;DR

This paper establishes sharp bounds for the Schwarzian and pre-Schwarzian norms of Ozaki close-to-convex functions, enhancing understanding of their geometric properties and extending to harmonic mappings with fixed analytic parts.

Contribution

It provides new sharp bounds for these norms in Ozaki close-to-convex functions and harmonic mappings, linking them to the second derivative at zero.

Findings

Sharp bounds for Schwarzian and pre-Schwarzian norms derived.

Bounds for distortion and growth theorems obtained.

Pre-Schwarzian norm bounds for harmonic mappings with fixed analytic part established.

Abstract

The primary objective of this paper is to derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives in the Ozaki close-to-convex functions $f$, expressed in terms of their value $f^{\prime\prime}(0)$, in particular, when the quantity is equal to zero. Additionally, we obtain sharp bounds for distortion and growth theorems. We will also derive the sharp bound of pre-Schwarzian norm for a certain class of harmonic mappings whose analytic part is fixed.