On close-to-convex functions

TL;DR

This paper introduces a new subclass of close-to-convex functions in the unit disk and derives sharp estimates for various geometric function theory problems such as Fekete-Szeg"{o} problem, growth, distortion, and convexity radius.

Contribution

It defines the subclass  nd provides sharp estimates for key geometric function theory problems.

Findings

Sharp Fekete-Szeg estimates for the subclass.

Growth and distortion theorems established.

Radius of convexity and pre-Schwarzian norm bounds obtained.

Abstract

We consider a new subclass $\widetilde{\mathcal{K}}_u$ of close-to-convex functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$. For this class, we obtain sharp estimates of the Fekete-Szeg\"{o} problem, growth and distortion theorem, radius of convexity and estimate of the pre-Schwarzian norm.