On close-to-convex functions

TL;DR

The paper introduces the new subclass $\widetilde{\mathcal{K}}_u$ of close-to-convex functions in the unit disk and leverages a representation $f'(z)=\frac{g(z)}{z}(1+\omega(z))$ with $g\in\mathcal{S}_u^*$ and $\omega\in\mathcal{B}_0$ to derive sharp geometric and analytic bounds. It delivers sharp growth and distortion results, identifies the radius of convexity as $s=\frac{3-\sqrt{5}}{2}$, and provides exact Fekete-Szegö bounds $|a_3-\lambda a_2^2|$ across all real $\lambda$, along with a sharp pre-Schwarzian norm bound $||P_f||\le \frac{9}{4}$. Extremals are constructed using Blaschke products and the extremal function $f(z)=ze^{z}$ (and its integral representations) to demonstrate sharpness. These results extend the theory of close-to-convex and $\mathcal{K}_u$-type classes in $\mathbb{D}$ and enrich the toolkit for coefficient problems and geometric function theory in the unit disk.

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-Szegö problem, growth and distortion theorem, radius of convexity and estimate of the pre-Schwarzian norm.

Theorems & Definitions (10)

• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 5.1
• proof
• Theorem 6.1
• proof