Region of variability for functions with positive real part

TL;DR

This paper determines the region of variability for integrals of functions with positive real part in the unit disk, extending to subclasses of univalent functions and providing graphical illustrations for various parameters.

Contribution

It explicitly characterizes the variability region for integrals of a broad class of functions with positive real part, including subclasses of univalent functions, with detailed parameter analysis.

Findings

Explicit variability regions for integrals of functions with positive real part.

Extension of variability results to subclasses of univalent functions.

Graphical representations of the variability regions for different parameters.

Abstract

For $\gamma\in\IC$ such that $|\gamma|<\pi/2$ and $0\leq\beta<1$, let ${\mathcal P}_{\gamma,\beta} $ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re\,} \left (e^{i\gamma}P(z)\right)>\beta\cos\gamma \quad \mbox{ in ${\mathbb D}$}. $$ For any fixed $z_0\in\mathbb{D}$ and $\lambda\in\overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,\lambda)$ for $\int_0^{z_0}P(\zeta)\,d\zeta$ when $P$ ranges over the class $$ \mathcal{P}(\lambda) = \left\{ P\in{\mathcal P}_{\gamma,\beta} :\, P'(0)=2(1-\beta)\lambda e^{-i\gamma}\cos\gamma \right\}. $$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.