On certain analytic functions defined by differential inequality

TL;DR

This paper investigates the geometric properties of a family of analytic functions defined by a specific differential inequality, deriving radii of convexity, starlikeness, and close-to-convexity, and explores their generalizations.

Contribution

It introduces new results on the radii of geometric properties for functions satisfying a differential inequality and extends the analysis to a generalized family with a parameter.

Findings

Determined radii of convexity, starlikeness, and close-to-convexity for the functions.

Established properties of the generalized family with parameter bblambdab2.

Provided bounds and conditions for geometric function properties.

Abstract

For the family of analytic functions $f(z)$ in the open unit disk $\mathbb{D}$ with $f(0)=f'(0)-1=0$, satisfying the differential equation \begin{equation*} zf'(z) - f(z) = \dfrac{1}{2} z^2 \phi(z), \quad |\phi(z)| \leq 1, \end{equation*} we obtain radii of convexity, starlikeness, and close-to-convexity of partial sums of $f(z)$. We also study the generalization of this family having the form \begin{equation*} zf'(z)-f(z) = \lambda z^2 \phi(z), \quad |\phi(z)| \leq 1, \end{equation*} where $\lambda > 0,$ and obtain some useful properties of these functions.