Geometric properties of Cesaro averaging operators

TL;DR

This paper investigates the geometric properties of analytic functions in the unit disc using positivity of trigonometric sums with generalized Vietoris coefficients, and applies findings to Cesàro type polynomials.

Contribution

It introduces new coefficient conditions to characterize geometric properties of analytic functions and generalized Cesàro polynomials.

Findings

Derived conditions on coefficients for geometric properties

Established positivity criteria for trigonometric sums

Analyzed geometric properties of generalized Cesàro polynomials

Abstract

In this paper, using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on the coefficient $\{a_k\}$ to characterize the geometric properties of the corresponding analytic function $f(z)=z+\displaystyle\sum_{k=2}^{\infty} a_kz^k$ in the unit disc $\mathbb{D}$. As an application we also find geometric properties of a generalized Ces\`aro type polynomials.