# On generalized Ces\`aro stable functions

## TL;DR

This paper introduces a generalized concept of Cesàro stable functions using a new type of Cesàro mean, and explores their properties, including applications to convex functions and related conjectures.

## Contribution

It extends the theory of Cesàro stable functions by defining a new generalized mean and investigates their geometric properties and conjectures.

## Key findings

- Generalized Cesàro stable functions are introduced using type $(b-1;c)$ means.
- Convex functions of order $rac{1}{2} ext{ to }1$ have Cesàro means that are close-to-convex.
- Two conjectures related to these generalized functions are proposed and partially discussed.

## Abstract

The notion of Ces\`aro stable function is generalized by introducing Ces\`aro mean of type $(b-1;c)$ which give rise to a new concept of generalized Ces\`aro stable function. As an application of generalized Ces\`aro stable functions we also prove for a convex function of order $\lambda\in[1/2,1)$, its Ces\`aro mean of type $(b-1;c)$ is close-to-convex of order $\lambda$. Further two conjectures are also posed in the direction of generalized Ces\`aro stable function. Some particular cases of these conjectures are also discussed.