# First order differential subordination for functions with positive real part

## TL;DR

This paper establishes sharp bounds on a parameter for analytic functions with positive real part, ensuring their subordination to starlike functions through first order differential inequalities, refining existing results.

## Contribution

It introduces sharp estimates for parameters in differential subordination conditions that guarantee functions with positive real part are subordinate to known starlike functions.

## Key findings

- Derived sharp bounds on β for subordination conditions.
- Extended previous results with sharper estimates.
- Confirmed subordination to well-known starlike functions.

## Abstract

Sharp estimates on $\beta$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever $1+\beta z p'(z), \,\,1+\beta z p'(z)/p(z), \,\,\mbox{or}\,\,1+\beta z p'(z)/p^{2}(z)$ is subordinate to $\sqrt{1+z}$. Our results provide sharp version of previously known results.