# On sharp bounds of certain Close-to-Convex functions

## TL;DR

This paper establishes sharp bounds for initial inverse coefficients of specific close-to-convex functions, using a general formula for the fourth coefficient within the Carathéodory class, enhancing understanding of their coefficient constraints.

## Contribution

It introduces a general formula for the fourth coefficient of Carathéodory functions and derives sharp bounds for inverse coefficients of certain close-to-convex functions.

## Key findings

- Derived a general formula for the fourth coefficient.
- Obtained sharp upper bounds for inverse coefficients.
- Analyzed functions satisfying specific real part inequalities.

## Abstract

We derive general formula for the fourth coefficient of the functions belonging to the Carath\'{e}odory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities: $\RE((1-z)f'(z))>0,$ $\RE((1-z^2)f'(z))>0,$ $\RE((1-z+z^2)f'(z))>0$ and $\RE((1-z)^2f'(z))>0$.