# On The Coefficient Conjecture of Clunie and Sheil-Small

## TL;DR

This paper proves a relation between analytic and co-analytic parts of harmonic univalent functions, verifies the coefficient conjecture of Clunie and Sheil-Small, and derives distortion bounds for this class.

## Contribution

It introduces a proof of the coefficient conjecture for harmonic univalent functions and establishes a relation between their analytic and co-analytic parts.

## Key findings

- Verified the coefficient conjecture of Clunie and Sheil-Small.
- Established a relation between analytic and co-analytic parts via second dilatation.
- Derived distortion bounds for harmonic univalent functions.

## Abstract

In this paper, we first prove relation between analytic and co-analytic part of the class harmonic univalent functions S_H(S):={f = h+\overline g|h is element of S} by means of second dilatation is constant. Next, we verify the coefficient conjecture of Clunie and Sheil-Small for class of harmonic univalent functions. Finally, we obtain distortion bounds of this class.