A Class of Analytic Functions associated with Sine Hyperbolic Functions

TL;DR

This paper introduces a new class of analytic functions related to the hyperbolic sine, providing coefficient bounds, conjectures, and differential subordination results to characterize these functions.

Contribution

It defines a novel class of functions subordinate to 1+ sinh(z) and establishes coefficient inequalities, conjectures for general coefficients, and bounds for the Hankel determinant.

Findings

Sharp bounds for the first five coefficients

A conjecture for the general nth coefficient

Bounds for the third Hankel determinant

Abstract

We introduce a class of analytic functions subordinate to the function $1+\sinh \left( z\right) $ and obtain various necessary and sufficient conditions for functions to be in the class. These conditions mainly comprise of the coefficient inequalities involving convolution. Further, we have obtained sharp five initial coefficients, a conjecture for the general nth coefficient and the third Hankel determinant bounds for the functions in this class. Also derived certain differential subordination implication results involving $1+\sinh \left( z\right)$.