New extensions for a theorem by Mocanu

TL;DR

This paper extends Mocanu's theorem on analytic functions by exploring cases where higher derivatives at zero vanish, providing new sufficient conditions for these functions within the unit disk.

Contribution

It introduces novel extensions of Mocanu's theorem for functions with higher-order zero derivatives at zero, broadening the scope of the original results.

Findings

Derived new sufficient conditions for functions with higher-order zero derivatives.

Generalized Mocanu's theorem to include functions with multiple initial derivatives zero.

Provided proofs and examples illustrating the extended conditions.

Abstract

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=f"(0)=0, P. T. Mocanu (Mathematica (Cluj), 42(2000)) has considered some sufficient arguments of f'(z)+zf"(z) for |\arg(zf'(z)/f(z))|<\pi\mu/2. The object of the present paper is to discuss those probrems for f(z) with f"(0)=f"'(0)=...=f^{(n)}(0)=0 and f^{(n+1)}(0) \ne 0.