Some classes of order \alpha\ for second-order differential inequalities

TL;DR

This paper investigates sufficient conditions for analytic functions in the unit disk to belong to certain classes of order alpha, extending previous work on starlikeness criteria.

Contribution

It introduces new sufficient conditions for functions to be in classes of order alpha, generalizing earlier results on starlikeness for analytic functions.

Findings

Derived new criteria for classes of order alpha

Extended previous starlikeness results

Provided conditions applicable to broader function classes

Abstract

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008)) have considered some sufficient problems for starlikeness. The object of the present paper is to discuss some sufficient problems for f(z) to be in some classes of order \alpha.