ON A SUBSET OF BAZILEVICH FUNCTIONS IDENTIFIED BY THE THREE-LEAF FUNCTION, MILLER-ROSS FUNCTION

Abstract

A significant portion of the collection of analytic-univalent functions of the type
$$
h(\zeta) = \zeta + \sum_{n=rm+1}^{\infty} a_n\zeta^n
$$
whose definition is found in the unit disk $$\Omega:=\{z:|z|<1\},$$ is investigated in this work. Several subsets of the well-known set of Bazilevi\v{c} functions are included in this new set. The new set and its findings are developed using the Miller-Ross function, the Schwarz function, some multiplier operators, and some mathematical ideas such as subordination, set theory, infinite series generation, and convolution of some geometric expressions. Among the main achievements are the estimates for the coefficient bounds and the Fekete-Szeg\"o functional. Generally speaking, the new set reduces to a number of known subsets with some supposedly unique results when some parameters are altered inside their declaration intervals.