A Brief Study of Certain Class of Harmonic Functions of Bazilevič Type

Abstract

We define and investigate a new subclass of Bazilevič type harmonic univalent functions using a linear operator. We investigated the harmonic structures in terms of its coefficient conditions, extreme points, distortion bounds, convolution, and convex combination. So, also, we discussed the subordination properties for the functions in this class.

Harmonic functions of class of Bazilevi´c type Related to new derivative operator

“…The remaining condition needs to investigate the function ( ) which belongs to the class ℱ , ( , , ). By (9) and 10, we have…”

“…Has been remarked that the linear operator which is defined in (4) is generalized many operators by giving specific values to the parameters which studied by several earlier authors as follows: (i) If = 0 and = 1, then operator (4) reduces A. T. Oladipo and D. Breaz operator [9]. (ii) If = 0, = 1 and = 1, then operator (4) reduces to the Salagean derivative operator [11].…”

“…Starting point in the study of functions characterized in (9) was discovered in 1955 by Bazilevi´c [2], when the Bazilevi´c function defined in by the form:…”

“…The class of harmonic functions have been studied by many authors for variant properties. By the earlier papers for contributors such as [ [3], [4], [5], [6], [8], [9]] and [10] regarding of the theory of analytic functions which have a wide application in many physical problem: as electrostatic potential in heat conduction, fluid flows, and theory of fractals constitute practical examples.…”

Coefficient Bounds for Bazilevic Functions Associated with Modified Sigmoid Function

“…However, by simplifying (1.6) it is quite possible to understand and investigate the family better. It should be noted that with special choices of parameters ε α , and the function ) (z g , the family ) , ( ε α B cracks down to some well-known subclasses of univalent functions (see [3] for details). For instance, if we let 0 = ε then (1.6) immediately yields…”