A new class of harmonic univalent functions by the generalized Salagean operator

Abstract

A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form , f h g = + where h and g are analytic in U. We define and investigate a new class SHP ( , ) λ α β by generalized Salagean operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class SHP ( , ) λ α β . These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.

Properties of a Class of -Harmonic Functions

“…Note that a necessary and sufficient condition for given by (13) and (14) to be in ( , , ) is that the condition (12) should be satisfied. This is equivalent to Re{ ( )/ ( )} ≥ 0, where…”

“…When = 1, we get the Salagean differential operator. For ( ) = ℎ( )+ ( ) given by (2), Li and Liu [14] defined the following generalized Salagean operator in :…”

“…When = 1, we get the generalized Salagean operator for harmonic univalent functions defined by Li and Liu [14].…”

“…Define ( , , ) := ( , , ) ∩ . The main object of the paper is to introduce a class of -harmonic functions by using the generalized Salagean operator which was defined by Li and Liu [14]. We investigate necessary and sufficient coefficient conditions, extreme points, distortion bounds, and convex combination of the class.…”