A NEW SUBCLASS OF HARMONIC MAPPINGS WITH POSITIVE REAL PART

Abstract

Complex-valued harmonic functions that are univalent and sense pre-

serving in the unit disk U can be written in the form $f=h +\bar{g}$, where

h and g are analytic in U. In this paper, we introduce a class $HP(\beta, \alpha)$,

$(\alpha \ge 0, 0 \le \beta< 1)$ of all functions $f=h+\bar{g}$ for which $\Re e\{\alpha z(h'(z)+g'(z))+h(z)+g(z))\}\ge\beta, f(0)=1$. We give suffcient coeffcient conditions

for normalized harmonic functions to be in $HP(\beta, \alpha)$. These conditions are

also shown to be necessary when the coeffcients are negative. This leads to

distortion bounds and extreme points.

Keywords

• Harmonic mappings,extreme points,distortion bounds

References

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