A NEW SUBCLASS OF HARMONIC MAPPINGS WITH POSITIVE REAL PART

Authors : Sibel YALÇIN, Metin ÖZTÜRK, Mümin YAMANKARADENİZ

Pages : 13-18

View : 26 | Download : 7

Publication Date : 2001-12-01

Article Type : Research Paper

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 $HPinsert ignore into journalissuearticles values(\beta, \alpha);$, $insert ignore into journalissuearticles values(\alpha \ge 0, 0 \le \beta< 1);$ of all functions   $f=h+\bar{g}$  for which $\Re e\{\alpha zinsert ignore into journalissuearticles values(h`insert ignore into journalissuearticles values(z);+g`insert ignore into journalissuearticles values(z););+hinsert ignore into journalissuearticles values(z);+ginsert ignore into journalissuearticles values(z););\}\ge\beta, finsert ignore into journalissuearticles values(0);=1$ . We give suffcient coeffcient conditions for normalized harmonic functions to be in $HPinsert ignore into journalissuearticles values(\beta, \alpha);$. 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