SOME INEQUALITIES WHICH HOLD FOR STARLIKE LOG-HARMONIC MAPPINGS OF ORDER α


Ozkan H. E., Aydoğan S. M.

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, vol.16, no.3, pp.478-485, 2014 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 16 Issue: 3
  • Publication Date: 2014
  • Journal Name: JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.478-485
  • Istanbul Kültür University Affiliated: Yes

Abstract

Let H(D) be the linear space of all analytic functions defined on the open disc D = {z vertical bar vertical bar z vertical bar < 1}. A log-harmonic mappings is a solution of the nonlinear elliptic partial differential equation <(f)over bar>((z) over bar) = w (f) over bar /f f(z) where w(z) is an element of H(D) is second dilatation such that vertical bar w(z)vertical bar < 1 for all z is an element of D. It has been shown that if f is a non-vanishing log-harmonic mapping, then f can be expressed as f(z) = h(z)<(g(z))over bar> where h(z) and g(z) are analytic function in D. On the other hand, if f vanishes at z = 0 but it is not identically zero then f admits following representation f(z) = z vertical bar z vertical bar(2 beta) h(z)<(g(z))over bar> where Re beta > -1/2, h and g are analytic in D, g(0) = 1, h(0) not equal 0. Let f = z vertical bar z vertical bar(2 beta) h (g) over bar be a univalent log-harmonic mapping. We say that f is a starlike log-harmonic mapping of order alpha if partial derivative(arg f(re(i theta)))/partial derivative theta = Rezf(z)-(z) over barf((z) over bar)/f > alpha, 0 <= alpha < 1. (for all z is an element of U) and denote by S-lh*(alpha) the set of all starlike log-harmonic mappings of order alpha. The aim of this paper is to define some inequalities of starlike log-harmonic functions of order alpha (0 <= alpha <= 1).