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

作者：Ozkan H Esra^*; Aydogan Melike

来源：Journal of Computational Analysis and Applications, 2014, 16(3): 478-485.

• Starlike log-harmonic functions
• univalent functions
• distortion theorem
• Marx-Strohhacker inequality

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).

• 出版日期2014-4