Publication: Growth theorems for perturbated starlike log-harmonic mappings of complex order
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Date
2009
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Abstract
Let H(D) be the linear space of all analytic functions defined on
the open unit disc D = {z ∈ C : |z| < 1}. A sense-preserving logharmonic
mapping is the solution of the non-linear elliptic partial
differential equation fz¯ = wfz
¡
f /f¢
, where w(z) ∈ H(D) is the
second dilatation of f such that |w(z)| < 1 for every z ∈ D. It has
been shown that if f is a non-vanishing log-harmonic mapping, then
f can be expressed as f = h(z)g(z), where h(z) and g(z) are analytic
in D. If f vanishes at z = 0 but it is not identically zero, then f
admits the representation f = z|z|
2βh(z)g(z), where Reβ > −1/2,
h(z) and g(z) are analytic in D, g(0) = 1, h(0) 6= 0 (see [1], [2], [3]).
Let f = zh(z)g(z) be a univalent log-harmonic mapping. We say
that f is a starlike log-harmonic mapping of complex order b (b 6= 0,
complex) if
Re ½
1 +
1
b
µ
zfz − zf¯ z¯
f
− 1
¶¾ > 0, z ∈ D.
The class of all starlike log-harmonic mappings of complex order b
is denoted by S
∗
LH(1 − b). We also note that if zh(z) is a starlike
function of complex order b, then the starlike log-harmonic mapping
f = zh(z)g(z) will be called a perturbated starlike log-harmonic
mapping of complex order b, and the family of such mappings will
be denoted by S
∗
LH(p)(1 − b).
The aim of this paper is to obtain the growth theorems for the
perturbated starlike log-harmonic mappings of complex order.
Description
Keywords
Analytic functions, harmonic functions, log-harmonic functions of complex order