A Note on Logharmonic Mappings
نویسنده
چکیده
where (a) m is nonnegative integer, (b) β= a(0)(1+a(0))/(1−|a(0)|2) and therefore, β >−1/2, (c) h and g are analytic in U , g(0)= 1, and h(0)≠ 0. Univalent logharmonic mappings on the unit disc have been studied extensively. For details see [1, 2, 3, 4, 5, 6, 7, 8]. Suppose that f is a univalent logharmonic mapping defined on the unit disc U . Then, if f(0) = 0, the function F(ζ) = log(f (eζ)) is univalent and harmonic on the half plane {ζ : Reζ < 0}. For more details on univalent harmonic mappings defined in the unit disc U , see [9, 10, 11, 12]. In this note, we consider the problem of minimizing the moments of order p over a subclass of the class logharmonic mappings defined over the unit disc U . It is interesting to note that the extremal functions are univalent starlike logharmonic mappings.
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