A Factorization Theorem for Logharmonic Mappings
نویسندگان
چکیده
We give the necessary and sufficient condition on sense-preserving logharmonic mapping in order to be factorized as the composition of analytic function followed by a univalent logharmonic mapping. Let D be a domain of C and denote by H(D) the linear space of all analytic functions defined on D. A logharmonic mapping is a solution of the nonlinear elliptic partial differential equation f z = a f f f z , (1) where a ∈ H(D) and |a(z)| < 1 for all z ∈ D. If f does not vanish on D, then f is of the form f = H · G, (2) where H and G are locally analytic (possibly multivalued) functions on D. On the other hand, if f vanishes at z 0 , but is not identically zero, then f admits the local represenation f (z) = z − z 0 m z − z 0 2βm h(z)g(z), (3) where (a) m is a 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 a neighbourhood of z 0. In particular, if D is a simply connected domain, then f admits a global representation of the form (3) (see, e.g., [2]). Univalent logharmonic mappings defined on the unit disk U have been studied extensively (for details, see, e.g., [1, 2, 3, 4, 5, 6]). In the theory of quasiconformal mappings, it is proved that for any measurable function µ with |µ| < 1, the solution of Beltrami equation f z = µf z can be factorized in the form f = ψ • F, where F is a univalent quasiconformal mapping and ψ is an analytic function (see [8]). Moreover, for sense-preserving
منابع مشابه
A Note on Logharmonic Mappings
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)) ...
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تاریخ انتشار 2002