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On the univalence of functions with logharmonic Laplacian. (English) Zbl 1179.31001
Summary: We prove that Radó’s theorem holds for functions of the form $$F(z) = r^2L(z)$$, where $$L$$ is logharmonic. We show that if $$F$$ is of the form $$F(z) = r^2L(z)$$, $$|z| < 1$$, where $$L(z) = h(z)\overline {g(z)}$$ is logharmonic, then $$F$$ is starlike iff $$\psi (z) = h(z)/g(z)$$ is starlike. In addition, when $$F(z) = r^2L(z) + H(z)$$, $$|z| < 1$$, where $$L$$ is logharmonic and $$H$$ is harmonic, we give sufficient conditions for $$F$$ to be locally univalent.

##### MSC:
 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
##### Keywords:
harmonic function; logharmonic function; starlike function
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##### References:
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