Properties of Logharmonic Mappings

The main aim of this paper is to investigate properties of certain class of logharmonic mappings. Initially, we establish the argument principle of sense-preserving mappings F = h g ¯ + H G ¯ , where h , g , H and G are analytic functions. As applications, we obtain a direct extension of Rouché’s theorem, open mapping theorem and minimal area image of sense-preserving logharmonic mappings. Furthermore, we present an estimate for the modulus of the partial derivative of K -quasiconformal logharmonic mappings, and by using it, we get a Schwarz type lemma of K -quasiconformal logharmonic mappings. Finally, we obtain estimates of h ′ ′ ( 0 ) 2 and g ′ ′ ( 0 ) 2 for univalent sense-preserving logharmonic mappings f = h g ¯ in the unit disk.