Normality criterion for logharmonic mapping
Molla Basir Ahamed, Sanju Mandal
TL;DR
The paper investigates normality for logharmonic mappings in the unit disk, introducing a Marty-type criterion and extending the Zalcman–Pang lemma and Lohwater–Pommerenke theorem to the logharmonic setting, with an application of the Zalcman–Pang lemma. It proves that a family of mappings of the form $f(z)=z|z|^{2\beta}h(z)\overline{g(z)}$ is normal precisely when the spherical derivative $f^{\#}$ is locally bounded, equivalently when no nonconstant logharmonic limit $F$ arises under rescaling with $F^{\#}(\xi)\le F^{\#}(0)=1$. A practical corollary shows that if $f^{\#}(z)>\\\ ext{epsilon}$ on $\mathbb{D}$ for some $\epsilon>0$, then every member is normal, and the results extend known normality criteria from meromorphic and harmonic cases to the logharmonic setting. These findings advance the understanding of boundary behavior and stability of normal logharmonic mappings.
Abstract
In this paper, we present several necessary and sufficient conditions for a logharmonic mapping to be normal i.e., we establish Marty's criterion, Zalcman-Pang lemma and the Lohwater-Pommerenke theorem for logharmonic mappings, along with an application of the Zalcman-Pang lemma.