The Generalized Janowski Starlike and Close-to-Starlike Log-Harmonic Mappings

Abstract

Motivated by the success of the Janowski starlike function, we consider here closely related functions for log-harmonic mappings of the form 𝑓(𝑧)=𝑧ℎ(𝑧)𝑔(𝑧) defined on the open unit disc 𝑈. The functions are in the class of the generalized Janowski starlike log-harmonic mapping, 𝑆∗lh(𝐴,𝐵,𝛼), with the functional 𝑧ℎ(𝑧) in the class of the generalized Janowski starlike functions, 𝑆∗(𝐴,𝐵,𝛼). By means of these functions, we obtained results on the generalized Janowski close-to-starlike log-harmonic mappings, 𝐶𝑆𝑇lh(𝐴,𝐵,𝛼).