Abstract
Motivated by the success of the Janowski starlike function, we consider here closely related functions for log‐harmonic mappings of the form defined on the open unit disc U. The functions are in the class of the generalized Janowski starlike log‐harmonic mapping, , 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).
Highlights
The class S∗ A, B was investigated by Janowski 1 in early 1970, and since various other subclasses in relation with this Janowski class have been introduced and studied
The Janowski starlike log-harmonic univalent functions were first introduced by Polatoglu and Deniz 11
1.1 where the second dilatation function a ∈ H U set of all analytic functions defined on U such that |a z | < 1 for all z ∈ U
Summary
The class S∗ A, B was investigated by Janowski 1 in early 1970, and since various other subclasses in relation with this Janowski class have been introduced and studied. Motivated by 11 , the class of the generalized Janowski log-harmonic starlike functions was introduced in 12. Let S∗ A, B, α denote the class of the generalized Janowski starlike functions of the analytic functions s z z s2z2 · · · such that s z ∈ S∗ A, B, α if and only if zs z sz pz. For univalent log-harmonic mapping f z zh z g z with g 0 1 and h 0 / 0, f is in the class of the generalized Janowski starlike log-harmonic mapping denoted by S∗lh A, B, α if p z − 1−. We consider the log-harmonic mapping f z zh z g z in the generalized Janowski starlike functions with the functional zh z ∈ S∗ A, B, α. We study the class of generalized Janowski close-to-starlike
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