The Bohr radius for starlike logharmonic mappings

Abstract

This paper studies the class consisting of univalent logharmonic mappings f (z) =are analytic in U and ϕ(z) = zh(z)/g(z) is a normalized starlike analytic function. A representation theorem for these mappings is obtained, which yields sharp distortion estimates, and a sharp Bohr radius.

Some properties of univalent log-harmonic mappings

“…In a manner similar in the proof of [11,Theorem 2], we see that the function Φ r (l) is continuous and is even in the interval |l| ≤ π/2. Hence…”

“…If α = 0, then Theorem 3.1 reduces to Theorem 3 in [11]. If α → 1, then r H = r G = 1/3, and r f = 3 − 2 √ 2 which is same as Bohr's radius of the subordinating family of univalent functions (see [9, Theorem 1]) which we recall for a ready reference below.…”

Notes on the starlike log-harmonic mappings of order alpha

“…where Re{β} > −1/2, h and g are in H(∆), h(0) = 0 and g(0) = 1 (see [3]). The class of functions of this form has been studied extensively by many works, see for example [1]- [6] and [9]. We continue the discussion with an example.…”

Modifications of Bohr's inequality in various settings

“…For instance, Kayumov and Ponnusamy [19,20] determined the Bohr radius for the class of p-symmetric analytic functions with multiple zeros at the origin, and introduced the notion of p-Bohr radius for harmonic functions and obtained the p-Bohr radius for the class of odd harmonic bounded functions (see also [4,22]) while in [21] the same authors discussed powered Bohr radius, originally discussed by Djakov and Ramanujan [14]. Aytuna and Djakov [6] studied the Bohr property of bases for holomorphic functions, and Ali et al [3] discussed the Bohr radius for the class of starlike logharmonic mappings. For further studies on the Bohr phenomenon, we refer to the survey articles [2,16] and the references therein.…”