The Bohr inequality for certain harmonic mappings

Abstract

Let $\varphi$ be analytic and univalent ({\it i.e.,} one-to-one) in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ such that $\varphi (\mathbb{D})$ has positive real part, is symmetric with respect to the real axis, starlike with respect to $\varphi (0)=1,$ and ${\varphi }^{\prime }(0)>0$. A function $f\in \mathcal{C}(\varphi )$ if $1+z{f}^{\prime \prime }(z)/f^{\prime }(z)\prec \varphi (z),$ and $f\in {\mathcal{C}}_c(\varphi )$ if $2(z{f}^{\prime }(z){)}^{\prime }/(f(z)+\overline{f(\bar{z})}{)}^{\prime }\prec \varphi (z)$ for $z\in \mathbb{D}$. In this article, we consider the classes $\mathcal{H}\mathcal{C}(\varphi )$ and ${\mathcal{H}\mathcal{C}}_c(\varphi )$ consisting of harmonic mappings $f=h+\overline{g}$ of the form $$ h(z)=z+ \sum \limits_{n=2}^{\infty} a_{n}z^{n} \quad \mbox{and} \quad g(z)=\sum \limits_{n=2}^{\infty} b_{n}z^{n} $$ in the unit disk $\mathbb{D}$, where $h$ belongs to $\mathcal{C}(\varphi )$ and ${\mathcal{C}}_c(\varphi )$ respectively, with the dilation $g^{\prime }(z)=\alpha z{h}^{\prime }(z)$ and $|\alpha |<1$. Using the Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we find the radius $R_f<1$ such that Bohr inequality $$|z|+\sum _{n=2}^{\infty }(|a_n|+|b_n|)|z{|}^n\le d(f(0),\partial f(\mathbb{D}))$$ holds for $|z|=r\le R_f$ for the classes $\mathcal{H}\mathcal{C}(\varphi )$ and ${\mathcal{H}\mathcal{C}}_c(\varphi )$ . As a consequence of these results, we obtain several interesting corollaries on Bohr inequality for the aforesaid classes.

18 pages, 4 figures

References in corpus (1)

• Bohr phenomenon for certain close-to-convex analytic functions

Cited by in corpus (2)

• Bohr inequalities for unimodular bounded functions on simply connected domains
• Bohr radius for certain close-to-convex harmonic mappings