1 Introduction

Harmonic mappings play the natural role in parameterizing minimal surfaces in the context of differential geometry. Planner harmonic mappings have application not only in the differential geometry but also in the various field of engineering, physics, operations research, and other intriguing aspects of applied mathematics. The theory of harmonic functions has been used to study and solve fluid flow problems (see [Reference Aleman and Constantin10]). The theory of univalent harmonic functions having prominent geometric properties like starlikeness, convexity, and close-to-convexity appear naturally while dealing with planner fluid dynamical problems. For instance, the fluid flow problem on a convex domain satisfying an interesting geometric property has been extensively studied by Aleman and Constantin [Reference Aleman and Constantin10]. With the help of geometric properties of harmonic mappings, Constantin and Martin [Reference Constantin and Martin19] have obtained a complete solution of classifying all two-dimensional fluid flows.

In this paper, our purpose is to investigate several Bohr-type inequalities which will be harmonic analog of the inequalities for bounded analytic functions. Below, we first recall the classical result of Harald Bohr discovered in $ 1914 $ , which generates intensive research activity, what is called Bohr phenomenon.

Let $ \mathcal {B} $ denote the class of analytic functions in the unit disk $ \mathbb {D}:=\{z\in \mathbb {C} : |z|<1\} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ such that $ |f(z)|<1 $ in $ \mathbb {D} $ . In the study of Dirichlet series, in $ 1914 $ , Bohr [Reference Bohr17] discovered the following interesting phenomenon that if $ f\in \mathcal {B} $ , then its associated majorant series:

$$ \begin{align*} M_f(r):=\sum_{n=0}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; |z|=r\leq \frac{1}{3}. \end{align*} $$

The constant $ 1/3 $ , known as the Bohr radius for the class $ \mathcal {B} $ , is the best possible. Furthermore, for $ \psi _a(z)={(a-z)}/{(1-az)},\; a\in [0,1), $ it follows easily that $ {M_{\psi _a}(r)>1 }$ if, and only if, $ r>1/(1+2a) $ , and hence the radius $ 1/3 $ is optimal as $ a\rightarrow 1 $ . For the recent advancements of the Bohr-type inequalities, we refer to the articles [Reference Huang, Liu and Ponnusamy26, Reference Kumar35, Reference Liu38, Reference Ponnusamy, Vijayakumar and Wirths44, Reference Ponnusamy, Vijayakumar and Wirths45] and references therein.

The Bohr inequality can be written in the following distance formulation:

(1.1) $$ \begin{align} d\left(\sum_{n=0}^{\infty}|a_nz^n|,|a_0|\right)=\sum_{n=1}^{\infty}|a_nz^n|\leq 1- |f(0)|=d(f(0),\partial \mathbb{D}), \end{align} $$

for $ |z|=r\leq 1/3 $ and the constant $ 1/3 $ is independent on the coefficients of the Taylor series of $ f $ , where $ d $ is the Euclidean distance and $ \partial f(\mathbb {D}) $ is the boundary of $ f(\mathbb {D}) $ . In view of the distance form (1.1), the notion of the Bohr phenomenon can be generalized to a class $ \mathcal {G} $ consisting of analytic functions $ f $ in $ \mathbb {D} $ which take values in a given domain $ \Omega \subseteq \mathbb {C} $ such that $ f(\mathbb {D})\subset \Omega $ and the class $ \mathcal {G} $ is said to satisfy the Bohr phenomenon if there exists largest radius $ r_{\Omega }\in (0,1) $ such that the inequality (1.1) holds for $ |z|=r\leq r_{\Omega } $ and for all functions $ f\in \mathcal {G} $ . The largest radius $ r_{\Omega } $ is called the Bohr radius for the class  $ \mathcal {G} $ .

A complex-valued function $ f(z)=u(x,y)+iv(x,y) $ is called harmonic in $ \mathbb {D} $ if both $ u $ and $ v $ satisfy the Laplace’s equation $ \bigtriangledown ^2 u=0 $ and $ \bigtriangledown ^2 v=0 $ , where

$$ \begin{align*} \bigtriangledown^2:=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}. \end{align*} $$

It is well known that under the assumption $ g(0)=0 $ , the harmonic function $ f $ has the unique canonical representation $ f=h+\overline {g} $ , where $ h $ and $ g $ are analytic functions in $ \mathbb {D} $ , respectively called, analytic and co-analytic parts of $ f $ . If, in addition, $ f $ is univalent, then we say that $ f $ is univalent harmonic on a domain $ \Omega $ . A locally univalent harmonic mapping $ f=h+\overline {g} $ is sense-preserving whenever its Jacobian $ J_f(z):=|f_{z}(z)|^2-|f_{\bar {z}}(z)|=|h^{\prime }(z)|^2-|g^{\prime }(z)|^2>0 $ for $ z\in \mathbb {D} $ .

Let $ \mathcal {H} $ be the class of all complex-valued harmonic functions $ f=h+\bar {g} $ defined on the unit disk $ \mathbb {D} $ , where $ h $ and $ g $ are analytic $ \mathbb {D} $ with the normalization $ h(0)=h^{\prime }(0)-1=0 $ and $ g(0)=0 $ . Let $ \mathcal {H}_0 $ be defined by $ \mathcal {H}_0=\{f=h+\bar {g}\in \mathcal {H} : g^{\prime }(0)=0\}. $ Then each $ f=h+\bar {g}\in \mathcal {H}_0 $ has the following form:

(1.2) $$ \begin{align} h(z)=z+\sum_{n=2}^{\infty}a_nz^n\quad\mbox{and}\quad g(z)=\sum_{n=1}^{\infty}b_nz^n. \end{align} $$

In 2013, Ponnusamy et al. [Reference Ponnusamy, Yamamoto and Yanagihara46] considered the following class of harmonic mappings:

$$ \begin{align*} \mathcal{P}_{\mathcal{H}}^{0}=\{f=h+\bar{g} \in \mathcal{H} : {\operatorname{Re}\,} h^{\prime}(z)>|g^{\prime}(z)| \;\; \mbox{with} \;\; g^{\prime}(0)=0 \;\; \mbox{for}\;\; z \in \mathbb{D}\} \end{align*} $$

and motivated by the class $\mathcal {P}_{\mathcal {H}}^{0}$ , Li and Ponnusamy [Reference Li and Ponnusamy36] have studied the class $\mathcal {P}_{\mathcal {H}}^{0}(\alpha )$ defined by

$$ \begin{align*} \mathcal{P}_{\mathcal{H}}^{0}(\alpha)=\{f=h+\overline{g} \in \mathcal{H} : {\operatorname{Re}\,} (h^{\prime}(z)-\alpha)>|g^{\prime}(z)|,\; 0\leq\alpha<1,\; g^{\prime}(0)=0\; \mbox{for}\; z \in \mathbb{D}\}. \end{align*} $$

It is easy to see that the class $ \mathcal {H} $ reduces to the class $ \mathcal {A} $ of normalized analytic functions if the co-analytic part of $ f $ is zero. A function $ h\in \mathcal {A} $ is called close-to-convex in $ \mathbb {D} $ if the complement of $ h(\mathbb {D}) $ can be written as the union of nonintersecting half lines. A function $ h\in \mathcal {A} $ is said to be close-to-convex function of order $ \beta \; (0\leq \beta <1) $ if $ \mathrm {Re}\left (h^{\prime }(z)\right )>\beta $ for $ z\in \mathbb {D} $ (see [Reference Rajbala and Prajapat47]). For $ \alpha \geq 0 $ and $ 0\leq \beta <1 $ , let

$$\begin{align*}\mathcal{W}(\alpha, \beta)=\{h\in\mathcal{A} :\mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)\right)>\beta\;\; \mbox{in}\; \mathbb{D}\}. \end{align*}$$

The class $ \mathcal {W}(\alpha , \beta ) $ has been studied by Gao and Zhou [Reference Gao and Zhou23] for $ \beta <1 $ and $ \alpha>1 $ .

In $ 1977 $ , Chichra [Reference Chichra18] introduced the following class $ \mathcal {W}(\alpha ) $ for $ \alpha \geq 0 $ :

$$ \begin{align*} \mathcal{W}(\alpha)=\{ h\in\mathcal{A} : \mathrm{Re} \left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)>0\right)\;\; \mbox{in}\; \mathbb{D}\}. \end{align*} $$

Moreover, Chichra [Reference Chichra18] has proved that functions in the class $ \mathcal {W}(\alpha ) $ constitute a subclass of close-to-convex functions in $ \mathbb {D} $ . In $ 2014 $ , Nagpal and Ravichandran [Reference Nagpal and Ravinchandran42] studied a new subclass $ \mathcal {W}^{0}_{\mathcal {H}} $ of univalent harmonic mappings and obtained the coefficient bounds for functions in the class $ \mathcal {W}^{0}_{\mathcal {H}} $ , where

$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}=\{f=h+\bar{g}\in \mathcal{H} : \mathrm{Re}\left(h^{\prime}(z)+zh^{\prime\prime}(z)\right)> |g^{\prime}(z)+zg^{\prime\prime}(z)|\;\; \mbox{for}\; z\in\mathbb{D}\}. \end{align*} $$

In $ 2019 $ , Ghosh and Allu [Reference Ghosh and Allu24] studied the class $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha ) $ , where

$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}(\alpha)=\{f=h+\bar{g}\in \mathcal{H} : \mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)\right)> |g^{\prime}(z)+\alpha zg^{\prime\prime}(z)|\;\; \mbox{for}\; z\in\mathbb{D}\}. \end{align*} $$

Chichra [Reference Chichra18] has shown that if $ 0\leq \beta <\alpha $ , then $ \mathcal {W}(\alpha )\subset \mathcal {W}(\beta ) $ . Hence, it is easy to see that $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha )\subset \mathcal {W}^{0}_{\mathcal {H}}(\beta ) $ if $ 0\leq \beta <\alpha $ . Therefore, for $ \alpha \geq 1 $ , $\mathcal {W}^{0}_{\mathcal {H}}(\alpha )\subseteq \mathcal {W}^{0}_{\mathcal {H}}(1)$ . Since functions in $ \mathcal {W}^{0}_{\mathcal {H}}(1) $ are starlike, it follows that functions in $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha ) $ are starlike for $ \alpha \geq 1 $ . For $ \alpha \geq 0 $ and $ 0\leq \beta <1 $ , let

$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}(\alpha, \beta)=\{f=h+\overline{g}\in\mathcal{H}^{0} : \mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)-\beta\right)>|g^{\prime}(z)+\alpha zg^{\prime\prime}(z)|\;\; z\in\mathbb{D}\}. \end{align*} $$

It is known that functions in $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $ are close-to-convex harmonic mappings (see [Reference Rajbala and Prajapat47]).

From the following results, it is easy to see that functions in the class $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $ are univalent for $ \alpha \geq 0 $ and $ 0\leq \beta <1 $ , and they are closely related to functions in $ \mathcal {W}(\alpha ,\beta ) $ .

The sharp coefficient bounds and the sharp growth estimates for functions in the class $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $ have been studied in [Reference Rajbala and Prajapat47].

Lemma 1.2 [Reference Rajbala and Prajapat47]

Let $ f\in \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $ for $ \alpha \geq 0 $ , $ 0\leq \beta <1 $ and be of the form (1.2). Then, for any $ n\geq 2 $ ,

1. (i) $ |a_n|+|b_n|\leq \displaystyle \frac {2(1-\beta )}{n\left (1+\alpha (n-1)\right )} $ ;

2. (ii) $ ||a_n|-|b_n||\leq \displaystyle \frac {2(1-\beta )}{n\left (1+\alpha (n-1)\right )} $ ;

3. (iii) $ |a_n|\leq \displaystyle \frac {2(1-\beta )}{n\left (1+\alpha (n-1)\right )} $ .

All these inequalities are sharp for the function $ f=f_{\alpha , \beta } $ given by

(1.3) $$ \begin{align} f_{\alpha, \beta}(z)=z+\sum_{n=2}^{\infty}\frac{2(1-\beta){\bar{z}}^n}{n\left(1+\alpha(n-1)\right)}. \end{align} $$

Lemma 1.3 [Reference Rajbala and Prajapat47]

Let $ f\in \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $ and be of the form (1.2) with $ 0<\alpha \leq 1 $ and $ {0\leq \beta <1 }$ . Then

(1.4) $$ \begin{align} |z|+\sum_{n=2}^{\infty}\frac{2(1-\beta)(-1)^{n-1}|z|^n}{n\left(1+\alpha(n-1)\right)}\leq |f(z)|\leq |z|+\sum_{n=2}^{\infty}\frac{2(1-\beta)|z|^n}{n\left(1+\alpha(n-1)\right)}. \end{align} $$

Both the inequalities are sharp for the function $ f=f_{\alpha , \beta } $ given by (1.3).

Bohr–Rogosinski radius is an analogous to the Bohr radius which has been defined (see [Reference Rogosinski48]) as follows: If $ f\in \mathcal {B} $ is given by $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ , then for an integer $ N\geq 1 $ , we have $ |S_N(z)|<1 $ in the disk $ \{z\in \mathbb {C} : |z|<1/2\} $ and the radius $ 1/2 $ is the best possible, where $ S_N(z)=\sum _{n=0}^{N}a_nz^n $ denotes the $ N $ th partial sum of $ f $ . The radius $ r=1/2 $ is called the Rogosinski radius. For $ f\in \mathcal {B} $ is given by $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ , the Bohr–Rogosinski sum $ R^f_N(z) $ of $ f $ defined by

(1.5) $$ \begin{align} R^f_N(z):=|f(z)|+\sum_{n=N}^{\infty}|a_n|r^n,\;\; |z|=r. \end{align} $$

It is worth to notice that for $ N=1$ , the quantity (1.5) is related to the classical Bohr sum in which $ |f(0)| $ is replaced by $ |f(z)| $ . The inequality $ R^f_N(z)\leq 1 $ is called Bohr–Rogosinski inequality. If $ B $ and $ R $ denote the Bohr radius and Bohr–Rogosinski radius, respectively, then $ B\leq R $ because $ |S_N(z)|\leq \sum _{n=0}^{N}|a_n||z|^n\leq \sum _{n=0}^{\infty }|a_n||z|^n $ .

In $ 2005 $ , Aizenberg et al. [Reference Aizenberg, Elin and Shoikhet8] generalized the Bohr–Rogosinski inequality for the holomorphic mappings of the open unit ball into an arbitrary convex domain as well as studied the multidimensional analog of Rogosinski’s theorem with some applications. In $ 2009 $ , Aizenberg [Reference Aizenberg, Gotliv and Vidras9] proved that the abscissas of Bohr and Rogosinski for ordinary Dirichlet series, mapping the right half-plane into the bounded convex domain $ G\subset \mathbb {C} $ are independent of the domain. In $ 2012 $ , Aizenberg [Reference Aizenberg7] studied Bohr and Rogosinski radii for Hardy classes $ H^p $ of holomorphic functions in the unit disk $ \mathbb {D} $ as well as the Bohr and Rogosinski radii for the mappings of Reinhradt domains in $ \mathbb {C}^n $ into Reinhardt domains in $ \mathbb {C}^n $ . In $ 2020 $ , Alkhaleefah et al. [Reference Alkhaleefah, Kayumov and Ponnusamy12] obtained Bohr–Rogosinski radius for analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ in unit disk $ \mathbb {D} $ replacing the coefficient $ a_n $ of the power series by the derivatives $ f^{(n)}(z)/n! $ . Recently, Huang et al. [Reference Huang, Liu and Ponnusamy25] have generalized and improved some refined versions of Bohr–Rogosinski inequalities which have been studied by Liu et al. [Reference Liu, Liu and Ponnusamy39]. In $ 2021 $ , Kayumov et al. [Reference Kayumov, Khammatova and Ponnusamy29] investigated Bohr–Rogosinski phenomenon for analytic functions defined on $ \mathbb {D} $ in a general setting and derived Bohr–Rogosinski radii for Cesáro operators on the space of bounded analytic functions. For the extensive study of Bohr phenomenon including recent developments, we refer the reader to the articles [Reference Abu Muhanna1–Reference Ahamed, Allu and Halder5, Reference Alkhaleefah, Kayumov and Ponnusamy11, Reference Allu and Halder14–Reference Boas and Khavinson16, Reference Das20–Reference Evdoridis, Ponnusamy and Rasila22, Reference Huang, Liu and Ponnusamy27, Reference Ismagilov, Kayumov and Ponnusamy28, Reference Kayumov and Ponnusamy30–Reference Kayumov and Ponnusamy32, Reference Kayumov, Ponnusamy and Shakirov34, Reference Lin, Liu and Ponnusamy37, Reference Liu and Ponnusamy40, Reference Liu and Ponnusamy41, Reference Paulsen and Singh43].

In $ 2017 $ , Kayumov and Ponnusamy [Reference Kayumov and Ponnusamy33] obtained the following result on the Bohr–Rogosinski radius for the analytic functions in the unit disk $ \mathbb {D} $ .

Theorem 1.1 [Reference Kayumov and Ponnusamy33]

Suppose that $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ is analytic in the unit disk $ \mathbb {D} $ and $ |f(z)|<1 $ in $ \mathbb {D} $ . Then

$$ \begin{align*} |f(z)|+\sum_{n=N}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq R_N, \end{align*} $$

where $ R_N $ is the positive root of the equation $ 2(1+r)r^N-(1-r^2)=0 $ . The radius $ R_N $ is the best possible. Moreover,

$$ \begin{align*} |f(z)|^2+\sum_{n=N}^{\infty}|a_n|r^n\leq 1\;\;\mbox{for}\;\; r\leq R^{\prime}_N, \end{align*} $$

where $ R^{\prime }_N $ is the positive root of the equation $ (1+r)r^N-(1-r^2)=0 $ . The radius $ R^{\prime }_N $ is the best possible.

Recently, Kayumov and Ponnusamy [Reference Kayumov and Ponnusamy33] have obtained another interesting result concerning Bohr–Rogosinski radius considering some power of $ z $ in $ f(z) $ as the following.

Theorem 1.2 [Reference Kayumov and Ponnusamy33]

Suppose that $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ is analytic in $ \mathbb {D} $ such that $ |f(z)|\leq 1 $ in $ \mathbb {D} $ . Then, for each $ m, N\in \mathbb {N} $ ,

$$ \begin{align*} |f(z^m)|+\sum_{n=N}^{\infty}|a_n|r^n\leq 1\;\; \mbox{for}\;\; r\leq R_{m,N}, \end{align*} $$

where $ R_{m,N} $ is the positive root of of the equation $ {2r^N\left (1+r^m\right )-(1-r)\left (1-r^m\right )=0,} $ and the number $ R_{m,N} $ cannot be improved. Moreover, $ \lim _{N\rightarrow \infty }R_{m,N}=1 $ and $\lim _{m\rightarrow \infty }R_{m,N}=A_N, $ where $ A_N $ is the positive root of the equation $ 2r^N=1-r $ . Also, $ A_1=1/3 $ and $ A_2=1/2 $ .

Study of sharp refine Bohr–Rogosinski inequalities for functions in the class $ \mathcal {B} $ is an important topic in the geometric function theory. The refine Bohr–Rogosinski inequality for the class $ \mathcal {B} $ is extensively studied by Liu et al. [Reference Liu, Liu and Ponnusamy39] and Ponnusamy et al. [Reference Ponnusamy, Vijayakumar and Wirths44]. For $ N=1 $ , it is easy to see that $ R_1=\sqrt {5}-2 $ and $ R^{\prime }_1=1/3 $ . However, recently, Liu et al. [Reference Liu, Liu and Ponnusamy39] proved the following interesting result showing that the two constants $ R_1 $ and $ R^{\prime }_1 $ can be improved for any individual function in $ \mathcal {B} $ (in the context of Theorem 1.1 with $ N=1 $ ).

Theorem 1.3 [Reference Liu, Liu and Ponnusamy39]

Suppose that $ f\in \mathcal {B} $ and $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ . Then

$$ \begin{align*} A(z):=|f(z)|+\sum_{n=1}^{\infty}|a_n|r^n+\left(\frac{1}{1+|a_1|}+\frac{r}{1-r}\right)\sum_{n=1}^{\infty}|a_n|^2r^{2n}\leq 1 \end{align*} $$

for $ |z|=r\leq r_{a_0}=2/\left (3+|a_0|+\sqrt {5}\left (1+|a_0|\right )\right ) $ . The radius $ r_{a_0} $ is best possible and $ {r_{a_0}>\sqrt {5}-2} $ . Moreover,

$$ \begin{align*} B(z):=|f(z)|^2+\sum_{n=1}^{\infty}|a_n|r^n+\left(\frac{1}{1+|a_1|}+\frac{r}{1-r}\right)\sum_{n=1}^{\infty}|a_n|^2r^{2n}\leq 1 \end{align*} $$

for $ |z|=r\leq r^{\prime }_{a_0} $ , where $ r^{\prime }_{a_0} $ is the unique positive root of the equation

$$ \begin{align*} \left(1-|a_0|^3\right)r^3-\left(1+2|a_0|\right)r^2-2r+1=0. \end{align*} $$

The radius $ r^{\prime }_{a_0} $ is best possible and $ 1/3<r^{\prime }_{a_0}<1/(2+|a_0|) $ .

Motivated from the papers [Reference Ahamed, Allu and Halder6, Reference Allu and Halder13], the main objective of this paper is to find the sharp Bohr–Rogosinski radius which are the harmonic analog of Theorems 1.1–1.3 for the class $ \mathcal {W}^0_{\mathcal {H}}(\alpha , \beta ) $ for $ \alpha \geq 0 $ and $ 0\leq \beta <1 $ . In Section 2, we state the main results of the paper and the proof of the main results will be discussed in Section 3.