On Certain Bounds of Harmonic Univalent Functions

Abstract

Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk $\mathbb{U}=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}$ can be written as a sum $f=h+\overline{g}$, where h and g are analytic functions in $\mathbb{U}$ and are called the analytic part and the co-analytic part of f, respectively. In this paper, the harmonic shear $f=h+\overline{g}\in S_{\mathcal{H}}$ and its rotation $f^{\mu }$ by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$ are considered. Bounds are established for this rotation $f^{\mu }$, specific inequalities that define the Jacobian of $f^{\mu }$ are obtained, and the integral representation is determined.

1. Introduction

We begin with the fundamental property that, for a holomorphic function, both the imaginary and real parts are harmonic functions. These functions have considerable significance for Geometric Function Theory. It is in the commonly known context of this theory that the present investigation considers the harmonic functions associated with the rotation by $\mu$, continuing the studies on the bounds that characterize the harmonic functions.

Considering $\mathbb{U}=\left\{z\in \mathbb{C}:\left|z\right|<1\right\},$ the functions of the form $f=u+iv$, where u and v are real-valued harmonic functions in $\mathbb{U}$, form the class denoted by $\mathcal{H}$ of complex-valued harmonic functions. Another way to express the functions f $\in \mathcal{H}$ is by using the analytic functions h and g in $\mathbb{U},$ writing $f=h+\overline{g},$ and referring to h as the analytic component of f and to g as the co-analytic component of f.

For a function $f=h+\overline{g}\in \mathcal{H}$, the Jacobian is represented by $J_f\left(z\right)=|h^{\prime }\left(z\right){|}^2-{\left|g^{\prime }\left(z\right)\right|}^2$. If and only if $J_f\left(z\right)>0$ in $\mathbb{U}$, Lewy [1] states that harmonic functions are sense-preserving and locally univalent in $\mathbb{U}$. This is equivalent to

$${\omega }_f\left(z\right)={\displaystyle \frac{g^{\prime }\left(z\right)}{h^{\prime }\left(z\right)}},z\in \mathbb{U},\mathrm{analytic}\mathrm{in}\mathbb{U},\mathrm{such}\mathrm{that}{|\omega }_f\left(z\right)|<1\mathrm{for}\mathrm{all}z\in \mathbb{U}.$$

(1)

The dilation of f is the function ${\omega }_f\left(z\right)$. Details on this notion can be found in [2].

Given the normalization requirements $h\left(0\right)=g\left(0\right)=0$ and $h^{\prime }\left(0\right)=1$, the class $S_{\mathcal{H}}$ represents all the harmonic functions $f=h+\overline{g}\in \mathcal{H}$ that are univalent and sense-preserving in $\mathbb{U}$. If $f=h+\overline{g}\in S_{\mathcal{H}},$ we have

$$h\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n\mathrm{and}g\left(z\right)=\sum _{n=1}^{\infty }{b}_n{z}^n\left(\left|b_1\right|<1;z\in \mathbb{U}\right).$$

(2)

Imposing the condition $g^{\prime }\left(0\right)=0\left(\mathrm{or}{\omega }_f\left(0\right)=0\right)$, $S_{\mathcal{H}}^0$ denotes the subclass of functions $f=h+\overline{g}\in S_{\mathcal{H}}$ satisfying it. Further, the subclass of functions $f\in S_{\mathcal{H}}^0$ that map $\mathbb{U}$ onto a convex region is denoted by $K_{\mathcal{H}}^0.$

Recently, harmonic functions have been thoroughly investigated from different points of view. Fluid flow problems have been studied and resolved using harmonic mapping techniques (see [3]). A subclass of $S_{\mathcal{H}}$ concerning functions with bounded radius rotation was the focus of the research presented in [4]. As generalizations of analytic mappings, the concept of differential subordination was investigated for harmonic mappings in [5,6]. New subclasses of harmonic functions associated with differential inequalities of different types of operators have also been introduced [7,8,9].

In [10], Clunie and Sheil-Small investigated the class $S_{\mathcal{H}}$, as well as its relevant geometric subclasses for which certain coefficient bounds were obtained, thus initiating the study of harmonic functions. Since then, interest in the study of the class of functions $S_{\mathcal{H}}$ has increased. Appropriate analyses of the class and its relevant subclasses have yielded interesting publications concerning coefficient conditions, extreme points, and convolution and convex combination properties. An early study by Avci and Zlotkiewicz [11] provided sufficient coefficient conditions such that a function $f=h+\overline{g}\in \mathcal{H}$ is in the class $K_{\mathcal{H}}^0$. Silverman [12] determined the sufficient conditions imposed on the coefficients of the functions $f=h+\overline{g}\in \mathcal{H}$ to map onto convex or starlike domains, further proving that the same conditions are necessary if the functions have negative coefficients. Silverman and Silvia [13], Jahangiri [14], and Frasin [15] investigated different subclasses of harmonic functions. Ahuja et al. [16] investigated certain convolution properties for harmonic univalent functions, while Jahangiri et al. [17] used the Alexander integral transform to investigate the construction of sense-preserving, univalent, and close-to-convex harmonic functions. Yalçın [18] and Caglar et al. [19] investigated the properties of generalized classes of harmonic univalent functions using a modified Sălăgean operator. The connections of harmonic mappings with hypergeometric functions were studied by Ahuja [20,21].

Jahangiri [22] was the first to introduce the q-analogue of complex harmonic functions and studied various geometric properties. Recently, certain subclasses of $S_{\mathcal{H}}$ associated with operators and q-operators have been discussed by several researchers [23,24,25].

Very recently, interesting applications of harmonic functions using power series with coefficients following a form of probability distribution have been considered by many authors in the literature, for example, El-Ashwah and Kota [26] and Frasin et al. [27]. Meanwhile, subclasses of multivalent harmonic functions were considered in the publication of Oros et al. [28].

Motivated by the work carried out by Polatoğlu et al. [29] and Sharma and Mishra [30], the present investigation focuses on the harmonic functions $f=h+\overline{g}\in \mathcal{H}$ associated with the rotation $f^{\mu }$ by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$, for which bounds are established. The Jacobian is characterized by particular inequalities, and the integral representation is established.

The main concept used for this investigation concerns the convexity in a certain direction. Related studies on convexity can be seen in [31]. Also, on local convexity, one can see [32].

The property of convexity in the direction $\gamma \in [0,\pi )$ of a domain $\Omega$ $\subset \mathbb{C}$ is given by the set $\Omega$ ∩$\{t+r{e}^{i\gamma }:r\in \mathbb{R}\}$, which is either empty or a connected set. For a function to have the property of being convex in the direction $\gamma$, it must be a univalent function mapping $\mathbb{U}$ onto a domain that is convex in the same direction.

Shear construction, often known as shearing, is a technique used to generate a univalent harmonic mapping starting with the related conformal map studied by Clunie and Sheil-Small ([10]) in 1984. The following generalized result is obtained using this technique to generate a harmonic univalent map that is convex in a specified direction:

Lemma 1 

([10]). A locally univalent harmonic function $f=h+\overline{g}$ in $\mathbb{U}$ is a univalent harmonic mapping of $\mathbb{U}$ onto a domain convex in a direction φ if and only if $h-e^{2i\phi }g$ is a univalent analytic mapping of $\mathbb{U}$ onto a domain convex in the direction $\phi \in [0,\pi ).$

Hengartner and Schober [33] investigated the analytic functions $\psi \left(z\right)$ that exhibit convexity along the imaginary axis, employing a normalization that fundamentally requires that the right and left boundaries of $\psi \left(\mathbb{U}\right)$ correspond to the images of 1 and $-1$. The normalization can be formulated as follows: there exist points $z_n^{\prime }$ converging to 1 and $z_n^{″}$ converging to $-1$ such that

$$\underset{n\to \infty }{lim}\Re e\left\{\psi \left(z_n^{\prime }\right)\right\}=\underset{\left|z\right|<1}{sup}\Re e\left\{\psi \left(z\right)\right\}$$

(3)

and

$$\underset{n\to \infty }{lim}\Re e\left\{\psi \left(z_n^{″}\right)\right\}=\underset{\left|z\right|<1}{inf}\Re e\left\{\psi \left(z\right)\right\}.$$

With $\mathcal{CIA}$ denoting the class of domains $\mathcal{D}$ exhibiting convexity along the imaginary axis with $\psi \left(\mathbb{U}\right)=\mathcal{D}$, considering the given normalization (3), the following result is obtained:

Theorem 1 

([33]). If ψ is analytic and non-constant for $\left|z\right|<1$, we have

$$\Re e\left\{\left(1-z^2\right){\psi }^{\prime }\left(z\right)\right\}\ge 0,$$

for $\left|z\right|<1$ if and only if the following conditions are satisfied:

(i) 

ψ is univalent on $\mathbb{U}$;

(ii) 

$\psi \left(\mathbb{U}\right)\in \mathcal{CIA}$;

(iii) 

$\psi \left(z\right)$ is normalized by (3).

Using the characterization from Theorem 1, the next theorem was established by Hengartner and Schober in [33].

Theorem 2 

([33]). If $\psi \left(z\right)$ is analytic for $\left|z\right|<1$ and satisfies

$$\Re e\left\{\left(1-z^2\right){\psi }^{\prime }\left(z\right)\right\}\ge 0,$$

for $\left|z\right|\le r\left(r<1\right),$

$${\displaystyle \frac{\left(1-r\right)\left|{\psi }^{\prime }\left(0\right)\right|}{\left(1+r\right)\left(1+r^2\right)}}\le \left|{\psi }^{\prime }\left(z\right)\right|\le {\displaystyle \frac{\left|{\psi }^{\prime }\left(0\right)\right|}{{\left(1-r\right)}^2}}.$$

The upper bound is sharp for $\psi \left(z\right)={\displaystyle \frac{z}{1-z}}$, and the lower bound is sharp for $\psi \left(z\right)={\displaystyle \frac{i}{2}}log{\displaystyle \frac{{\left(1-iz\right)}^2}{\left(1-z^2\right)}}.$

Royster and Ziegler [34] expanded on the research conducted by Hengartner and Schober [33] by examining the points $e^{i\left(u-v\right)}$ and $e^{i\left(u+v\right)}$, which are designated as the right and left extremes of $\psi \left(\mathbb{U}\right)$, and established the following theorem:

Theorem 3 

([34]). Let $\psi \left(z\right)$ be a non-constant function regular in $\mathbb{U}.$ The function $\psi \left(z\right)$ maps univalently onto a domain Ω convex in the direction of the imaginary axis if and only if there are numbers u and $v,0\le u<2\pi$ and $0\le v<\pi$ such that

$$\Re e\left\{-i{e}^{iu}\left(1-2cosv{e}^{-iu}z+e^{-2iu}{z}^2\right){\psi }^{\prime }\left(z\right)\right\}\ge 0,z\in \mathbb{U}.$$

(4)

Furthermore, $\psi \left(e^{i\left(u-v\right)}\right)$ and $\psi \left(e^{i\left(u+v\right)}\right)$ are the right and left extremes, respectively, of $\Omega .$

Employing the normalization (4), Royster and Ziegler introduced a class $\Gamma$ of normalized analytic univalent functions that map $\mathbb{U}$ onto domains convex in the direction of the imaginary axis. It was established that $\psi \in \Gamma$ if and only if $\psi \left(0\right)=0$ and ${\psi }^{\prime }\left(0\right)=1,$ with $\psi \left(z\right)$ satisfying (4) for specifically chosen u and v. Denote $\Gamma =\Gamma \left(u,v\right)$ and $\Gamma \left(v\right)=\underset{u}{\cup }\Gamma \left(u,v\right).$ The bounds of $\left|{\psi }^{\prime }\left(z\right)\right|$ are given as follows:

Theorem 4 

([34]). If $\psi \in \Gamma \left(v\right),$ for $\left|z\right|\le r\left(r<1\right),$

$$\left|{\psi }^{\prime }\left(z\right)\right|\ge {\displaystyle \frac{1+r}{\left(1+r\right)\left(1+2r\left|cosv\right|+r^2\right)}}$$

(5)

and

$$\left|{\psi }^{\prime }\left(z\right)\right|\le \left\{\begin{array}{cc}{\displaystyle \frac{1-r}{\left(1-r\right)\left(1-2r\left|cosv\right|+r^2\right)}},& r<{\displaystyle \frac{1-sinu}{\left|cosv\right|}},\\ {\displaystyle \frac{1}{sinv{\left(1-r\right)}^2}},& {\displaystyle \frac{1-sinu}{\left|cosv\right|}}\le r<1.\end{array}\right.$$

(6)

Inequality (6) should be interpreted to mean that the top inequality is used for all r when v is 0 or $\pi ,$ and the bottom inequality is used for all r when $v={\displaystyle \frac{\pi }{2}}.$

Lemma 2 

([35]). Let f be an analytic function in $\mathbb{D}$ with $f\left(0\right)=0$ and $f^{\prime }\left(0\right)\ne 0.$ Suppose also that

$$\phi \left(z\right)={\displaystyle \frac{z}{\left(1+z{e}^{i\theta }\right)\left(1+z{e}^{-i\theta }\right)}}\left(\theta \in \mathbb{R};z\in \mathbb{D}\right).$$

(7)

If

$$\Re \left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{\phi \left(z\right)}}\right)>0\left(z\in \mathbb{D}\right),$$

(8)

f is convex in the direction of the real axis.

All functions $\varphi$ that are analytic in $\mathbb{U},$ satisfying $\varphi \left(0\right)=0$ and ${\varphi }^{\prime }\left(0\right)=1$, form the class denoted by $\mathcal{A}$. Let h and g be of the form (2), and let ${\displaystyle \frac{h\left(z\right)-g\left(z\right)}{1-b_1}}\in \mathcal{A}$ be a function that exhibits convexity along the real axis. Then, $f=h+\overline{g}\in S_{\mathcal{H}}$ are the horizontal shears of ${\displaystyle \frac{h\left(z\right)-g\left(z\right)}{1-b_1}}$, and $f^{\mu }$ are the rotations of f by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right).$

Definition 1 

([30]). The rotations of f and ϕ by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$, denoted by $f^{\mu }$ and ${\varphi }^{\mu }$, are given, respectively, by

$$f^{\mu }\left(z\right)=\overline{\mu }f\left(\mu z\right)\mathit{and}{\varphi }^{\mu }\left(z\right)=\overline{\mu }\varphi \left(\mu z\right).$$

In this paper, the harmonic shear $f=h+\overline{g}\in S_{\mathcal{H}}$ and its rotation $f^{\mu }$ by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$ are considered, and results based on certain bounds related to the rotation function $f^{\mu }$ are derived.

We note that certain results on bounds for the function $f=h+\overline{g}\in S_{\mathcal{H}},$ convex in the direction of the real axis, were previously obtained by Polatoğlu et al. [29] and Schaubroeck [36].

In the next section, some new results are proved through a proposition, with an associated corollary and two lemmas used as tools for the proofs of the main results.

2. Preliminaries

If $\varphi \in \mathcal{A}$ is convex in the direction of the real axis, then ${\varphi }^{\mu }$ is convex in the direction of $\overline{\mu }$. We first prove the following proposition based on the rotation by $\mu$.

Proposition 1. 

Consider $f=h+\overline{g}\in S_{\mathcal{H}},$ with h and g having the form seen in (2), satisfying

$${\displaystyle \frac{h\left(z\right)-g\left(z\right)}{1-b_1}}=\varphi \left(z\right),$$

(9)

when $\varphi \in$$\mathcal{A}$ is convex in the direction of the real axis. Then, $f^{\mu }=H+\overline{G}\in S_{\mathcal{H}}$ when $H\left(z\right)=\overline{\mu }h\left(\mu z\right)$ and $G\left(z\right)=\mu g\left(\mu z\right)$ satisfy ${\displaystyle \frac{H\left(z\right)-{\overline{\mu }}^{2}G\left(z\right)}{1-b_1}}={\varphi }^{\mu }\left(z\right),$ and it is convex in the direction of $\overline{\mu }.$

Proof. 

Consider $f=h+\overline{g}\in S_{\mathcal{H}}.$ We have

$$f^{\mu }\left(z\right)=\overline{\mu }f\left(\mu z\right)=\overline{\mu }h\left(\mu z\right)+\overline{\mu g\left(\mu z\right)}=H+\overline{G}.$$

Also, from (9), we obtain

$${\displaystyle \frac{H\left(z\right)-{\overline{\mu }}^{2}G\left(z\right)}{1-b_1}}={\varphi }^{\mu }\left(z\right),$$

(10)

which is a convex function in the direction of $\overline{\mu }.$ By using Lemma 1 for shearing, the harmonic function $f^{\mu }=H+\overline{G}$ is generated, which is convex in the direction of $\overline{\mu }.$ Additionally, we obtain

$$\begin{array}{ccc}\hfill \left|{\omega }_{f^{\mu }}\left(z\right)\right|& =& \left|{\displaystyle \frac{G^{\prime }\left(z\right)}{H^{\prime }\left(z\right)}}\right|\hfill \\ & =& \left|{\displaystyle \frac{{\mu }^2{g}^{\prime }\left(\mu z\right)}{h^{\prime }\left(\mu z\right)}}\right|=\left|{\displaystyle \frac{g^{\prime }\left(\mu z\right)}{h^{\prime }\left(\mu z\right)}}\right|=\left|{\omega }_f\left(\mu z\right)\right|<1,\hfill \end{array}$$

which proves that $f^{\mu }\in S_{\mathcal{H}}.$ This proves the proposition. □

Remark 1. 

If $b_1=0$, the harmonic functions f and $f^{\mu }$ involved in Proposition 1 are members of the class $S_{\mathcal{H}}^0$.

With $\mu =e^{i\gamma }$ and $\varphi \left(z\right)={\displaystyle \frac{z}{1-z}}$ in Proposition 1, particularly chosen, the following outcome is derived:

Corollary 1. 

Consider $f=h+\overline{g}\in S_{\mathcal{H}},$ with h and g having the form seen in (2), satisfying

$${\displaystyle \frac{h\left(z\right)-g\left(z\right)}{1-b_1}}={\displaystyle \frac{z}{1-z}}\left(z\in \mathbb{U}\right),$$

(11)

which is a convex function in the direction of the real axis. Consider $\gamma \in \left[0,\pi \right).$ Then, $e^{-i\gamma }f\left(e^{i\gamma }z\right)=:f_{\gamma }=\mathcal{H}+\overline{\mathcal{G}}\in S_{\mathcal{H}}$ when $\mathcal{H}\left(z\right)=e^{-i\gamma }h\left(e^{i\gamma }z\right)$ and $\mathcal{G}\left(z\right)=e^{i\gamma }h\left(e^{i\gamma }z\right)$ satisfy ${\displaystyle \frac{\mathcal{H}\left(z\right)-e^{-2i\gamma }\mathcal{G}\left(z\right)}{1-b_1}}={\displaystyle \frac{z}{1-e^{i\gamma }z}}$, and this function is convex in the direction of $e^{-i\gamma }$(or in the direction of $-\gamma$).

The following lemma is necessary for the proofs of the main results. The concept of differential subordination is applied, as is known in geometric function theory, meaning that with $f,g\in \mathcal{A}$, function f is subordinate to function g, written as $f\left(z\right)\prec g\left(z\right)$, if and only if there exists a function $\omega \in \mathcal{A}$ satisfying $\omega \left(0\right)=0,\left|\omega \right(z\left)\right|<1$ for $z\in \mathbb{U}$ and $f\left(z\right)=g\left(\omega \right(z\left)\right).$ When g is univalent in $\mathbb{U},$ the subordination is characterized by

$$f\left(z\right)\prec g\left(z\right)⟺f\left(0\right)=g\left(0\right)\mathrm{and}f\left(\mathbb{U}\right)\subset g\left(\mathbb{U}\right).$$

(12)

Lemma 3. 

Let $f=h+\overline{g}\in S_{\mathcal{H}},$ where h and g are of the form (2), be such that

$${\displaystyle \frac{h\left(z\right)-g\left(z\right)}{1-b_1}}=\varphi \left(z\right),$$

(13)

where $\varphi \in \mathcal{A}$ is convex in the direction of the real axis. Let $f^{\mu }$ be a rotation of f by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$, and let ${\omega }_{f^{\mu }}$ be its dilation. Then, for $\left|z\right|\le r\left(r<1\right),$

$${\displaystyle \frac{|b_1|-r}{1-|b_1|r}}\le \left|{\omega }_{f^{\mu }}\left(z\right)\right|\le {\displaystyle \frac{r+|b_1|}{1+|b_1|r}},$$

(14)

$${\displaystyle \frac{(1-r)(1-|b_1\left|\right)}{1+|b_1|r}}\le 1-\left|{\omega }_{f^{\mu }}\left(z\right)\right|\le \left\{\begin{array}{cc}{\displaystyle \frac{(1+r)(1-|b_1\left|\right)}{1-|b_1|r}},& r<|b_1|,\\ {\displaystyle \frac{(1-r)(1+|b_1\left|\right)}{1-|b_1|r}},& r>|b_1|,\end{array}\right.$$

(15)

$$\left.\begin{array}{cc}{\displaystyle \frac{(1-r)(1+|b_1\left|\right)}{1-|b_1|r}},& r<|b_1|\\ {\displaystyle \frac{(1+r)(1-|b_1\left|\right)}{1-|b_1|r}},& r>|b_1|\end{array}\right\}\le 1+\left|{\omega }_{f^{\mu }}\left(z\right)\right|\le {\displaystyle \frac{(1+r)(1+|b_1\left|\right)}{1+|b_1|r}},$$

(16)

$${\displaystyle \frac{{(1-r)}^2(1-|b_1{|}^2)}{1-|b_1{|}^2{r}^2}}\le 1-{\left|{\omega }_{f^{\mu }}\left(z\right)\right|}^2\le \left\{\begin{array}{cc}{\displaystyle \frac{{(1+r)}^2(1-|b_1{|}^2)}{1-|b_1{|}^2{r}^2}},& r<|b_1|,\\ {\displaystyle \frac{(1-r^2)(1+|b_1{\left|\right)}^2}{1-|b_1{|}^2{r}^2}},& r>|b_1|.\end{array}\right.$$

(17)

Proof. 

From Proposition 1, we write

$$f^{\mu }\left(z\right)=H+\overline{G},$$

and using (10), we obtain

$${\displaystyle \frac{H^{\prime }\left(z\right)\left(1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)\right)}{1-b_1}}={\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right),$$

which shows that

$${\omega }_{f^{\mu }}\left(0\right)=b_1{\mu }^2.$$

Hence, there exists a Schwarz-class function $w\left(z\right)$ given by

$$w\left(z\right)={\displaystyle \frac{{\omega }_{f^{\mu }}\left(z\right)-b_1{\mu }^2}{1-\overline{b_1}{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)}}$$

which gives the representation of the dilation function ${\omega }_{f^{\mu }}\left(z\right)$ as follows:

$${\omega }_{f^{\mu }}\left(z\right)={\displaystyle \frac{b_1{\mu }^2+w\left(z\right)}{1+\overline{b_1}{\overline{\mu }}^{2}w\left(z\right)}}$$

and therefore, it implies that

$${\omega }_{f^{\mu }}\left(z\right)\prec p\left(z\right),$$

(18)

where

$$p\left(z\right)={\displaystyle \frac{b_1{\mu }^2+z}{1+\overline{b_1}{\overline{\mu }}^{2}z}}$$

(19)

is univalent in $\mathbb{U}$. Observe that the bilinear transformation $p\left(z\right)$ given by (19) maps a circle $\left|z\right|=r\left(r<1\right)$ onto a circle

$$\left|p\left(z\right)-{\displaystyle \frac{b_1{\mu }^2\left(1-r^2\right)}{1-|b_1{|}^2{r}^2}}\right|={\displaystyle \frac{r\left(1-|b_1{|}^2\right)}{1-|b_1{|}^2{r}^2}}.$$

Thus, in view of the subordination (18), for $\left|z\right|=r\left(r<1\right),$

$$\left|{\omega }_{f^{\mu }}\left(z\right)-{\displaystyle \frac{b_1{\mu }^2\left(1-r^2\right)}{1-|b_1{|}^2{r}^2}}\right|\le {\displaystyle \frac{r\left(1-|b_1{|}^2\right)}{1-|b_1{|}^2{r}^2}}$$

which derives the results (14), (15), (16), and (17), taking both possibilities whether $r<|b_1|$ or $r>|b_1|$. □

Remark 2. 

If we take $h-g=z$ and $\omega \left(z\right)=(1+z)/2$ with $b_1=1/2$, we obtain the sharp bounds of Lemma (2).

The results proved in this section are further applied to obtain the main results. The function $f=h+\overline{g}\in S_{\mathcal{H}}$ and $f^{\mu }$, its rotation by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$, are the objects of this investigation. The bounds of $f^{\mu }$ are given in the first two theorems proved, each with an associated corollary. Connections with previously established results are highlighted in a remark. Furthermore, the bounds established for $f^{\mu }$ are applied to obtain estimates of the Jacobian of $f^{\mu }$. Finally, the integral representation of $f^{\mu }$ is provided, and a simple example is given for $f^{\mu }$.

3. Main Results

The bounds of the rotation function $f^{\mu }$ by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$ are first obtained.

Theorem 5. 

Let $f=h+\overline{g}\in S_{\mathcal{H}}$ be the same as that considered in Lemma 3 with the condition (13). Let $f^{\mu }$ and ${\varphi }^{\mu }$ be rotations of f and $\varphi ,$ respectively, by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right).$ Then, for $\left|z\right|\le r\left(r<1\right),$

$${\displaystyle \frac{\left|1-b_1\right|(1+|b_1\left|r\right)|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)|}{(1+|b_1\left|\right)(1+r)}}\le \left|f_z^{\mu }\left(z\right)\right|\le {\displaystyle \frac{\left|1-b_1\right|(1+|b_1\left|r\right)|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)|}{(1-|b_1\left|\right)(1-r)}}$$

(20)

and

$${\displaystyle \frac{\left|1-b_1\right|||b_1|-r\left|\right(1+|b_1\left|r\right)|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)|}{(1+|b_1\left|\right)(1+r)(1-|b_1\left|r\right)}}\le \left|f_{\overline{z}}^{\mu }\left(z\right)\right|\le {\displaystyle \frac{\left|1-b_1\right|\left(r+|b_1|\right)\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\right|}{(1-|b_1\left|\right)(1-r)}},$$

(21)

Proof. 

From Proposition 1, we have

$$f^{\mu }\left(z\right)=H\left(z\right)+\overline{G\left(z\right)}$$

and from Equation (10), we have

$${\displaystyle \frac{H^{\prime }\left(z\right)\left(1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)\right)}{1-b_1}}={\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right).$$

Hence,

$$f_z^{\mu }\left(z\right)=H^{\prime }\left(z\right)={\displaystyle \frac{(1-b_1){\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)}{1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)}}$$

(22)

and

$$f_{\overline{z}}^{\mu }\left(z\right)=G^{\prime }\left(z\right)={\displaystyle \frac{(1-b_1){\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right){\omega }_{f^{\mu }}\left(z\right)}{1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)}}.$$

(23)

Since the analytic function ${\omega }_{f^{\mu }}\left(z\right)$ satisfies the condition $|{\omega }_{f^{\mu }}\left(z\right)|<1$, for every $z\in \mathbb{D}$, we have

$${\displaystyle \frac{\left|1-b_1\right|\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\right|}{1+|{\omega }_{f^{\mu }}\left(z\right)|}}\le \left|f_z^{\mu }\left(z\right)\right|\le {\displaystyle \frac{\left|1-b_1\right|\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\right|}{1-|{\omega }_{f^{\mu }}\left(z\right)|}},$$

(24)

$${\displaystyle \frac{\left|1-b_1\right||{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\left|\right|{\omega }_{f^{\mu }}\left(z\right)|}{1+|{\omega }_{f^{\mu }}\left(z\right)|}}\le \left|f_{\overline{z}}^{\mu }\left(z\right)\right|\le {\displaystyle \frac{\left|1-b_1\right||{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\left|\right|{\omega }_{f^{\mu }}\left(z\right)|}{1-|{\omega }_{f^{\mu }}\left(z\right)|}}$$

(25)

By using (15), (16), and (14), we obtain the required result. □

Theorem 6. 

Let $f=h+\overline{g}\in S_{\mathcal{H}}$ be the same as that considered in Lemma 2 with the condition (13). Let $f^{\mu }$ be a rotation of f by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right).$ Then, for $z=r{e}^{i\theta }\left(0\le r<1,\theta \in \mathbb{R}\right),$

$$|f^{\mu }\left(z\right)|\le {\displaystyle \frac{\left|1-b_1\right|(1+|b_1\left|\right)}{1-|b_1|}}{\int }_0^r{\displaystyle \frac{1+\rho }{1-\rho }}\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(\rho e^{i\theta }\right)\right|d\rho .$$

Proof. 

From Proposition 1, we have

$$\begin{array}{ccc}\hfill f^{\mu }\left(z\right)& =& H\left(z\right)+\overline{G\left(z\right)}\hfill \\ & =& {\int }_0^r{H}^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }\mathrm{d}\rho +\overline{{\int }_0^r{G}^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }\mathrm{d}\rho }\hfill \\ & =& {\int }_0^r{H}^{\prime }\left(\rho e^{i\theta }\right)e^{i\theta }\mathrm{d}\rho +{\int }_0^r\overline{G^{\prime }\left(\rho e^{i\theta }\right)}{e}^{-i\theta }\mathrm{d}\rho \hfill \\ & =& {\int }_0^r{f}_z^{\mu }\left(\rho e^{i\theta }\right)e^{i\theta }\mathrm{d}\rho +{\int }_0^r{f}_{\overline{z}}^{\mu }\left(\rho e^{i\theta }\right)e^{-i\theta }\mathrm{d}\rho \hfill \end{array}$$

hence,

$$\left|f^{\mu }\left(z\right)\right|\le {\int }_0^r\left|f_z^{\mu }\left(\rho e^{i\theta }\right)\right|\mathrm{d}\rho +{\int }_0^r\left|f_{\overline{z}}^{\mu }\left(\rho e^{i\theta }\right)\right|\mathrm{d}\rho$$

which, upon applying (20) and (21), yields the result. □

From Theorem 6, we obtain the following result:

Corollary 2. 

Let $f=h+\overline{g}\in S_{\mathcal{H}}^0$ with

$$h\left(z\right)-g\left(z\right)=\varphi \left(z\right),$$

(26)

where $\varphi \in \mathcal{A}$ is convex in the direction of the real axis. Let $f^{\mu }$ be a rotation of f by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right).$ Then, for $z=r{e}^{i\theta }\left(0\le r<1,\theta \in \mathbb{R}\right),$

$$|f^{\mu }\left(z\right)|\le {\int }_0^r{\displaystyle \frac{1+\rho }{1-\rho }}\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(\rho e^{i\theta }\right)\right|d\rho .$$

Remark 3. 

By applying Theorem 6, we can obtain the upper bounds of the rotation function $f^{\mu }\left(z\right)$ by taking several special forms of $\varphi \left(z\right)$. The descriptions of a few are as follows:

(i) 

If $\varphi \left(z\right)={\displaystyle \frac{z}{1-z}},$ then $\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(\rho e^{i\theta }\right)\right|=\left|{\displaystyle \frac{1}{{(1-\mu \rho e^{i\theta })}^2}}\right|\le {\displaystyle \frac{1}{{\left(1-\rho \right)}^2}}.$

(ii) 

If $\varphi \left(z\right)={\displaystyle \frac{1}{2}}log{\displaystyle \frac{1+z}{1-z}},$ then $\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(\rho e^{i\theta }\right)\right|=\left|{\displaystyle \frac{1}{1-{\mu }^2{\left(\rho e^{i\theta }\right)}^2}}\right|\le {\displaystyle \frac{1}{1-{\rho }^2}}.$

(iii) 

If $\varphi \left(z\right)={\displaystyle \frac{z}{1+cz+z^2}},$ then

$$\begin{array}{ccc}\hfill \left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(\rho e^{i\theta }\right)\right|& =& {\displaystyle \frac{1-{\mu }^2{\left(\rho e^{i\theta }\right)}^2}{{(1+c\mu \rho e^{i\theta }+{\mu }^2{\left(\rho e^{i\theta }\right)}^2)}^2}}\hfill \\ & \le & {\displaystyle \frac{2}{{\left(1-\left|c\right|\rho -{\rho }^2\right)}^2}}\hfill \end{array}$$

$\left(c\in \left[-2,2\right]\right).$

In Corollary 2, upon taking $\varphi \left(z\right)={\displaystyle \frac{z}{1-z}},$ we obtain the following result:

Corollary 3. 

Let $f=h+\overline{g}\in S_{\mathcal{H}}$ with

$$h\left(z\right)-g\left(z\right)={\displaystyle \frac{z}{1-z}}.$$

(27)

Let $f^{\mu }$ be a rotation of f by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right).$ Then, for $\left|z\right|=r\left(r<1\right),$

$$|f^{\mu }\left(z\right)|\le {\displaystyle \frac{r}{{\left(1-r\right)}^2}}.$$

Remark 4. 

Corollary 3 shows that the upper bound ${\displaystyle \frac{r}{{\left(1-r\right)}^2}}$ of $|f^{\mu }\left(z\right)|$ is independent of $\mu .$ For $\mu =1,$ this upper bound was obtained by Schaubroeck [36] [Theorem 1.4, p. 629].

Theorem 7. 

Let $f=h+\overline{g}\in S_{\mathcal{H}}$ be the same as that considered in Lemma 2 with the condition (13). Let $f^{\mu }$ be a rotation of f by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right).$ Then, for any $\left|z\right|\le r\left(r<1\right),$

$$J_{f^{\mu }}\left(z\right)\le {\displaystyle \frac{{\left|1-b_1\right|}^2{\left(1+|b_1|\right)}^2\left(1+|b_1|r\right)\left(1+r\right){\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\right|}^2}{{\left(1-|b_1|\right)}^2\left(1-|b_1|r\right)\left(1-r\right)}}$$

and in the case where $r<|b_1|$,

$$J_{f^{\mu }}\left(z\right)\ge {\displaystyle \frac{{\left|1-b_1\right|}^2\left(1-|b_1{|}^2\right)\left(1+|b_1|r\right){\left(1-r\right)}^2{\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\right|}^2}{{\left(1+|b_1|\right)}^2\left(1-|b_1|r\right){\left(1+r\right)}^2}},$$

and in the case where $r>|b_1|$,

$$J_{f^{\mu }}\left(z\right)\ge {\displaystyle \frac{{\left|1-b_1\right|}^2{\left(1-|b_1|\right)}^2\left(1+|b_1|r\right)\left(1-r^2\right){\left|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)\right|}^2}{{\left(1+|b_1|\right)}^2\left(1-|b_1|r\right){\left(1+r\right)}^2}}.$$

Proof. 

Since

$$\begin{array}{ccc}\hfill J_{f^{\mu }}\left(z\right)& =& {\left|H^{\prime }\left(z\right)\right|}^2-{\left|G^{\prime }\left(z\right)\right|}^2\hfill \\ & =& {\left|H^{\prime }\left(z\right)\right|}^2\left(1-{\left|{\omega }_{f^{\mu }}\left(z\right)\right|}^2\right)\hfill \\ & =& {\left|f_z^{\mu }\left(z\right)\right|}^2\left(1-{\left|{\omega }_{f^{\mu }}\left(z\right)\right|}^2\right),\hfill \end{array}$$

we obtain the result by substituting the bounds from (17) and (20). □

Let $f=h+\overline{g}\in S_{\mathcal{H}}$ be the same as that considered in Lemma 2 with the condition (13). Let $f^{\mu }$ and ${\varphi }^{\mu }$ be rotations of f and $\varphi ,$ respectively, by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$, and let ${\omega }_{f^{\mu }}$ be the dilation of $f^{\mu }.$ Then, from Proposition 1, $f^{\mu }=H+\overline{G}\in S_{\mathcal{H}}$, and from (22) and (23), we obtain

$$H^{\prime }\left(z\right)={\displaystyle \frac{\left(1-b_1\right){\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)}{1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)}}\mathrm{and}{G}^{\prime }\left(z\right)={\displaystyle \frac{\left(1-b_1\right){\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right){\omega }_{f^{\mu }}\left(z\right)}{1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(z\right)}}.$$

Hence, the integral representation of $f^{\mu }$ is given by

$$f^{\mu }\left(z\right)=\left(1-b_1\right)\left[{\int }_0^z{\displaystyle \frac{{\left({\varphi }^{\mu }\right)}^{\prime }\left(\xi \right)}{1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(\xi \right)}}\mathrm{d}\xi +\overline{{\int }_0^z{\displaystyle \frac{{\left({\varphi }^{\mu }\right)}^{\prime }\left(\xi \right){\omega }_{f^{\mu }}\left(\xi \right)}{1-{\overline{\mu }}^2{\omega }_{f^{\mu }}\left(\xi \right)}}\mathrm{d}\xi }\right].$$

If $\varphi \left(z\right)={\displaystyle \frac{z}{1-z}},$ we obtain ${\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)={\displaystyle \frac{1}{{\left(1-\mu z\right)}^2}},$ and taking ${\omega }_{f^{\mu }}\left(z\right)={\displaystyle \frac{b_1{\mu }^2+{\mu }^{3}z}{1+b_1\mu z}},$ we obtain $H^{\prime }\left(z\right)={\displaystyle \frac{1+b_1\mu z}{{\left(1-\mu z\right)}^3}}$ and $G^{\prime }\left(z\right)={\displaystyle \frac{b_1{\mu }^2+{\mu }^{3}z}{{\left(1-\mu z\right)}^3}}$, which, upon integrating with the normalization, gives

$H\left(z\right)={\displaystyle \frac{z-\frac{1-b_1}{2}\mu z^2}{{\left(1-\mu z\right)}^2}}$ and $G\left(z\right)={\displaystyle \frac{b_1{\mu }^{2}z+\frac{1-b_1}{2}{\mu }^3{z}^2}{{\left(1-\mu z\right)}^2}}.$ Thus, an example of $f^{\mu }$ is given by

$$f^{\mu }\left(z\right)={\displaystyle \frac{z-\frac{1-b_1}{2}\mu z^2}{{\left(1-\mu z\right)}^2}}+\overline{{\displaystyle \frac{b_1{\mu }^{2}z+\frac{1-b_1}{2}{\mu }^3{z}^2}{{\left(1-\mu z\right)}^2}}}.$$

In particular, if in Lemma 2, we substitute $b_1=0,\varphi \left(z\right)={\displaystyle \frac{z}{1-z}}$ into the condition (13) and ${\omega }_f\left(z\right)=z,$ we obtain the harmonic shear

$$f={\displaystyle \frac{z-\frac{1}{2}{z}^2}{{\left(1-z\right)}^2}}+\overline{{\displaystyle \frac{\frac{1}{2}{z}^2}{{\left(1-z\right)}^2}}}.$$

4. Conclusions

In the present investigation, we study harmonic functions $f=h+\overline{g}\in \mathcal{H}$ associated with the rotation $f^{\mu }$ by $\mu \left(\mu \in \mathbb{C},\left|\mu \right|=1\right)$, for which bounds are established, the Jacobian is characterized by particular inequalities, and the integral representation is established. Observe that corresponding to the rotation function ${\varphi }^{\mu }$, which is convex in the direction of $\overline{\mu }$, we may define a function $\psi \left(z\right)$ by

$$\psi \left(z\right)=i\mu {\varphi }^{\mu }\left(z\right)$$

which is convex in the direction of the imaginary axis. Hence, we may also use the bounds of $|{\left({\varphi }^{\mu }\right)}^{\prime }\left(z\right)|=\left|{\psi }^{\prime }\left(z\right)\right|$ from Theorems 2 and 4 in our results in Theorems 5–7 to obtain further results corresponding to Theorems 5–7. Another possible future line of research involves further studies concerning the theory of differential subordination, given the properties of the subordination chains for obtaining other bounds for the rotation $f^{\mu }$.