Estimation of the Bounds of Some Classes of Harmonic Functions with Symmetric Conjugate Points

Define $\mathcal{A}$ as a class of analytic functions h of the form where $\mathbb{U}=\{z\in \mathbb{C}$: $\left|z\right|<1\}$.

Let $\mathcal{S},{\mathcal{S}}^{*}$, and $\mathcal{K}$ be the subclasses of $\mathcal{A}$, which are composed of univalent functions, starlike functions and convex functions, respectively ([1,2]).

Let $\mathcal{P}$ denote the class of analytic functions p with a positive real part on $\mathbb{U}$ of the following form:

The function $p\in \mathcal{P}$ is called the Carathéodory function.

Suppose that the functions F and G are analytic in $\mathbb{U}$. The function F is said to be subordinate to the function G if there exists a function $\omega$ satisfying $\omega \left(0\right)=0$ and $\left|\omega \right(z\left)\right|<1(z\in \mathbb{U})$, such that $F\left(z\right)=G\left(\omega \right(z\left)\right)(z\in \mathbb{U})$. Note that $F\left(z\right)\prec G\left(z\right)$. In particular, if G is univalent in $\mathbb{U}$, the following conclusion follows (see [1]):

In 1994, Ma and Minda [3] introduced the classes ${\mathcal{S}}^{*}\left(\varphi \right)$ and $\mathcal{K}\left(\varphi \right)$ of starlike functions and convex functions by using the subordination. The function $h\left(z\right)\in {\mathcal{S}}^{*}\left(\varphi \right)$ if and only if $\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}\prec \varphi \left(z\right)$ and the function $h\left(z\right)\in \mathcal{K}\left(\varphi \right)$ if and only if $1+\frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}\prec \varphi \left(z\right)$, where $h\in \mathcal{A}$ and $\varphi \in \mathcal{P}$.

Let $\varphi \left(z\right)=\frac{1+Az}{1+Bz}$ and $-1\le B<A\le 1$. The classes ${\mathcal{S}}^{*}\left(\frac{1+Az}{1+Bz}\right)={\mathcal{S}}^{*}(A,B)$ and $\mathcal{K}\left(\frac{1+Az}{1+Bz}\right)=\mathcal{K}(A,B)$, which are the classes of Janowski starlike and convex functions, respectively (refer to [4]). ${\mathcal{S}}^{*}\left(\frac{1+z}{1-z}\right)={\mathcal{S}}^{*}$ and $\mathcal{K}\left(\frac{1+z}{1-z}\right)=\mathcal{K}$ are known for the classes of starlike and convex function, respectively.

In 1959, Sakaguchi [5] introduced the class ${\mathcal{S}}_s^{*}$ of starlike functions with respect to symmetric points. The function $h\in {\mathcal{S}}_s^{*}$ if and only if

In 1987, El Ashwah and Thomas [6] introduced the classes ${\mathcal{S}}_c^{*}$ and ${\mathcal{S}}_{sc}^{*}$ of starlike functions with respect to conjugate points and symmetric conjugate points as follows:

Similarly to the previous section, the classes ${\mathcal{S}}_c^{*}$ and ${\mathcal{S}}_{sc}^{*}$ can be further generalized to the classes ${\mathcal{S}}_{sc}^{*}\left(\varphi \right)$ and ${\mathcal{K}}_{sc}\left(\varphi \right)$.

The function $h\left(z\right)$ belongs to ${\mathcal{S}}_{sc}^{*}\left(\varphi \right)$ if and only if $\frac{2z{h}^{\prime }\left(z\right)}{h\left(z\right)-\overline{h}(-\overline{z})}\prec \varphi \left(z\right)$ holds true and $h\left(z\right)$ belongs to ${\mathcal{K}}_{sc}\left(\varphi \right)$ if and only if $\frac{2{\left(z{h}^{\prime }\left(z\right)\right)}^{\prime }}{{(h\left(z\right)-\overline{h}(-\overline{z}))}^{\prime }}\prec \varphi \left(z\right)$ holds true, where $h\in \mathcal{A}$ and $\varphi \in \mathcal{P}$.

If the function $h\in \mathcal{A}$ meets the following criteria: $Re\left(\frac{h\left(z\right)}{z{h}^{\prime }\left(z\right)}\right)>\alpha (0\le \alpha <1)$, then h is said to be in the class of the reciprocal starlike functions of order $\alpha$, which is represented by $h\in R{S}^{*}\left(\alpha \right)$.

In contrast to the classical starlike function class $S^{*}\left(\alpha \right)$ of order $\alpha$, the reciprocal starlike function class of order $\alpha$ maps the unit disk to a starlike region within a disk with $\left(\frac{1}{2\alpha },0\right)$ as the center and $\frac{1}{2\alpha }$ as the radius ([7]). In particular, the disk is large when $0<\alpha <\frac{1}{2}$. Therefore, the study of the class of reciprocal starlike functions has aroused the research interest of most scholars [8,9,10,11,12,13,14,15]. In 2012, Sun et al. [8] extended the reciprocal starlike function to the class of meromorphic univalent function.

For the analytic functions $h\left(z\right)$ and $g\left(z\right)(z\in \mathbb{U})$, let $S_H$ be a class of harmonic mappings, which has the following form (see [16,17]): where

Specifically, h is referred to as the analytical part, and g is known as the co-analytic part of f.

It is known that the function $f=h+\overline{g}$ is locally univalent and sense-preserving in $\mathbb{U}$ if and only if $|h^{\prime }\left(z\right)|>|g^{\prime }\left(z\right)|$ (see [18]).

Based on these results, it is possible to obtain the geometric properties of the co-analytic part by means of the analytic part of the harmonic function.

In the past few years, different subclasses of $S_H$ have been studied by several authors as follows.

In 2007, Klimek and Michalski [19] investigated the subclass $S_H$ with $h\in \mathcal{K}$.

In 2014, Hotta and Michalski [20] investigated the subclass $S_H$ with $h\in \mathcal{S}$.

In 2015, Zhu and Huang [21] investigated the subclasses of $S_H$ with $h\in {\mathcal{S}}^{*}(\frac{1+(1-2\beta )z}{1-z})$ and $h\in \mathcal{K}(\frac{1+(1-2\beta )z}{1-z})$.

Combined with the above studies, by using the subordination relationship, this paper further constructs the reciprocal structure harmonic function class with symmetric conjugate points as follows.

In this paper, we will discuss the harmonic Bloch constant and the norm of the pre-Schwarzian derivative for the classes.

For $f=h+\overline{g}\in S_H$, the harmonic Bloch constant of f is where is the hyperbolic distance between z and w, and $z,w\in \mathbb{U}$. If ${\mathcal{B}}_f<\infty$, then f is called the Bloch harmonic function. By (6), Colonna [22] proved that

Recently, many authors have studied the Bloch constant of harmonic functions (see [23,24]).

Let f be the analytic and locally univalent function in $\mathbb{U}$, and the pre-Schwarzian derivative of f is and the norm of $T_f$ is defined as

Unlike the case of analytic functions, the pre-Schwarzian derivative of harmonic functions allows a variety of different definitions (see [25,26,27]).

In [27], Chuaqui–Duren–Osgood gives the following definition of the pre-Schwarzian derivative the harmonic function $f=h+\overline{g}$: where $\lambda =\left|h^{\prime }\right|+\left|g^{\prime }\right|$. In fact, it is easy to see that the above definition is consistent with the classical pre-Schwarzian derivative of an analytic function.

In 2022, Xiong et al. [24] rewrite the pre-Schwarzian derivative as follows: and the norm of the pre-Schwarzian derivative of the harmonic function f can be defined in terms of (9).

In this paper, we will give an inequality of f belonging to the class ${HRK}_{sc}^{\alpha }(-\frac{1}{2},-1)$ with respect to its pre-Schwarzian derivative. In particular, the bounds of the norm of the pre-Schwarzian derivative of f in the class ${HRK}_{sc}^{\frac{1}{2}}(-\frac{1}{2},-1)$ is also determined.

To obtain our results, we need the following Lemmas.

According to the subordination relationship, we obtain the distortion theorem of the classes $R{S}_{sc}^{*}(A,B)$ and $R{K}_{sc}^{*}(A,B)$.

First, we will find the Bloch constants for the class $HR{S}_{sc}^{*,\alpha }(A,B)$.

In particular, let $\alpha =\frac{1}{2},A=-\frac{5}{24},B=-\frac{1}{2}$ in Theorem 1, we can obtain the following result.

In particular, let $\alpha =\frac{1}{2},A=0,B=-\frac{1}{2}$ in Theorem 1, we can obtain the following result.

Next, we will find the Bloch constants for the class $HR{K}_{sc}^{\alpha }(A,B)$.