Categories of Harmonic Functions in the Symmetric Unit Disk Linked to the Bessel Function

Abstract

Here in this paper, we establish the basic inclusion relations among the harmonic class $\mathbb{HF}(\sigma ,\eta )$ with the classes $S_{\mathcal{HF}}^{*}$ of starlike harmonic functions and $K_{\mathcal{HF}}$ of convex harmonic functions defined in open symmetric unit disk U. Moreover, we investigate inclusion connections for the harmonic classes $\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$ and $\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$ of harmonic functions by applying the operator $\mathrm{\Lambda }$ associated with the Bessel function. Furthermore, several special cases of the main results are obtained for the particular case $\sigma =0$.

1. Introduction

Bessel functions were studied by Euler, Lagrange, and members of the Bernoulli family. However, they were first systematically used by Friedrich Wilhelm Bessel in his analysis of three-body motion, where Bessel functions emerged in the series expansion of planetary perturbations. Bessel functions are named after Friedrich Wilhelm Bessel (1784–1846); however, Daniel Bernoulli is generally credited with first introducing the concept in 1732. He used the zero-order function as a solution to the problem of an oscillating chain hanging from one end. By 1764, Leonhard Euler had employed Bessel functions of both zero and integral orders in his analysis of the vibrations of a stretched membrane. This research was further developed by Lord Rayleigh in 1878, who demonstrated that Bessel functions are particular cases of Laplace functions [1].

Harmonic functions play a fundamental role in applied mathematics, arising naturally in fields such as fluid dynamics, electrostatics, and potential theory. In classical differential geometry, these functions were extensively studied due to their ability to furnish isothermal (or conformal) parameters for minimal surfaces. For comprehensive insights into differential geometry and its wide-ranging applications, see [2,3], along with the references therein.

Let $\mathbb{HF}$ be the family of all harmonic functions of the form $\hslash =\Im +\overline{\aleph }$ defined in the open symmetric unit disk $\mathrm{Y}=\left\{\xi \in \mathbb{C}:\left|\xi \right|<1\right\}$ for which $\hslash \left(0\right)={\hslash }^{\prime }\left(0\right)-1=0$, where

$$\Im \left(\xi \right)=\xi +\sum _{\iota =2}^{\infty }{x}_{\iota }{\xi }^{\iota },\aleph \left(\xi \right)=\sum _{\iota =1}^{\infty }{y}_{\iota }{\xi }^{\iota },\left|y_1\right|<1(\xi \in \mathrm{Y}),$$

(1)

are analytic in $\mathrm{Y}$. Moreover, let ${\mathbb{S}}_{\mathbb{HF}}$ denote the family of functions $f=\Im +\overline{\aleph }$ that are harmonic univalent and sense preserving in $\mathrm{Y}$. Note that the family ${\mathbb{S}}_{\mathbb{HF}}=\mathbb{S}$ if $\aleph =0.$

Also, we let ${\mathbb{S}}_{\mathbb{HF}}^0$ be the subclass of ${\mathbb{S}}_{\mathbb{HF}}$ defined as follows:

$${\mathbb{S}}_{\mathbb{HF}}^0=\{\hslash =\Im +\overline{\aleph }\in {\mathbb{S}}_{\mathbb{HF}}:{\aleph }^{\prime }\left(0\right)=y_1=0\}.$$

The classes ${\mathbb{S}}_{\mathbb{HF}}^0$ and ${\mathbb{S}}_{\mathbb{HF}}$ were first studied in [4].

A sense-preserving harmonic mapping $f\in {\mathbb{S}}_{\mathbb{HF}}^0$ is classified as belonging to the class ${\mathbb{S}}_{\mathbb{HF}}$ if the image $\hslash \left(Y\right)$ is starlike with respect to the origin. If $f\in {\mathbb{S}}_{\mathbb{HF}}^{*}$, it is known as a harmonic starlike mapping in Y. Similarly, a function f defined in Y is considered in the class ${\mathbb{K}}_{\mathbb{HF}}$ if $f\in {\mathbb{S}}_{\mathbb{HF}}^0$ and the image $\hslash \left(Y\right)$ forms a convex domain. The function ${\mathbb{K}}_{\mathbb{HF}}$ is called harmonic convex in Y. Analytically, we have

$$\hslash \in {\mathbb{S}}_{\mathbb{HF}}^{*}\mathrm{iff}arg\left({\displaystyle \frac{\partial }{\partial \nu }}\hslash \left(r{e}^{i\nu }\right)\right)\ge 0,$$

and

$$\hslash \in {\mathbb{K}}_{\mathbb{HF}}\mathrm{iff}{\displaystyle \frac{\partial }{\partial \nu }}\left\{arg\left({\displaystyle \frac{\partial }{\partial \nu }}arg\hslash \left(r{e}^{i\nu }\right)\right)\right\}\ge 0,$$

$$\xi =r{e}^{i\nu }\in \mathrm{Y},\nu \in [0,2\pi ],r\in [0,1].$$

For more details of these classes, see [5,6,7,8,9,10,11,12].

Let ${\mathcal{T}}_{\mathbb{HF}}$ be the class of functions in ${\mathbb{S}}_{\mathbb{HF}}$ that may be expressed as $\hslash =\Im +\overline{\aleph }$, where

$$\Im \left(\xi \right)=\xi -\sum _{\iota =2}^{\infty }|x_{\iota }|{\xi }^{\iota },$$

(2)

$$\aleph \left(\xi \right)=\sum _{\iota =1}^{\infty }|y_{\iota }|{\xi }^{\iota },\left|y_1\right|<1.$$

For $\varrho \in [0,1)$, let

$${\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)=\left\{\hslash \in \mathbb{HF}:\mathfrak{Re}\left({\displaystyle \frac{f^{\prime }\left(\xi \right)}{{\xi }^{\prime }}}\right)\ge \varrho ,\xi =r{e}^{i\nu }\in \mathrm{Y}\right\},$$

and

$${\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)=\left\{\hslash \in \mathbb{HF}:\mathfrak{Re}\left({\displaystyle \frac{f^{″}\left(\xi \right)}{{\xi }^{″}}}\right)\ge \varrho ,\xi =r{e}^{i\nu }\in \mathrm{Y}\right\},$$

where

$${\xi }^{\prime }={\displaystyle \frac{\partial }{\partial \nu }}(\xi =r{e}^{i\nu }),{\xi }^{″}={\displaystyle \frac{\partial }{\partial \nu }}\left({\xi }^{\prime }\right),{\hslash }^{\prime }\left(\xi \right)={\displaystyle \frac{\partial }{\partial \nu }}\hslash \left(r{e}^{i\nu }\right),{\hslash }^{″}\left(\xi \right)={\displaystyle \frac{\partial }{\partial \nu }}\left({\hslash }^{\prime }\left(\xi \right)\right).$$

Define

$$\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)={\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)\cap {\mathcal{T}}_{\mathbb{HF}}and\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)={\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)\cap {\mathcal{T}}_{\mathbb{HF}}.$$

For more details about the classes ${\mathcal{T}}_{\mathbb{HF}}$, ${\mathfrak{N}}_{\mathbb{HF}}$, $\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$, ${\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$, and $\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$, each was defined and studied in [10,13].

Let $\mathbb{HF}(\sigma ,\eta )$ be the class of functions $f\in \mathbb{HF}$ that satisfy [14].

$$\mathfrak{Re}\left\{{\Im }^{\prime }\left(\xi \right)+{\aleph }^{\prime }\left(\xi \right)+3\sigma \xi ({\Im }^{″}\left(\xi \right)+{\aleph }^{″}\left(\xi \right))+\sigma {\xi }^3({\Im }^{‴}\left(\xi \right)+{\aleph }^{‴}\left(\xi \right))\right\}>\eta ,$$

(3)

for more $\sigma \ge 0$ and $\eta \in [0,1)$. For $\sigma =0$, we obtain the class $\mathbb{HF}\left(\eta \right)$, which satisfies

$$\mathfrak{Re}\{{\Im }^{\prime }\left(\xi \right)+{\aleph }^{\prime }\left(\xi \right)\}>\eta .$$

(4)

2. Bessel Function

Now, we represent the standard Bessel function of the first kind, often denoted as $w=w_{p,b,c}$, which is a solution to a particular second-order linear homogeneous differential equation

$${\xi }^2{\omega }^{″}\left(\xi \right)+b\xi {\omega }^{\prime }\left(\xi \right)+[c{\xi }^2-p^2+(1-b)p]\omega \left(\xi \right)=0,$$

(5)

where $p,b,$ and $c\in \mathbb{C}$. For solutions to such a generalized Bessel-type equation, they can often be expressed in terms of special functions such as

$$\omega \left(\xi \right)={\omega }_{p,b,c}\left(\xi \right)=\sum _{\iota =0}^{\infty }{\displaystyle \frac{{(-1)}^{\iota }{c}^{\iota }}{\iota !\Gamma (p+\iota +\frac{b+1}{2})}}{\left({\displaystyle \frac{\xi }{2}}\right)}^{2\iota +p},(\xi \in \mathbb{C}).$$

(6)

A generalized form of the Bessel Equation (5) may be expressed in a way that includes modifications to its coefficients, allowing it to represent a broader class of special functions. These include, but are not limited to, the modified Bessel functions and the spherical Bessel functions. The generalized Bessel Equation (5) extends the classical Bessel differential equation to a broader form. A particular solution to this generalized Equation (6) is called the generalized Bessel function of the first kind of order p. Although the series defining the function ${\omega }_{p,b,c}\left(\xi \right)$ converges everywhere in the unit disk Y, the function is generally not univalent there. It is especially worth mentioning that, when $b=c=1$, we obtain the Bessel function ${\omega }_{p,1,1}\left(\xi \right)=j_p$, and for $c=-1,b=1$, the function ${\omega }_{p,1,-1}$ becomes the modified Bessel function $I_p$. Now, consider the function $u_{p,b,c}$ defined by the transformation

$$u_{p,b,c}\left(\xi \right)=2^p\Gamma \left(p+{\displaystyle \frac{b+1}{2}}\right){\xi }^{-p/2}{\omega }_{p,b,c}\left({\xi }^{1/2}\right).$$

For $a\ne 0,-1,-2,\dots$, the Pochhammer symbol (or rising factorial) ${\left(a\right)}_{\ell }$ is defined by

$${\left(a\right)}_{\ell }={\displaystyle \frac{\Gamma (a+\ell )}{\Gamma \left(a\right)}}=\left\{\begin{array}{c}1,if\ell =0\\ a(a+1)...(a+\ell -1),if\ell =1,2,3,....\end{array}\right.$$

where ℓ is a non-negative integer, and $\Gamma$ is the Euler Gamma function.

For the function $u_{p,b,c}$, we obtain the following representation

$$u_{p,b,c}\left(\xi \right)=\sum _{\iota =0}^{\infty }{\displaystyle \frac{{(-c/4)}^{\iota }}{{\left(p+\frac{b+1}{2}\right)}_{\iota }}}{\displaystyle \frac{{\xi }^{\iota }}{\iota !}},$$

(7)

where $p+(b+1)/2\ne 0,-1,-2,\dots$. This function is analytic on $\mathbb{C}$ and satisfies the second-order linear differential equation

$$4{\xi }^2{u}^{″}\left(\xi \right)+2(2p+b+1)\xi u^{\prime }\left(\xi \right)+c\xi u\left(\xi \right)=0.$$

In the context of the present study, $u\left(1\right)$ denotes the value of the kernel function $u_{p,b,c}\left(\xi \right)$ at the boundary $\left|\xi \right|=1$ of the unit disk. Geometrically, this value determines the scaling of the image of a harmonic mapping under the transformation $\mathrm{\Lambda }.$ More precisely, $u\left(1\right)$ acts as a multiplicative factor that influences the boundary amplitude of the transformed function. Consequently, it directly affects the size and possible geometric properties (such as starlikeness or convexity) of the image domain, which explains its central role in the inequalities controlling the inclusion relations established in this paper.

We give the following notation for convenience:

$$u_{p,b,c}=u_p,t=p+{\displaystyle \frac{b+1}{2}}.$$

For complex parameters $c_1,c_2,t_1,t_2(t_1,t_2\ne 0,-1,-2,...)$, we define the functions ${\Theta }_1\left(\xi \right)=\xi u_{p_1}\left(\xi \right)$ and ${\Theta }_2\left(\xi \right)=\xi u_{p_2}\left(\xi \right)$.

The convolution (or Hadamard product) of $\Im \left(\xi \right)$ and $\aleph \left(\xi \right)$ (written as $(\Im \ast \aleph )\left(\xi \right))$ is defined as

$$(\Im \ast \aleph )\left(\xi \right)=\xi +\sum _{\ell =2}^{\infty }{x}_{\ell }{y}_{\ell }{\xi }^{\ell }.$$

The concept of convolution arose from the integral

$$f\left(r^2{e}^{i\nu }\right)=(\Im \ast \aleph )\left(r^2{e}^{i\nu }\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }\Im \left(r{e}^{i(\nu -s)}\right)\aleph \left(r{e}^{is}\right)ds,\left|r\right|<1$$

and has proved very resourceful in dealing with certain problems of the theory of analytic and univalent functions, especially the closure of families of functions under certain transformations. This is since many a transformation of ℑ is expressible as a convolution of ℑ with some other analytic function, sometimes with predetermined behavior. It is natural, therefore, to desire to investigate the convolution properties of many classes of functions.

Corresponding to these functions, we introduce the following convolution operator

$$\mathrm{\Lambda }\equiv \mathrm{\Lambda }\left({\displaystyle \genfrac{}{}{0pt}{}{t_1,c_1}{t_2,c_2}}\right):\mathbb{H}\to \mathbb{H}$$

defined by

$$\begin{array}{ccc}\hfill \mathrm{\Lambda }\left({\displaystyle \genfrac{}{}{0pt}{}{t_1,c_1}{t_2,c_2}}\right)\hslash & =& \hslash \ast ({\Theta }_1+\overline{{\Theta }_2})=\Im \left(\xi \right)\ast {\Theta }_1\left(\xi \right)+\overline{\aleph \left(\xi \right)\ast {\Theta }_2\left(\xi \right)}\hfill \\ & =& \Im \left(\xi \right)\ast {\int }_0^{\xi }{u}_{p_1}\left(s\right)ds+\overline{\aleph \left(\xi \right)\ast {\int }_0^{\xi }{u}_{p_2}\left(s\right)ds}\hfill \end{array}$$

for any function $\hslash =\Im +\overline{\aleph }$ in $\mathbb{H}$ [11,15].

Letting

$$\mathrm{\Lambda }\left({\displaystyle \genfrac{}{}{0pt}{}{t_1,c_1}{t_2,c_2}}\right)\hslash \left(\xi \right)=H\left(\xi \right)+\overline{G\left(\xi \right)},$$

(8)

where

$$H\left(\xi \right)=\xi +\sum _{\iota =2}^{\infty }{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}{x}_{\iota }{\xi }^{\iota },G\left(\xi \right)=\sum _{\iota =1}^{\infty }{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}{y}_{\iota }{\xi }^{\iota }.$$

(9)

In the literature, several authors studied the relations between different subclasses of analytic and univalent functions by using hypergeometric functions [13,14], the Mittag–Leffler function [16,17,18], and the generalized Bessel function [19,20], and there has also been recent research on distribution series [21,22,23,24].

In this study, we will recall some lemmas that will be useful in proving the main results. Following the work performed in [15], we establish some inclusion relations of the harmonic class $\mathbb{HF}(\sigma ,\eta )$ with the classes ${\mathbb{S}}_{\mathbb{HF}}^{*}$, ${\mathbb{K}}_{\mathbb{HF}}$, ${\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$, and ${\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$ by applying the operator $\mathrm{\Lambda }$ related with Bessel function. Finally, for $\sigma =0$, several special cases of our main results are obtained.

3. Useful Lemmas

In order to establish connections between harmonic starlike mappings and harmonic convex mappings with the class $\mathbb{HF}(\sigma ,\eta )$, we need the following results in Lemma 1 [14], Lemma 2 [5], Lemma 3 [13], Lemma 4 [4], and Lemma 5 [4].

Lemma 1. 

Ref. [14]. Let $\hslash =\Im +\overline{\aleph }$ where ℑ and ℵ are given by (1) and suppose that $\sigma \ge 0$, $\eta \in [0,1)$ and

$$\sum _{\iota =2}^{\infty }\iota [1+\sigma ({\iota }^2-1)]|x_{\iota }|+\sum _{\iota =1}^{\infty }\iota [1+\sigma ({\iota }^2-1)]\left|y_{\iota }\right|\le 1-\eta .$$

(10)

Then, ℏ is harmonic, sense-preserving univalent function in $\mathrm{Y}$ and $\hslash \in \mathbb{HF}(\sigma ,\eta )$. Moreover, if $\hslash \in \mathbb{HF}(\sigma ,\eta )$, then

$$|x_{\iota }|\le {\displaystyle \frac{1-\eta }{\iota [1+\sigma ({\iota }^2-1)]}},\iota \ge 2,$$

(11)

and

$$|y_{\iota }|\le {\displaystyle \frac{1-\eta }{\iota [1+\sigma ({\iota }^2-1)]}},\iota \ge 1.$$

(12)

The result (10) is sharp.

Lemma 2. 

Ref. [5]. Let $\hslash =\Im +\overline{\aleph }$ where ℑ and ℵ are given by (2) and suppose that $\varrho \in [0,1)$. Then, $\hslash \in \mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$ if and only if

$$\sum _{\iota =2}^{\infty }\iota |x_{\iota }|+\sum _{\iota =1}^{\infty }\iota \left|y_{\iota }\right|\le 1-\varrho .$$

(13)

Moreover, if $\hslash \in \mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$, then

$$|x_{\iota }|\le {\displaystyle \frac{1-\varrho }{\iota }},\iota \ge 2,$$

(14)

and

$$|y_{\iota }|\le {\displaystyle \frac{1-\varrho }{\iota }},\iota \ge 1.$$

(15)

The result (13) is sharp.

Lemma 3. 

Ref. [13] Let $\hslash =\Im +\overline{\aleph }$ where ℑ and ℵ are given by (2) and suppose that $\varrho \in [0,1)$. Then, $\hslash \in \mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$ if and only if

$$\sum _{\iota =2}^{\infty }{\iota }^2|x_{\iota }|+\sum _{\iota =1}^{\infty }{\iota }^2\left|y_{\iota }\right|\le 1-\varrho .$$

(16)

Moreover, if $\hslash \in \mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)$, then

$$|x_{\iota }|\le {\displaystyle \frac{1-\varrho }{{\iota }^2}},\iota \ge 2,$$

(17)

and

$$|y_{\iota }|\le {\displaystyle \frac{1-\varrho }{{\iota }^2}},\iota \ge 1.$$

(18)

The result (16) is sharp.

Lemma 4. 

Ref. [4]. If $\hslash =\Im +\overline{\aleph }\in$${\mathbb{S}}_{\mathbb{HF}}^{*}$ where ℑ and ℵ are given by (1) with $y_1=0$, then

$$|x_{\iota }|\le {\displaystyle \frac{(2\iota +1)(\iota +1)}{6}}\mathrm{and}\left|y_{\iota }\right|\le {\displaystyle \frac{(2\iota -1)(\iota -1)}{6}}.$$

(19)

Lemma 5. 

Ref. [4]. If $\hslash =\Im +\overline{\aleph }\in {\mathbb{K}}_{\mathbb{HF}}$ where ℑ and ℵ are given by (1) with $y_1=0$, then

$$|x_{\iota }|\le {\displaystyle \frac{\iota +1}{2}}and\left|y_{\iota }\right|\le {\displaystyle \frac{\iota -1}{2}}.$$

(20)

Throughout the sequence, we use the following:

$$\mathrm{\Lambda }(\hslash )=\mathrm{\Lambda }\left({\displaystyle \genfrac{}{}{0pt}{}{t_1,c_1}{t_2,c_2}}\right)\hslash .$$

Let

$$\Theta \left(\xi \right)=\xi u_p\left(\xi \right)=\xi +\sum _{\iota =2}^{\infty }{\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}{\xi }^{\iota }$$

$$\Theta \left(1\right)=u_p\left(1\right)=1+\sum _{\iota =2}^{\infty }{\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}$$

(21)

and

$${\Theta }^{\prime }\left(\xi \right)=\xi u_p^{\prime }\left(\xi \right)+u_p\left(\xi \right)=1+\sum _{\iota =2}^{\infty }\iota {\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}{\xi }^{\iota -1}$$

$${\Theta }^{\prime }\left(1\right)=u_p^{\prime }\left(1\right)+u_p\left(1\right)-1=\sum _{\iota =2}^{\infty }\iota {\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}$$

(22)

$${\Theta }^{″}\left(\xi \right)=\xi u_p^{″}\left(\xi \right)+2u_p^{\prime }\left(\xi \right)=\sum _{\iota =2}^{\infty }\iota (\iota -1){\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}{\xi }^{\iota -2}$$

$${\Theta }^{″}\left(1\right)=u_p^{″}\left(1\right)+2u_p^{\prime }\left(1\right)=\sum _{\iota =2}^{\infty }\iota (\iota -1){\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}$$

(23)

$${\Theta }^{‴}\left(\xi \right)=\xi u_p^{‴}\left(\xi \right)+3u_p^{″}\left(\xi \right)=\sum _{\iota =2}^{\infty }\iota (\iota -1)(\iota -2){\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}}{\xi }^{\iota -3}$$

$${\Theta }^{‴}\left(1\right)=u_p^{‴}\left(1\right)+3u_p^{″}\left(1\right)=\sum _{\iota =2}^{\infty }\iota (\iota -1)(\iota -2){\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}},$$

(24)

$${\Theta }^{\left(4\right)}\left(\xi \right)=\xi u_p^{\left(4\right)}\left(\xi \right)+4u_p^{‴}\left(\xi \right)=\sum _{\ell =2}^{\infty }\ell (\ell -1)(\ell -2)(\ell -3){\displaystyle \frac{{(-c/4)}^{\ell -1}}{{\left(t\right)}_{\ell -1}(\ell -1)!}}{\xi }^{\ell -4}$$

$${\Theta }^{\left(4\right)}\left(1\right)=u_p^{\left(4\right)}\left(1\right)+4u_p^{‴}\left(1\right)=\sum _{\ell =2}^{\infty }\ell (\ell -1)(\ell -2)(\ell -3){\displaystyle \frac{{(-c/4)}^{\ell -1}}{{\left(t\right)}_{\ell -1}(\ell -1)!}},$$

(25)

$${\Theta }^{\left(5\right)}\left(\xi \right)=\xi u_p^{\left(5\right)}\left(\xi \right)+5u_p^{\left(4\right)}\left(\xi \right)=\sum _{\ell =2}^{\infty }\ell (\ell -1)(\ell -2)(\ell -3)(\ell -4){\displaystyle \frac{{(-c/4)}^{\ell -1}}{{\left(t\right)}_{\ell -1}(\ell -1)!}}{\xi }^{\ell -5}$$

$${\Theta }^{\left(5\right)}\left(1\right)=u_p^{\left(5\right)}\left(1\right)+5u_p^{\left(4\right)}\left(1\right)=\sum _{\ell =2}^{\infty }\ell (\ell -1)(\ell -2)(\ell -3)(\ell -4){\displaystyle \frac{{(-c/4)}^{\ell -1}}{{\left(t\right)}_{\ell -1}(\ell -1)!}},$$

(26)

and in general, we have

$$\begin{array}{ccc}\hfill {\Theta }^{\left(j\right)}\left(1\right)& =& u_p^{\left(j\right)}\left(1\right)+j{u}_p^{(j-1)}\left(1\right)\hfill \\ & =& \sum _{\iota =2}^{\infty }\iota (\iota -1)(\iota -2)\dots (\iota -(j-1)){\displaystyle \frac{{(-c/4)}^{\iota -1}}{{\left(t\right)}_{\iota -1}(\iota -1)!}},j=1,2,\dots .\hfill \end{array}$$

(27)

4. Inclusion Relations of the Class $\mathbb{HF}\mathbf{(}\mathit{\sigma }\mathbf{,}\mathit{\eta }\mathbf{)}$

In this section, we will obtain the inclusion relations of harmonic classes $\mathbb{HF}(\sigma ,\eta )$ with the classes ${\mathbb{S}}_{\mathbb{HF}}^{*}$, ${\mathbb{K}}_{\mathbb{HF}}$, $\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}$, and $\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}$, respectively, using Bessel function. Also, unless otherwise mentioned, we suppose that $t_1,t_2>0$, $c_1,c_2<0$, $\sigma >0$, and $\eta ,\varrho \in [0,1)$.

Theorem 1. 

If

$$\begin{array}{ccc}& & 2\sigma \left[u_{p_1}^{\left(5\right)}\left(1\right)+u_{p_2}^{\left(5\right)}\left(1\right)\right]+3\sigma \left[11u_{p_1}^{\left(4\right)}\left(1\right)+9u_{p_2}^{\left(4\right)}\left(1\right)\right]+(159\sigma +2)u_{p_1}^{\left(3\right)}\left(1\right)\hfill \\ & & +(246\sigma +15)u_{p_1}^{\left(2\right)}\left(1\right)+(90\sigma +24)u_{p_1}^{\prime }\left(1\right)+6(u_p\left(1\right)-1)+(99\sigma +2)u_{p_2}^{\left(3\right)}\left(1\right)\hfill \\ & & +(102\sigma +9)u_{p_2}^{\left(2\right)}\left(1\right)+(18\sigma +6)u_{p_2}^{\prime }\left(1\right)\hfill \\ & \le & 6(1-\eta ),\hfill \end{array}$$

(28)

then

$$\mathrm{\Lambda }\left({\mathbb{S}}_{\mathbb{HF}}^{*}\right)\subset \mathbb{HF}(\sigma ,\eta ).$$

Proof. 

Let $\hslash =\Im +\overline{\aleph }$∈${\mathbb{S}}_{\mathbb{HF}}^{*}$ where ℑ and ℵ are of the form (1) with $y_1=0$. We need to show that $\mathrm{\Lambda }\left(f\right)=H+\overline{G}$∈$\mathbb{HF}(\sigma ,\eta )$, which is given by (9) with $y_1=0$. In view of Lemma 1, we need to prove that

$$\Xi (u_{p_1},u_{p_2},\sigma )\le 1-\eta ,$$

where

$$\begin{array}{ccc}\hfill \Xi (u_{p_1},u_{p_2},\sigma )& =& \sum _{\iota =2}^{\infty }\iota (1+\sigma ({\iota }^2-1)){\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\left|x_{\iota }\right|\hfill \\ & & +\sum _{\iota =2}^{\infty }\iota (1+\sigma ({\iota }^2-1)){\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\left|y_{\iota }\right|.\hfill \end{array}$$

(29)

Using the inequalities (19) of Lemma 4, we obtain

$$\begin{array}{c}\Xi (u_{p_1},u_{p_2},\sigma )\le {\displaystyle \frac{1}{6}}\left\{\sum _{\iota =2}^{\infty }(2\iota +1)(\iota +1)(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ \hfill +\left.\sum _{\iota =2}^{\infty }(2\iota -1)(\iota -1)(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}\end{array}$$

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{6}}\left\{\sum _{\iota =2}^{\infty }[2\sigma {\iota }^5+3\sigma {\iota }^4+(2-\sigma ){\iota }^3+(3-3\sigma ){\iota }^2+(1-\sigma )\iota ]{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ & & +\left.\sum _{\iota =2}^{\infty }[2\sigma {\iota }^5-3\sigma {\iota }^4+(2-\sigma ){\iota }^3+(3\sigma -3){\iota }^2+(1-\sigma )\iota ]{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}.\hfill \end{array}$$

(30)

Writing

$${\iota }^2=\iota (\iota -1)+\iota ,$$

(31)

$${\iota }^3=\iota (\iota -1)(\iota -2)+3\iota (\iota -1)+\iota ,$$

(32)

$${\iota }^4=\iota (\iota -1)(\iota -2)(\iota -3)+6\iota (\iota -1)(\iota -2)+7\iota (\iota -1)+\iota ,$$

(33)

and

$$\begin{array}{cc}\hfill {\iota }^5& =\iota (\iota -1)(\iota -2)(\iota -3)(\iota -4)+10\iota (\iota -1)(\iota -2)(\iota -3)\hfill \\ \hfill & +25\iota (\iota -1)(\iota -2)15\iota (\iota -1)+\iota ,\hfill \end{array}$$

(34)

in (30), we have

$$\begin{array}{c}\Xi (u_{p_1},u_{p_2},\sigma )\le {\displaystyle \frac{1}{6}}\left\{\sum _{\iota =2}^{\infty }[2\sigma \iota (\iota -1)(\iota -2)(\iota -3)(\iota -4)+23\sigma \iota (\iota -1)(\iota -2)(\iota -3)\right.\hfill \\ \hfill +\left.(67\sigma +2)\iota (\iota -1)(\iota -2)+(45\sigma +9)\iota (\iota -1)+6\iota ]{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right\}\end{array}$$

$$\begin{array}{ccc}& & +{\displaystyle \frac{1}{6}}\left\{\sum _{\iota =2}^{\infty }[2\sigma \iota (\iota -1)(\iota -2)(\iota -3)(\iota -4)+17\sigma \iota (\iota -1)(\iota -2)(\iota -3)\right.\hfill \\ & & +\left.(31\sigma +2)\iota (\iota -1)(\iota -2)+(9\sigma +3)\iota (\iota -1)]{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}.\hfill \end{array}$$

(35)

Using the Equations (21)–(26), we have

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{6}}\left\{[2\sigma {\Theta }_1^{\left(5\right)}\left(1\right)+23\sigma {\Theta }_1^{\left(4\right)}\left(1\right)(67\sigma +2){\Theta }_1^{\left(3\right)}\left(1\right)+(45\sigma +9){\Theta }_1^{\left(2\right)}\left(1\right)+6{\Theta }_1^{\prime }\left(1\right)\right.\hfill \\ & & +\left.2\sigma {\Theta }_2^{\left(5\right)}\left(1\right)+17\sigma {\Theta }_2^{\left(4\right)}\left(1\right)+(31\sigma +2){\Theta }_2^{\left(3\right)}\left(1\right)+(9\sigma +3){\Theta }_2^{\left(2\right)}\left(1\right)]\right\}.\hfill \end{array}$$

(36)

$$\begin{array}{c}\Xi (u_{p_1},u_{p_2},\sigma )\le {\displaystyle \frac{1}{6}}\left\{2\sigma u_{p_1}^{\left(5\right)}\left(1\right)+33\sigma u_{p_1}^{\left(4\right)}\left(1\right)+(159\sigma +2)u_{p_1}^{\left(3\right)}\left(1\right)\right.\hfill \\ +(246\sigma +15)u_{p_1}^{\left(2\right)}\left(1\right)+(90\sigma +24)u_{p_1}^{\prime }\left(1\right)+6(u_{p_1}\left(1\right)-1)\\ +2\sigma u_{p_2}^{\left(5\right)}\left(1\right)+27\sigma u_{p_2}^{\left(4\right)}\left(1\right)+(99\sigma +2)u_{p_2}^{\left(3\right)}\left(1\right)\\ \hfill +\left.(102\sigma +9)u_{p_2}^{\left(2\right)}\left(1\right)+(18\sigma +6)u_{p_2}^{\prime }\left(1\right)\right\}.\end{array}$$

Now, $\Xi (u_{p_1},u_{p_2},\sigma )\le 1-\eta$ if (28) holds. □

Theorem 2. 

If

$$\begin{array}{ccc}& & \sigma \left[u_{p_1}^{\left(4\right)}\left(1\right)+u_{p_2}^{\left(4\right)}\left(1\right)\right]+11\sigma u_{p_1}^{\left(3\right)}\left(1\right)+(30\sigma +1)u_{p_1}^{\left(2\right)}\left(1\right)+(18\sigma +4)u_{p_1}^{\prime }\left(1\right)\hfill \\ & & +2(u_{p_1}\left(1\right)-1)+9\sigma u_{p_2}^{\left(3\right)}\left(1\right)+(18\sigma +1)u_{p_2}^{\left(2\right)}\left(1\right)+(6\sigma +2)u_{p_2}^{\prime }\left(1\right)\hfill \\ & \le & 2(1-\eta ),\hfill \end{array}$$

(37)

then

$$\mathrm{\Lambda }\left({\mathbb{K}}_{\mathbb{HF}}\right)\subset \mathbb{HF}(\sigma ,\eta ).$$

Proof. 

Let $\hslash =\Im +\overline{\aleph }$∈${\mathbb{K}}_{\mathbb{HF}}$ where ℑ and ℵ are of the form (2) with $y_1=0$. We need to show that $\mathrm{\Lambda }\left(f\right)=H+\overline{G}$∈$\mathbb{HF}(\sigma ,\eta )$, which is given by (9) with $y_1=0$. In view of Lemma 1, we need to prove that

$$\Xi (u_{p_1},u_{p_2},\sigma )\le 1-\eta ,$$

where $\Xi (u_{p_1},u_{p_2},\sigma )$ as given in (29). Using the inequalities (20) of Lemma 5, we obtain

$$\begin{array}{c}\Xi (u_{p_1},u_{p_2},\sigma )\le {\displaystyle \frac{1}{2}}\left\{\sum _{\iota =2}^{\infty }(\iota +1)(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ \hfill +\left.\sum _{\iota =2}^{\infty }(\iota -1)(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}\end{array}$$

$$\begin{array}{c}={\displaystyle \frac{1}{2}}\left\{\sum _{\iota =2}^{\infty }\left[\sigma {\iota }^4+\sigma {\iota }^3+(1-\sigma ){\iota }^2+(1-\sigma )\iota \right]{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ \hfill +\left.\sum _{\iota =2}^{\infty }\left[\sigma {\iota }^4-\sigma {\iota }^3+(1-\sigma ){\iota }^2+(\sigma -1)\iota \right]{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}.\end{array}$$

Using the Equations (31)–(33), we have

$$\begin{array}{c}\Xi (u_{p_1},u_{p_2},\sigma )\hfill \\ \le {\displaystyle \frac{1}{2}}\left\{\sum _{\iota =2}^{\infty }\left[\sigma \iota (\iota -1)(\iota -2)(\iota -3)+7\sigma \iota (\iota -1)(\iota -2)+(9\sigma +1)\iota (\iota -1)+2\iota \right]\right.\\ \times {\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\\ \hfill \left.+\sum _{\iota =2}^{\infty }\left[\sigma \iota (\iota -1)(\iota -2)(\iota -3)+5\sigma \iota (\iota -1)(\iota -2)+(3\sigma +1)\iota (\iota -1)\right]\times {\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}.\end{array}$$

$$\begin{array}{c}={\displaystyle \frac{1}{2}}\left\{\sigma {\Theta }_1^{\left(4\right)}\left(1\right)+7\sigma {\Theta }_1^{\left(3\right)}\left(1\right)+(9\sigma +1){\Theta }_1^{\left(2\right)}\left(1\right)+2{\Theta }_1^{\prime }\left(1\right)\right.\hfill \\ \hfill +\left.\sigma {\Theta }_2^{\left(4\right)}\left(1\right)+5\sigma {\Theta }_2^{\left(3\right)}\left(1\right)+(3\sigma +1){\Theta }_2^{\left(2\right)}\left(1\right)\right\}.\end{array}$$

Using the Equations (21)–(25), we have

$$\begin{array}{c}={\displaystyle \frac{1}{2}}\left\{\sigma u_{p_1}^{\left(4\right)}\left(1\right)+11\sigma u_{p_1}^{\left(3\right)}\left(1\right)+(30\sigma +1)u_{p_1}^{\left(2\right)}\left(1\right)+(18\sigma +4)u_{p_1}^{\prime }\left(1\right)+2(u_{p_1}\left(1\right)-1)\right.\hfill \\ \hfill +\left.\sigma u_{p_2}^{\left(4\right)}\left(1\right)+9\sigma u_{p_2}^{\left(3\right)}\left(1\right)+(18\sigma +1)u_{p_2}^{\left(2\right)}\left(1\right)+(6\sigma +2)u_{p_2}^{\prime }\left(1\right)\right\}.\end{array}$$

Now, $\Xi (u_{p_1},u_{p_2},\sigma )\le 1-\eta$ if (37) holds. □

The connection between $\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$ and $\mathbb{HF}(\sigma ,\eta )$ is given below in the next theorem.

Theorem 3. 

If

$$(1-\varrho )\left[\sigma \left(u_{p_1}^{″}\left(1\right)+u_{p_2}^{″}\left(1\right)\right)+3\sigma \left(u_{p_1}^{\prime }\left(1\right)+u_{p_2}^{\prime }\left(1\right)\right)+\left(u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-2\right)\right]\le 1-\eta -|y_1|,$$

then

$$\mathrm{\Lambda }\left(\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)\right)\subset \mathbb{HF}(\sigma ,\eta ).$$

Proof. 

Let $\hslash =\Im +\overline{\aleph }$∈$\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)$ where ℑ and ℵ are of the form (2) with $y_1=0$. In view of Lemma 1, it is enough to show that $\mathbb{P}(u_{p_1},u_{p_2},\sigma )\le 1-\eta$, where

$$\begin{array}{ccc}\hfill \mathbb{P}(u_{p_1},u_{p_2},\sigma )& =& \sum _{\iota =2}^{\infty }(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\left|x_{\iota }\right|\hfill \\ & & +|y_1|+\sum _{\iota =2}^{\infty }(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\left|y_{\iota }\right|.\hfill \end{array}$$

(38)

Using the inequalities (14) and (15) of Lemma 2, it follows that

$$\begin{array}{c}\mathbb{P}(u_{p_1},u_{p_2},\sigma )\le (1-\varrho )\left\{\sum _{\iota =2}^{\infty }\left[\sigma {\iota }^2+1-\iota \right]{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ \hfill +\left.\sum _{\iota =2}^{\infty }\left[\sigma {\iota }^2+1-\iota \right]{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}+\left|y_1\right|.\end{array}$$

$$\begin{array}{c}=(1-\varrho )\left\{\sum _{\iota =2}^{\infty }\left[\sigma \iota (\iota -1)+\sigma \iota +1-\sigma \right]{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ \hfill +\left.\sum _{\iota =2}^{\infty }\left[\sigma \iota (\iota -1)+\sigma \iota +1-\sigma \right]{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}+\left|y_1\right|.\end{array}$$

$$=(1-\varrho )\left[\sigma {\Theta }_1^{″}\left(1\right)+\sigma {\Theta }_1^{\prime }\left(1\right)+(1-\sigma ){\sigma }_1\left(1\right)+\sigma {\Theta }_2^{″}\left(1\right)+\sigma {\Theta }_2^{\prime }\left(1\right)+(1-\sigma ){\Theta }_2\left(1\right)\right]+\left|y_1\right|$$

Using the Equations (21)–(23), we have

$$\begin{array}{c}=(1-\varrho )\left[\sigma (u_{p_1}^{ȃ}\left(1\right)+u_{p_2}^{″}\left(1\right))+3\sigma (u_{p_1}^{\prime }\left(1\right)+u_{p_2}^{\prime }\left(1\right))+(u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-2)\right]+\left|y_1\right|\hfill \\ \hfill \le 1-\eta ,\end{array}$$

by the given hypothesis. □

Below, we prove that $\mathrm{\Lambda }\left(\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)\right)\subset \mathbb{HF}(\sigma ,\eta ).$

Theorem 4. 

If

$$\begin{array}{ccc}& & (1-\varrho )\left[\sigma \left(u_{p_1}^{\prime }\left(1\right)+u_{p_2}^{\prime }\left(1\right)\right)+\sigma \left(u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-2\right)+{\int }_0^1{u}_{p_1}\left(s\right)ds+{\int }_0^1{u}_{p_2}\left(s\right)ds\right]\hfill \\ & \le & 1-\eta -|y_1|,\hfill \end{array}$$

then

$$\mathrm{\Lambda }\left(\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)\right)\subset \mathbb{HF}(\sigma ,\eta ).$$

Proof. 

Making use of Lemma 1, we need only to prove that $\mathbb{P}(u_{p_1},u_{p_2},\sigma )\le 1-\eta$, where $\mathbb{P}(u_{p_1},u_{p_2},\sigma )$ as given in (38). Using the inequalities (17) and (18) of Lemma 3, it follows that

$$\begin{array}{c}\mathbb{P}(u_{p_1},u_{p_2},\sigma )=\sum _{\iota =2}^{\infty }(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\left|x_{\iota }\right|\hfill \\ \hfill +|y_1|+\sum _{\iota =2}^{\infty }(\iota +\sigma \iota ({\iota }^2-1)){\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\left|y_{\iota }\right|.\end{array}$$

$$\begin{array}{c}\le (1-\varrho )\left\{\sum _{\iota =2}^{\infty }\left(\sigma \iota +{\displaystyle \frac{1-\sigma }{\iota }}\right){\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}\right.\hfill \\ \hfill +\left.\sum _{\iota =2}^{\infty }\left(\sigma \iota +{\displaystyle \frac{1-\sigma }{\iota }}\right){\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right\}+\left|y_1\right|\end{array}$$

Using the Equations (21)and (22), we have

$$=(1-\varrho )\left[\sigma {\Theta }_1^{\prime }\left(1\right)+{\int }_0^1{\displaystyle \frac{{\varphi }_1\left(s\right)}{s}}ds+\sigma {\Theta }_2^{\prime }\left(1\right)+{\int }_0^1{\displaystyle \frac{{\Theta }_2\left(s\right)}{s}}ds\right]+\left|y_1\right|$$

$$\begin{array}{c}\hfill =(1-\varrho )\left[\sigma (u_{p_1}^{\prime }\left(1\right)+u_{p_2}^{\prime }\left(1\right))+\sigma (u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-2)+{\int }_0^1{u}_{p_1}\left(s\right)ds+{\int }_0^1{u}_{p_2}\left(s\right)ds\right]+\left|y_1\right|\hspace{1em}\le \hspace{1em}1-\eta ,\end{array}$$

by given hypothesis. □

Theorem 5. 

If

$$u_{p_1}\left(1\right)+u_{p_2}\left(1\right)\le 5-{\displaystyle \frac{|y_1|}{1-\eta }},$$

then

$$\mathrm{\Lambda }\left(\mathbb{HF}\right(\sigma ,\eta \left)\right)\subset \mathbb{HF}(\sigma ,\eta ).$$

Proof. 

Using the inequalities (11) and (12) of Lemma 1, we obtain

$$\begin{array}{ccc}\hfill \mathbb{P}(u_{p_1},u_{p_2},\sigma )& \le & (1-\eta )\left[\sum _{\iota =2}^{\infty }{\displaystyle \frac{{(-c_1/4)}^{\iota -1}}{{\left(t_1\right)}_{\iota -1}(\iota -1)!}}+\sum _{\iota =2}^{\infty }{\displaystyle \frac{{(-c_2/4)}^{\iota -1}}{{\left(t_2\right)}_{\iota -1}(\iota -1)!}}\right]+\left|y_1\right|\hfill \\ & =& (1-\eta )\left[({\Theta }_1\left(1\right)-1)+({\Theta }_2\left(1\right)-1)\right]+\left|y_1\right|\hfill \\ & =& (1-\eta )\left[u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-4\right]+\left|y_1\right|\le 1-\eta ,\hfill \end{array}$$

Using the Equation (21)

$$\begin{array}{ccc}\hfill \mathbb{P}(u_{p_1},u_{p_2},\sigma )& \le & (1-\eta )\left[u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-4\right]+\left|y_1\right|\hfill \\ & \le & 1-\eta ,\hfill \end{array}$$

by given condition. □

Corollary 1. 

If

$$\begin{array}{ccc}& & 2\left(u_{p_1}^{\left(3\right)}\left(1\right)+u_{p_2}^{\left(3\right)}\left(1\right)\right)+3\left(5u_{p_1}^{\left(2\right)}\left(1\right)+3u_{p_2}^{\left(2\right)}\left(1\right)\right)+24u_{p_1}^{\prime }\left(1\right)+6u_{p_2}^{\prime }\left(1\right)+6(u_{p_1}\left(1\right)-1)\hfill \\ & \le & 6(1-\eta )\hfill \end{array}$$

then

$$\mathrm{\Lambda }\left({\mathbb{S}}_{\mathbb{HF}}^{*}\right)\subset \mathbb{HF}\left(\eta \right).$$

Corollary 2. 

If

$$u_{p_1}^{\left(2\right)}\left(1\right)+u_{p_2}^{\left(2\right)}\left(1\right)+4u_{p_1}^{\prime }\left(1\right)+2u_{p_2}^{\prime }\left(1\right)+2(u_{p_1}\left(1\right)-1)\le 2(1-\eta )$$

then

$$\mathrm{\Lambda }\left({\mathbb{K}}_{\mathbb{HF}}\right)\subset \mathbb{HF}\left(\eta \right).$$

Corollary 3. 

If

$$(1-\varrho )\left[u_{p_1}\left(1\right)+u_{p_2}\left(1\right)-2\right]\le 1-\eta -|y_1|,$$

then

$$\mathrm{\Lambda }\left(\mathcal{T}{\mathfrak{N}}_{\mathbb{HF}}\left(\varrho \right)\right)\subset \mathbb{HF}\left(\eta \right).$$

Corollary 4. 

If

$$(1-\varrho )\left[{\int }_0^1{u}_{p_1}\left(s\right)ds+{\int }_0^1{u}_{p_2}\left(s\right)ds\right]\le 1-\eta -|y_1|,$$

then

$$\mathrm{\Lambda }\left(\mathcal{T}{\mathfrak{Q}}_{\mathbb{HF}}\left(\varrho \right)\right)\subset \mathbb{HF}\left(\eta \right).$$

Thus, the results obtained for $\sigma =0$ are reduced from the class $\mathbb{HF}(\sigma ,\eta )$ to the class $\mathbb{HF}\left(\eta \right)$.

5. Conclusions

Using the operator $\Lambda$, defined in Equation (8) and associated with the Bessel function, we have established several inclusion relations involving the harmonic function class $\mathbb{HF}(\sigma ,\eta )$ and other subclasses of harmonic analytic functions defined in the open symmetric unit disk $\mathrm{Y}$. Several corollaries of the main result when $\sigma =0$ are also provided. Following this study, new inclusion relations can be established for new harmonic subclasses of analytic functions through the application of the operator $\Lambda$. Furthermore, the $\Lambda$ operator can be compared with other known integral operators (e.g., Bernardi, Ruscheweyh, and Salagean operators). Relationships between classes can be established under the influence of these operators. Stability conditions can be determined for classes obtained with the $\Lambda$ operator. Harmonic transformations are frequently used, especially in image processing, signal processing, and potential theory.