Relations of Harmonic Starlike Function Subclasses with Mittag–Leffler Function

Abstract

In this study, the connection between certain subfamilies of harmonic univalent functions is established by utilizing a convolution operator involving the Mittag–Leffler function. The investigation reveals inclusion relations concerning harmonic $\gamma$-uniformly starlike mappings in the open unit disc, harmonic starlike functions and harmonic convex functions, highlighting the improvements given by the results presented here on previously published outcomes.

1. Introduction

Many physical problems can be modeled using harmonic univalent mappings. They also have applications in engineering and biology. Since harmonic mappings give minimal conformal parameters for surfaces, differential geometers also find them useful for their investigations where planar harmonic mappings allow for a more thorough study of a number of minimal surface characteristics, including the Gauss curvature.

Harmonic functions have significant implications in Geometric Function Theory starting with the basic property that the real and imaginary parts of a holomorphic function are harmonic functions. The topic of harmonic univalent functions was first introduced to Geometric Function Theory in 1984 by the renowned researchers Clunie and Sheil-Small [1]. In their research, they employed an advanced analytical approach and were successful in identifying feasible alternatives of the standard growth and distortion results, covering theorems, and coefficient problems in the context of planar harmonic mappings. They proposed new subclasses of univalent harmonic sense-preserving functions using the following well-known classes: $\mathcal{A}$, that includes the functions h written as $h\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k,$ analytic in the open unit disc $U=\{z\in C:|z|<1\}$, and satisfying the normalization condition $h\left(0\right)=h^{\prime }\left(0\right)-1=0$, and $\mathcal{H}$, the class of harmonic functions written as

$$f=h+\overline{g},$$

(1)

where

$$h\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k,g\left(z\right)=\sum _{k=1}^{\infty }{b}_k{z}^k,(z\in U),$$

(2)

$h,g\in \mathcal{A}$; hence, $f\left(z\right)\in \mathcal{H}$ written as

$$f\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k+\overline{\sum _{k=1}^{\infty }{b}_k{z}^k},(z\in U).$$

(3)

The subclass of $\mathcal{H}$ including univalent and sense-preserving functions in U is identified by $S_{\mathcal{H}}$. The sense-preserving characteristic implies that $\left|b_1\right|<1$, as one can easily demonstrate. If $f\in S_{\mathcal{H}}$, then ${\displaystyle \frac{f-\overline{b_{1}f}}{1-{\left|b_1\right|}^2}}\in S_{\mathcal{H}}$. The subclass of $S_{\mathcal{H}}$ denoted by $S_{\mathcal{H}}^0$ is described as

$$S_{\mathcal{H}}^0=\left\{f=h+\overline{g}\in S_{\mathcal{H}}:g^{\prime }\left(0\right)=b_1=0\right\}.$$

Initial research on $S_{\mathcal{H}}^0$ and $S_{\mathcal{H}}$ was conducted in [1]. Also, $S_{\mathcal{H}}^{*,0},C_{\mathcal{H}}^0$ and $K_{\mathcal{H}}^0$ illustrate the subclasses of $S_{\mathcal{H}}^0$ of harmonic functions that are starlike, close-to-convex and convex in U, respectively. To learn more about these classes’ definitions and characteristics, consult [2,3,4,5,6].

Harmonic functions $f=h+\overline{g}$, where

$$h\left(z\right)=z-\sum _{k=2}^{\infty }\left|a_k\right|z^k,g\left(z\right)=\sum _{k=1}^{\infty }\left|b_k\right|z^k,(z\in U),$$

(4)

are comprised in the class denoted by ${\mathcal{T}}_H$, which is a class that was introduced and investigated by Silverman [7].

For $0\le \alpha <1, 0\le r<1$ and $0\le \theta \le 2\pi$, Ahuja and Jahangiri studied in [4] the following classes:

$$N_{\mathcal{H}}\left(\alpha \right)=\left\{f\in \mathcal{H}:Re\left({\displaystyle \frac{f^{\prime }\left(z\right)}{z^{\prime }}}\right)\ge \alpha ,z=r{e}^{i\theta }\right\}$$

where

$$f^{\prime }\left(z\right)={\displaystyle \frac{\partial }{\partial \theta }}f\left(r{e}^{i\theta }\right)=i\left(z{h}^{\prime }\left(z\right)-\overline{z{g}^{\prime }\left(z\right)}\right),z^{\prime }={\displaystyle \frac{\partial }{\partial \theta }}\left(r{e}^{i\theta }\right),$$

and

$$\mathcal{T}{N}_{\mathcal{H}}\left(\alpha \right)=N_{\mathcal{H}}\left(\alpha \right)\cap {\mathcal{T}}_{\mathcal{H}}.$$

Further applications of harmonic functions in fractional and quantum calculus can be read in [8,9, 10], respectively.

Definition 1

([3]). A function $f=h+\stackrel{-}{g}$ is said to be a γ-uniformly harmonic starlike function in U if it satisfies the following condition:

$$Re\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{z^{\prime }[(1-\eta )z+\eta (h\left(z\right)+\overline{g\left(z\right)})]}}-\delta \right)\ge \gamma \left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{z^{\prime }[(1-\eta )z+\eta (h\left(z\right)+\overline{g\left(z\right)})]}}-1\right|$$

for $0\le \eta \le 1,0\le \delta <1,0\le \gamma <\infty$.

The family of this function is denoted by $G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Also, define $\mathcal{I}{G}_{\mathcal{H}}(\gamma ,\delta ,\eta )=G_{\mathcal{H}}(\gamma ,\delta ,\eta )\cap {\mathcal{T}}_{\mathcal{H}}$.

The above-defined class includes several simpler subclasses. We point out here some of these particular classes.

(a)

Taking $\gamma =0$ and $\eta =0$, the class $N_{\mathcal{H}}\left(\delta \right)$ is obtained, which was investigated by Ahuja and Jahangiri [4];

(b)

Letting $\gamma =0$ and $\eta =1$, the class $S_{\mathcal{H}}^{*}\left(\delta \right)$ follows, which was proposed by Jahangiri [11];

(c)

Considering $\eta =1$ and $\delta =0$, the class $G_{\mathcal{H}}^{*}\left(\gamma \right)$ emerges, which was employed by Rosy et al. [12];

(d)

Utilizing $\gamma =1,\delta =0,\eta =1$ and $g\left(z\right)\equiv 0$, the class $U{S}^{*}$ is determined, which was introduced by R$\varphi$nning [13];

(e)

Assuming $\eta =1$, the class studied by Porwal and Srivastava [14], $HU{S}^{*}(\gamma ,\delta )$, is obtained.

2. Materials and Methods

In this paper, motivated by the earlier studies of Porwal and Srivastava seen in [14], the subclass of harmonic univalent functions $f\in \mathcal{H}$ seen in Definition 1 is considered for studying inclusion relations implementing the Mittag–Leffler function. Certain known lemmas used for the investigation are listed below.

Lemma 1

([1]). If $f\in K_{\mathcal{H}}^0$ and $f=h+\overline{g}$ where h and g are given by (2) with $b_1=0$, then

$$\left|a_k\right|\le {\displaystyle \frac{k+1}{2}},\left|b_k\right|\le {\displaystyle \frac{k-1}{2}}(k\ge 1).$$

Lemma 2

([2]). If $f\in C_{\mathcal{H}}^0$ or $S_{\mathcal{H}}^{*,0}$ and $f=h+\overline{g}$ where h and g are given by (2) with $b_1=0$, then

$$\left|a_k\right|\le {\displaystyle \frac{(2k+1)(k+1)}{6}},\left|b_k\right|\le {\displaystyle \frac{(2k-1)(k-1)}{6}}(k\ge 1).$$

Lemma 3

([4]). If $f\in \mathcal{T}{N}_{\mathcal{H}}\left(\alpha \right)$ and $f=h+\overline{g}$ where h and g are given by (3), then

$$\left|a_k\right|\le {\displaystyle \frac{1-\alpha }{k}},\left|b_k\right|\le {\displaystyle \frac{1-\alpha }{k}}(k\ge 1,0\le \alpha <1).$$

Lemma 4

([3]). Let $0\le \eta \le 1, 0\le \delta <1$ and $0\le \gamma <\infty$. Also, let $f=h+\overline{g}$, where h and g are given by (2). If the following condition

$$\sum _{k=2}^{\infty }{\displaystyle \frac{k(\gamma +1)-\eta (\gamma +\delta )}{1-\delta }}\left|a_k\right|+\sum _{k=1}^{\infty }{\displaystyle \frac{k(\gamma +1)+\eta (\gamma +\delta )}{1-\delta }}\left|b_k\right|\le 1,$$

(5)

holds, then f is sense preserving and harmonic mapping in U and $f\in G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Lemma 5

([3]). Let $0\le \eta \le 1, 0\le \delta <1$ and $0\le \gamma <\infty$. Also, let $f=h+\overline{g}$, where h and g are given by (2). A function $f\in \mathcal{I}\left(G_{\mathcal{H}}(\gamma ,\delta ,\eta )\right)$ if and only if the condition (5) holds. Moreover, if $f\in \mathcal{I}\left(G_{\mathcal{H}}(\gamma ,\delta ,\eta )\right)$, then

$$\begin{array}{cc}& \left|a_k\right|\le {\displaystyle \frac{1-\delta }{k(\gamma +1)-\eta (\gamma +\delta )}}(k\ge 2),\hfill \\ & \left|b_k\right|\le {\displaystyle \frac{1-\delta }{k(\gamma +1)+\eta (\gamma +\delta )}}(k\ge 1).\hfill \end{array}$$

The function ${\mathbf{E}}_{\rho }\left(z\right)$ given by

$${\mathbf{E}}_{\rho }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(\rho k+1)}},(z,\rho \in \mathbb{C}\mathrm{with}Re\rho >0),$$

known as the Mittag–Leffler function, was introduced in [15]. Wiman [16] generalized the function ${\mathbf{E}}_{\rho }\left(z\right)$ by proposing the function ${\mathbf{E}}_{\rho ,\beta }\left(z\right)$ of the form

$${\mathbf{E}}_{\rho ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(\rho k+\beta )}},(z,\rho ,\beta \in \mathbb{C}\mathrm{with}Re\rho >0,Re\beta >0),$$

(6)

It should be noted that the function ${\mathbf{E}}_{\rho ,\beta }\left(z\right)$ includes numerous well-known functions as its specific examples, such as ${\mathbf{E}}_{\mathbf{0},\mathbf{0}}\left(z\right)={\sum }_{k=0}^{\infty }{z}^k,{\mathbf{E}}_{\mathbf{1},\mathbf{1}}\left(z\right)=e^z,{\mathbf{E}}_{\mathbf{1},\mathbf{2}}\left(z\right)={\displaystyle \frac{e^z-1}{z}},{\mathbf{E}}_{\mathbf{2},\mathbf{1}}\left(z^2\right)=coshz$, ${\mathbf{E}}_{\mathbf{2},\mathbf{1}}\left(-z^2\right)=cosz,{\mathbf{E}}_{\mathbf{2},\mathbf{2}}\left(z^2\right)={\displaystyle \frac{sinhz}{z}},{\mathbf{E}}_{\mathbf{2},\mathbf{2}}\left(-z^2\right)={\displaystyle \frac{sinz}{z}},{\mathbf{E}}_{\mathbf{4}}\left(z\right)={\displaystyle \frac{1}{2}}\left[cos{z}^{{\displaystyle \frac{1}{4}}}+cosh{z}^{{\displaystyle \frac{1}{4}}}\right]$ and

$${\mathbf{E}}_{\mathbf{3}}\left(z\right)={\displaystyle \frac{1}{2}}\left[e^{z^{{\displaystyle \frac{1}{3}}}}+2e^{-{\displaystyle \frac{1}{2}}{z}^{{\displaystyle \frac{1}{3}}}}cos\left({\displaystyle \frac{\sqrt{3}}{2}}{z}^{{\displaystyle \frac{1}{3}}}\right)\right].$$

It is of interest to note that by fixing $\rho ={\displaystyle \frac{1}{2}}$ and $\beta =1$, we obtain

$${\mathbf{E}}_{{\displaystyle \frac{1}{2}},\mathbf{1}}\left(z\right)=e^{z^2}·erfc(-z)=e^{z^2}\left(1+{\displaystyle \frac{2}{\sqrt{\pi }}}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k}{n!(2k+1)}}{z}^{2k+1}\right).$$

Mittag–Leffler functions have emerged as a crucial component of special functions theory in recent years. Over the past fifteen years, engineers and scientists have become much more interested in Mittag–Leffler functions and Mittag–Leffler-type functions because of their numerous applications in a variety of applied problems, including probability, statistical distribution theory, fluid flow, rheology, diffusive transport akin to diffusion, and electric networks. Because of its numerous uses in research and engineering, this function has grown in significance and popularity during the course of fractional calculus’s numerous advances over the past forty years. In the study of complex systems, random walks, Lévy fligts, the fractional generalization of kinetic equations, and super-diffusive transport, the Mittag–Leffler function is particularly helpful when examining differential and integral equations of fractional order. Numerous publications such as [17,18,19,20,21] contain a variety of Mittag–Leffler function and generalized Mittag–Leffler function distinctive characteristics. The Mittag–Leffler function ${\mathbf{E}}_{\rho ,\beta }\left(z\right)$ is not a member of $\mathcal{A}$, as we prompt. Consequently, Bansal and Prajapat [18] proposed the normalization of ${\mathbf{E}}_{\rho ,\beta }\left(z\right)$ as follows:

$${\mathcal{E}}_{\rho ,\beta }\left(z\right)=z\mathsf{\Gamma }\left(\beta \right){\mathbf{E}}_{\rho ,\beta }\left(z\right)=z+\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{z}^k,$$

for complex parameters $z,\rho ,\beta \in \mathbb{C}$, with $Re\rho >0,Re\beta >0$.

In our present study, the attention will be focused on the case of real-valued $\rho ,\beta$ and $z\in U$, and hence new linear operators are defined based on the convolution (or Hadamard) product as below:

$${\mathcal{E}}_{\rho ,\beta }\left(z\right)=z+\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{z}^k,\mathrm{and}{\mathcal{E}}_{\xi ,\phi }\left(z\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}{z}^k.$$

(7)

In the study by Murugusundaramoorthy et al. [22], the convolution operator $\mathsf{\Omega }\left(f\right)$ is defined as follows:

$$\mathcal{F}\left(z\right)=\mathsf{\Omega }f\left(z\right)=h\left(z\right)\ast {\mathcal{E}}_{\rho ,\beta }\left(z\right)+\overline{g\left(z\right)\ast {\mathcal{E}}_{\xi ,\phi }\left(z\right)}$$

$$=z+\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{a}_k{z}^k+\sum _{k=1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}{b}_k{z}^k$$

(8)

for real parameters $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$.

The literature on this specific topic examined inclusion relations between various subclasses of analytic and univalent functions using hypergeometric functions ([23,24]), and more recently, distribution series ([25,26,27]). A number of authors have lately examined inclusion results and mapping features for the families of harmonic univalent functions that apply different linear and nonlinear operators ([28,29,30]).

In our current study, motivated by the the previously mentioned research and by the recent study proposed in [31], we apply the convolution operator $\mathsf{\Omega }$ given by (8) involving the Mittag–Leffler function to uncover the connections between the classes $S_{\mathcal{H}}^{*,0},K_{\mathcal{H}}^0$ and ${\mathcal{G}}_{\mathcal{H}}(\lambda ,\alpha ,\gamma )$.

Over the course of the investigation, we make use of the following:

$${\mathcal{E}}_{\rho ,\beta }\left(z\right)=z+\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{z}^k,{\mathcal{E}}_{\rho ,\beta }\left(1\right)=1+\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}},$$

(9)

and

$${\mathcal{E}}_{\rho ,\beta }^{\prime }\left(z\right)=1+\sum _{k=2}^{\infty }{\displaystyle \frac{k\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{z}^{k-1},{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)-1=\sum _{k=2}^{\infty }{\displaystyle \frac{k\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}},$$

(10)

$${\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)=\sum _{k=2}^{\infty }{\displaystyle \frac{k(k-1)\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}},$$

(11)

$${\mathcal{E}}_{\rho ,\beta }^{\prime \prime \prime }\left(1\right)=\sum _{k=2}^{\infty }{\displaystyle \frac{k(k-1)(k-2)\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}.$$

(12)

3. Results

In this section, we will use the Mittag–Leffler function properties to find the inclusion relations between the harmonic class $G_{\mathcal{H}}(\gamma ,\delta ,\eta )$ and the classes $K_{\mathcal{H}}^0$ and $S_{\mathcal{H}}^{*}$, respectively.

Theorem 1.

Let $0\le \delta <1,0\le \eta \le 1$ and $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$ be real. If we have

$$(\gamma +1)\left[{\mathcal{E}}_{\rho ,\beta }^{″}\left(1\right)+{\mathcal{E}}_{\xi ,\phi }^{″}\left(1\right)\right]-\eta (\gamma +\delta )[\left.{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)-{\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)+{\mathcal{E}}_{\rho ,\beta }\left(1\right)+{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]$$

$$+2(\gamma +1){\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)\le 2[(1-\eta )(\gamma +\delta )+2(1-\delta )],$$

(13)

then $\mathsf{\Omega }\left(K_{\mathcal{H}}^0\right)\subset G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Proof. 

Consider $f=h+\overline{g}\in K_{\mathcal{H}}^0$ with h and g given by (2) and (3) with $b_1=0$. Using $b_1=0$ and (8), we must demonstrate that $\mathsf{\Omega }\left(f\right)=\mathcal{F}\left(z\right)\in G_{\mathcal{H}}(\gamma ,\delta ,\eta )$. Lemma 4 requires us to demonstrate that

$$\begin{array}{cc}\hfill {\Psi }_1=\sum _{k=2}^{\infty }[k(\gamma +1)& -\eta (\gamma +\delta )]\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{a}_k\right|\hfill \\ & +\sum _{k=2}^{\infty }[k(\gamma +1)+\eta (\gamma +\delta )]\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}{b}_k\right|\le 1-\delta .\hfill \end{array}$$

We now establish a relationship between $G_{\mathcal{H}}(\gamma ,\delta ,\eta )$ and the harmonic convex functions using Lemma 1.

$$\begin{array}{cc}\hfill {\Psi }_1\le {\displaystyle \frac{1}{2}}\left\{\sum _{k=2}^{\infty }[k(\gamma +1)\right.& -\eta (\gamma +\delta )](k+1){\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\hfill \\ & \left.+\sum _{k=2}^{\infty }[k(\gamma +1)+\eta (\gamma +\delta )](k-1){\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\right\}\hfill \end{array}$$

$$\begin{array}{cc}& =\sum _{k=2}^{\infty }\left[{\displaystyle \frac{k(\gamma +1)}{2}}-{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right](k+1){\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\hfill \\ & +\sum _{k=2}^{\infty }\left[{\displaystyle \frac{k(\gamma +1)}{2}}+{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right](k-1){\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\hfill \end{array}$$

$$=\sum _{k=2}^{\infty }[{\displaystyle \frac{(\gamma +1)}{2}}(k-1)(k-2)+\left(2(\gamma +1)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right)(k-1)+((\gamma +1)-\eta (\gamma +\delta ))]$$

$$·{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}$$

$$+\sum _{k=2}^{\infty }\left[{\displaystyle \frac{(\gamma +1)}{2}}(k-1)(k-2)+\left((\gamma +1)+{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right)(k-1)\right]{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}$$

$$\begin{array}{cc}& ={\displaystyle \frac{(\gamma +1)}{2}}\sum _{k=2}^{\infty }{\displaystyle \frac{(k-1)(k-2)\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}+\left(2(\gamma +1)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right)\sum _{k=2}^{\infty }{\displaystyle \frac{(k-1)\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\hfill \\ & +((\gamma +1)-\eta (\gamma +\delta ))\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}+{\displaystyle \frac{(\gamma +1)}{2}}\sum _{k=2}^{\infty }{\displaystyle \frac{(k-1)(k-2)\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\hfill \\ & +\left((\gamma +1)+{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right)\sum _{k=2}^{\infty }{\displaystyle \frac{(k-1)\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}.\hfill \end{array}$$

Now, by using Equations (9)–(11) we write

$$\begin{array}{cc}\hfill {\Psi }_1\le & {\displaystyle \frac{(\gamma +1)}{2}}\left[{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)-2{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)+2{\mathcal{E}}_{\rho ,\beta }\left(1\right)\right]+\left(2(\gamma +1)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right)\left[{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)-{\mathcal{E}}_{\rho ,\beta }\left(1\right)\right]\hfill \\ & +[(\gamma +1)-\eta (\gamma +\delta )]\left[{\mathcal{E}}_{\rho ,\beta }\left(1\right)-1\right]+{\displaystyle \frac{(\gamma +1)}{2}}\left[{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)-2{\mathcal{E}}_{\xi ,\varphi }^{\prime }\left(1\right)+2{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]\hfill \\ & +\left[(\gamma +1)+{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right]\left[{\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)-{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& {\displaystyle \frac{(\gamma +1)}{2}}{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)+\left((\gamma +1)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right){\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta ){\mathcal{E}}_{\rho ,\beta }\left(1\right)\hfill \\ & +{\displaystyle \frac{(\gamma +1)}{2}}{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)+{\displaystyle \frac{\eta }{2}}(\gamma +\delta ){\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta ){\mathcal{E}}_{\xi ,\phi }\left(1\right)-(\gamma +1)+\eta (\gamma +\delta )\le (1-\delta ).\hfill \end{array}$$

Under the specified condition, the last expression is bounded above by $(1-\delta )$. Therefore, Theorem 1 has complete proof. □

Remark 1.

By taking $\gamma =1$ in Theorem 1, the previous result given in ([32], Theorem 1) is imporved. Also, for $\eta =1$ and $\gamma =1$, letting $G=1$, the previous result given in ([22], Theorem 2.1) is improved.

Similarly to Theorem 1, we further provide the relation between the classes $S_{\mathcal{H}}^{*,0}$ and $G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Theorem 2.

Let $0\le \delta <1,0\le \eta \le 1$ and $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$ be real. If the inequality

$$2(\gamma +1){\mathcal{E}}_{\rho ,\beta }^{\prime \prime \prime }\left(1\right)+[9(\gamma +1)-2\eta (\gamma +\delta )]{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)+[6(\gamma +1)-5\eta (\gamma +\delta )]{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)$$

$$-\eta (\gamma +\delta ){\mathcal{E}}_{\rho ,\beta }\left(1\right)+2(\gamma +1){\mathcal{E}}_{\xi ,\phi }^{\prime \prime \prime }\left(1\right)+[3(\gamma +1)+2\eta (\gamma +\delta )]{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)-\eta (\gamma +\delta ){\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)$$

$$+\eta (\gamma +\delta ){\mathcal{E}}_{\xi ,\phi }\left(1\right)\le 6[2(\gamma +1)-(1+\eta )(\gamma +\delta )],$$

(14)

holds, then $\mathsf{\Omega }\left(S_{\mathcal{H}}^{*,0}\right)\subset G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Proof. 

Consider $f=h+\overline{g}\in S_{\mathcal{H}}^{*,0}$ with h and g given by (2) and (3) when $b_1=0$. It is necessary to prove that $\mathsf{\Omega }\left(f\right)=\mathcal{F}\left(z\right)\in G_{\mathcal{H}}(\gamma ,\delta ,\eta )$, which is given by (8) with $b_1=0$. According to Lemma 4, it is sufficient to prove that

$$\begin{array}{cc}\hfill {\Psi }_2=\sum _{k=2}^{\infty }[k(\gamma +1)& -\eta (\gamma +\delta )]\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{a}_k\right|\hfill \\ & +\sum _{k=2}^{\infty }[k(\gamma +1)+\eta (\gamma +\delta )]\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}{b}_k\right|\le 1-\delta .\hfill \end{array}$$

By Lemma 2, we write

$$\begin{array}{cc}& {\Psi }_2\le {\displaystyle \frac{1}{3}}\left\{\sum _{k=2}^{\infty }\left[{\displaystyle \frac{k(\gamma +1)}{2}}-{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right](2k+1)(k+1){\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\right.\hfill \\ & \left.+\sum _{k=2}^{\infty }\left[{\displaystyle \frac{k(\gamma +1)}{2}}+{\displaystyle \frac{\eta }{2}}(\gamma +\delta )\right](2k-1)(k-1){\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& {\displaystyle \frac{1}{3}}\left\{(\gamma +1)\sum _{k=2}^{\infty }(k-1)(k-2)(k-3){\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\right.\hfill \\ & +\left[{\displaystyle \frac{15}{2}}(\gamma +1)-\eta (\gamma +\delta )\right]\sum _{k=2}^{\infty }(k-1)(k-2){\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\hfill \\ \hfill +& \left[12(\gamma +1)-{\displaystyle \frac{9}{2}}\eta (\gamma +\delta )\right]\sum _{k=2}^{\infty }(k-1){\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\hfill \\ \hfill +& [3(\gamma +1)-3\eta (\gamma +\delta )]\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}\hfill \\ \hfill +& (\gamma +1)\sum _{k=2}^{\infty }(k-1)(k-2)(k-3){\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\hfill \\ \hfill +& \left[{\displaystyle \frac{9}{2}}(\gamma +1)+\eta (\gamma +\delta )\right]\sum _{k=2}^{\infty }(k-1)(k-2){\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\hfill \\ & \left.+\left[3(\gamma +1)+{\displaystyle \frac{3}{2}}\eta (\gamma +\delta )\right]\sum _{k=2}^{\infty }(k-1){\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\right\}.\hfill \end{array}$$

Now, by using Equations (9)–(12) we obtain

$$\begin{array}{cc}\hfill {\Psi }_2=& {\displaystyle \frac{1}{3}}\left\{(\gamma +1)\left[{\mathcal{E}}_{\rho ,\beta }^{\prime \prime \prime }\left(1\right)-3{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)+6{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)-6{\mathcal{E}}_{\rho ,\beta }\left(1\right)\right]\right.\hfill \\ & +\left[{\displaystyle \frac{15}{2}}(\gamma +1)-\eta (\gamma +\delta )\right]\left[{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)-2{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)+2{\mathcal{E}}_{\rho ,\beta }\left(1\right)\right]\hfill \\ & +\left[12(\gamma +1)-{\displaystyle \frac{9}{2}}\eta (\gamma +\delta )\right]\left[{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)-{\mathcal{E}}_{\rho ,\beta }\left(1\right)\right]\hfill \\ & +[3(\gamma +1)-3\eta (\gamma +\delta )]\left[{\mathcal{E}}_{\rho ,\beta }\left(1\right)-1\right]\hfill \\ & +(\gamma +1)\left[{\mathcal{E}}_{\xi ,\phi }^{\prime \prime \prime }\left(1\right)-3{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)+6{\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)-6{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]\hfill \\ & +\left[{\displaystyle \frac{9}{2}}(\gamma +1)+\eta (\gamma +\delta )\right]\left[{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)-2{\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)+2{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]\hfill \\ & \left.+\left[3(\gamma +1)+{\displaystyle \frac{3}{2}}\eta (\gamma +\delta )\right]\left[{\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)-{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]\right\}\hfill \end{array}$$

$$={\displaystyle \frac{1}{3}}\left\{(\gamma +1){\mathcal{E}}_{\rho ,\beta }^{\prime \prime \prime }\left(1\right)+\left[{\displaystyle \frac{9}{2}}(\gamma +1)-\eta (\gamma +\delta )\right]{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)+\left[3(\gamma +1)-{\displaystyle \frac{5\eta }{2}}(\gamma +\delta )\right]{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)\right.$$

$$-{\displaystyle \frac{\eta }{2}}(\gamma +\delta ){\mathcal{E}}_{\rho ,\beta }\left(1\right)-[3(\gamma +1)-3\eta (\gamma +\delta )]+(\gamma +1){\mathcal{E}}_{\xi ,\phi }^{\prime \prime \prime }\left(1\right)$$

$$\left.+\left[{\displaystyle \frac{3}{2}}(\gamma +1)+\eta (\gamma +\delta )\right]{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)-{\displaystyle \frac{\eta }{2}}(\gamma +\delta ){\mathcal{E}}_{\xi ,\phi }^{\prime }\left(1\right)+{\displaystyle \frac{\eta }{2}}(\gamma +\delta ){\mathcal{E}}_{\xi ,\phi }\left(1\right)\right\}\le 1-\delta .$$

Hence, the proof is finalized. □

Remark 2.

Taking $\gamma =1$ in Theorem 2, the result given in ([32], Theorem 2) is improved. Also, letting $\eta =1$ and $\gamma =1$ and putting $G=1$, the result given in ([22], Theorem 2.2) is improved.

Theorem 3.

Consider $0\le \delta <1,0\le \eta \le 1$ and $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$ as real. If the inequality

$${\mathcal{E}}_{\rho ,\beta }\left(1\right)+{\mathcal{E}}_{\xi ,\phi }\left(1\right)\le 2,$$

(15)

holds, then $\mathsf{\Omega }\left(\mathcal{I}{G}_{\mathcal{H}}(\gamma ,\delta ,\eta )\right)\subset G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Proof. 

Let $f=h+\overline{g}\in \mathcal{I}{G}_{\mathcal{H}}(\gamma ,\delta ,\eta )$ with h and g given by (2) and (8) when $b_1=0$. By Lemma 4, it is sufficient to prove that

$$\begin{array}{cc}\hfill {\Psi }_3=\sum _{k=2}^{\infty }[k(\gamma +1)& -\eta (\gamma +\delta )]\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}{a}_k\right|\hfill \\ & +\sum _{k=2}^{\infty }\left[k(\gamma +1)+\eta (\gamma +\delta ]\left|{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}{b}_k\right|\le 1-\delta \right..\hfill \end{array}$$

By Lemma 5, we write

$$\begin{array}{cc}\hfill {\Psi }_3& \le (1-\delta )\left\{\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}+\sum _{k=1}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k-1)+\phi )}}\right\}\hfill \\ & =(1-\delta )\left\{\sum _{k=2}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }\left(\rho \right(k-1)+\beta )}}+\sum _{k=0}^{\infty }{\displaystyle \frac{\mathsf{\Gamma }\left(\phi \right)}{\mathsf{\Gamma }\left(\xi \right(k)+\phi )}}\right\},\hfill \end{array}$$

and by using (9), we obtain that the expression above is equal to

$$(1-\delta )\left[{\mathcal{E}}_{\rho ,\beta }\left(1\right)-1+{\mathcal{E}}_{\xi ,\phi }\left(1\right)\right]$$

using the given condition; hence, the proof of the theorem is completed. □

Remark 3.

By taking $\gamma =1$ in Theorem 3, the result obtained in ([32], Theorem 3.3) is improved. Also, letting $\eta =1$ and $\gamma =1$ and putting $G=1$, the result given in ([22], Theorem 2.4) is improved.

Theorem 4.

Consider $0\le \delta <1,0\le \eta \le 1$ and $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$ as real numbers. If we have

$${\mathcal{E}}_{\rho ,\beta }\left(1\right)+{\mathcal{E}}_{\xi ,\phi }\left(1\right)\le 2,$$

(16)

then $\mathsf{\Omega }\left(\mathcal{I}{G}_{\mathcal{H}}(\gamma ,\delta ,\eta )\right)\subset \mathcal{I}{G}_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Proof. 

Since this theorem’s proof is analogous with the proof of Theorem 3, the specifics are not repeated here. □

Remark 4.

By taking $\eta =1$ and $\gamma =1$ in Theorem 4, and letting $G=1$, the result obtained in ([22], Theorem 2.5) is improved.

By taking $\gamma =1$ and $\eta =0$ in Theorems 1 and 2, the following results are obtained.

Corollary 1.

Consider $0\le \delta <1,0\le \eta \le 1$ and $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$ as real numbers. If we have

$${\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)+2{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)+{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)\le 3-\delta ,$$

then $\mathsf{\Omega }\left(K_{\mathcal{H}}^0\right)\subset G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

Corollary 2.

Consider $0\le \delta <1,0\le \eta \le 1$ and $\rho ,\beta ,\xi ,\phi (\rho ,\beta ,\xi ,\phi \notin \{0,-1,-2,\dots \left\}\right)$ as real numbers. If we have

$$4{\mathcal{E}}_{\rho ,\beta }^{\prime \prime \prime }\left(1\right)+18{\mathcal{E}}_{\rho ,\beta }^{\prime \prime }\left(1\right)+12{\mathcal{E}}_{\rho ,\beta }^{\prime }\left(1\right)+4{\mathcal{E}}_{\xi ,\phi }^{\prime \prime \prime }\left(1\right)+6{\mathcal{E}}_{\xi ,\phi }^{\prime \prime }\left(1\right)\le 6(3-\delta ),$$

then $\mathsf{\Omega }\left(S_{\mathcal{H}}^{*,0}\right)\subset G_{\mathcal{H}}(\gamma ,\delta ,\eta )$.

4. Discussion

The present investigation concerns the class of $\gamma$-uniformly harmonic starlike functions in U denoted by $G_{\mathcal{H}}(\gamma ,\delta ,\eta )$, introduced in [3], and the related class $\mathcal{I}{G}_{\mathcal{H}}(\gamma ,\delta ,\eta )$. The study uses as tools for proving the new results a linear operator $\mathsf{\Omega }\left(f\right)$ defined by Murugusundaramoorthy et al. in [22] involving the convolution concept and the Mittag–Leffler function. The inclusion relations connecting the classes of harmonic starlike functions in U, $S_{\mathcal{H}}^{*,0}$, harmonic convex functions in U, $K_{\mathcal{H}}^0$ and $\gamma$-uniformly harmonic starlike functions in U, $G_{\mathcal{H}}(\gamma ,\delta ,\eta )$ are established using known results given by Lemmas 1–5 along with the properties that the operator $\mathsf{\Omega }\left(f\right)$ given by (8) has due to the remarkable Mittag–Leffler function.

The new outcome adds knowledge regarding the theory of harmonic functions improving some previously established results found in [22,32], which were highlighted in Remarks 1–4.