Mapping properties of Janowski-type harmonic functions involving Mittag-Leffler function

1.   Introduction and preliminaries

Denote by $\mathcal{A},$ the family of analytic functions in the unit disc

whose members are

and normalized by $h(0) = h'(0)-1 = 0.$ Let continuous complex- valued harmonic function in a complex domain $\Omega$ be $f = u+iv,$ if both $u$ and $v$ are real and harmonic in $\Omega.$ We write $f = h+ \overline{g},$ where $h$ and $g$ are analytic in $\mathbf{C} \subset \Omega,$ a simply-connected domain. We appeal $h$ the analytic part and $g$ the co-analytic part of $f.$ Clunie and Sheil-Small ^[1] pointed out that $f$ to be locally univalent and sense preserving in $\mathbf{C}$ if and only if $| h'(z)| > | g'(z)|$ in $\mathbb{D}$.

Denote by $\mathrm{H}$ the family of all harmonic functions of the form $f = h+\overline{g}$, is given by,

where

are members of $\mathcal{A}.$ We let

Symbolize $\mathrm{S}_ \mathrm{H},$ the subclass of $\mathrm{H}$ that are univalent and orientation preserving in $\mathbb{D}$. Note that $\frac{f-\overline{\mathrm{B}_1 f}}{1-|{\mathrm{B}_1}|^2} \in S_ \mathrm{H}$ whenever $f\in \mathrm{S}_ \mathrm{H}.$ Due to Clunie and Sheil-Small ^[1] we also let

Further, let $\mathrm{K}_ \mathrm{H} ^0$, $\mathrm{ST}_ \mathrm{H} ^{0}$ and $\mathrm{C}_\mathrm{H}^0$ are the subclasses of $\mathrm{S}_\mathrm{H}^0$ which are harmonic convex, starlike and close-to-convex in $\mathbb{D}$ respectively. Also $\mathrm{T}^0_\mathrm{H},$ denotes typically real harmonic functions for further details refer ^[1,2,3,4].

We recall following lemmas proved in ^[1,2].

Lemma 1. (^[1,2]). If $f\in K_\mathrm{H}^0$ is assumed as in $(1.2)$ with $\mathrm{B}_1 = 0$, then

Lemma 2. ^[1,2] If $f \in \mathrm{ST}_\mathrm{H}^{0}$ (or $f\in \mathrm{C}_\mathrm{H}^{0}$) is assumed as in $(1.2)$ with $\mathrm{B}_1 = 0$, then

Lately, for $f \in \mathcal{S}_H$ whose members are given by (1.2), Dziok^[5] introduced a new class $\mathcal{S}^*_ \mathrm{H}(F, G)$ $(-G \leq F < G \leq 1)$ by

where $\mathfrak{D}_H f(z) = zh'(z)-\overline{zg'(z)}\quad \rm{and} \quad z\in \mathbb{D}.$ Also let

In particular $\mathcal{R}_\mathrm{H}(F, G)$, the class discussed by Dziok ^[6] is given by

Remark 1. We note that

$1).$ $\mathcal{S}^*(F, G) : = \mathcal{S}^*_H (F, G) \cap\mathcal{ A}$ was introduced by Janowski ^[7].

$2).$ For $0\leq\vartheta < 1$ the classes $\mathcal{S}^*_H(\vartheta) : = \mathcal{S}^*_ H(2\vartheta-1, 1)$ and $\mathcal{K}_ H(\vartheta) : = \mathcal{K}_H(2\vartheta-1, 1)$ were investigated by Jahangiri ^[8,9].

$3).$ Also $\mathcal{S}^*_ H : = \mathcal{S}^*_ H(0)$ and $\mathcal{K}_ H : = \mathcal{K}_ H(0)$ are the classes harmonic starlike and convex functions in $\mathbb{D}$ and $\mathcal{S}^*_ H(F, G) \subset \mathcal{S}^*_ H,$ $\mathcal{K}_H(F, G) \subset\mathcal{K}_ H.$

Lately, Dizok ^[5] gave the following coefficient conditions:

Lemma 3. ^[5] Let $f \in\mathrm{H}$ be assumed as in $(1.2)$, then $f \in\mathcal{S}^{*}_\mathrm{H}(F, G)$ if

where $\mathrm{A}_1 = 1; \mathrm{B}_1 = 0$ and $-G \leq F < G \leq 1.$

Lemma 4. ^[5] Let $f \in \mathcal{T}_\mathrm{H}$ be assumed as in $(1.3)$ and

where $\mathrm{A}_1 = 1; \mathrm{B}_1 = 0$ and $-G \leq F < G \leq 1.$

Remark 2. If $f\in \mathcal{S}^{*}_\mathrm{H}(F, G)$, then

Lemma 5. ^[5] Let $f \in \mathcal{T}_\mathrm{H}$ be assumed as in $(1.3)$ and

where $\mathrm{A}_1 = 1; \mathrm{B}_1 = 0$ and $-G \leq F < G \leq 1.$

Lemma 6. ^[6] Let $f \in \mathrm{H}$ be assumed as in $(1.2)$ then $f \in\mathcal{R}_\mathrm{H}(F, G)$ if

where $\mathrm{A}_1 = 1; \mathrm{B}_1 = 0$ and $-G \leq F < G \leq 1.$

Lemma 7. Let $f \in \mathcal{T}_\mathrm{H}$ be assumed as in $(1.3)$ and

where $\mathrm{A}_1 = 1; \mathrm{B}_1 = 0$ and $-G \leq F < G \leq 1.$

Remark 3. If $f\in \mathcal{R}_\mathrm{H}(F, G)$, then $|\mathrm{A}_j|\le \frac{G-F}{j(1+G)}$ and $|\mathrm{B}_j|\le \frac{G-F}{j(1+G)}, \; \; k\ge 2.$

Mittag-Leffler ^[10] defined a function $\mathbf{E}_{\alpha }(z)$ given by

was commonly known as the Mittag-Leffler function. Wiman ^[11] defined a more general function $\mathbf{E}_{\alpha, \beta }$ generalizing $\mathbf{E}_{\alpha }(z)$ was given by

Perceive that the function $\mathbf{E}_{\alpha, \beta }$ comprises many well-known functions as its exceptional case, for example, $\mathbf{E}_{0, 0 }(z) = \sum_{j = 0}^{\infty } z^j,$ $\mathbf{E}_{1, 1}(z) = e^{z},$ $\mathbf{E}_{1, 2}(z) = \frac{e^{z}-1}{z},$ $\mathbf{E}_{2, 1}(z^{2}) = \cosh z$, $\mathbf{E}_{2, 1}(-z^{2}) = \cos z,$ $\mathbf{E}_{2, 2}(z^{2}) = \frac{\sinh z}{z}$, $\mathbf{E}_{2, 2}(-z^{2}) = \frac{\sin z}{z},$ $\mathbf{E}_{4}(z) = \frac{1}{2}[\cos z^{1/4}+\cosh z^{1/4}]\ $ and $\mathbf{E}_{3}(z) = \frac{1}{2} [e^{z^{1/3}}+2e^{-\frac{1}{2}z^{1/3}}\cos (\frac{\sqrt{3}}{2}z^{1/3})].$ It is of interest to note that by fixing $\alpha = 1/2$ and $\beta = 1$ we get

The Mittag-Leffler function ascends logically in the elucidation of fractional order differential and integral equations, and primarily in the investigations of fractional generalization of kinetic equation, random walks, Lévy flights, super-diffusive transport and in the study of complex systems. Numerous properties of Mittag-Leffler function and generalized Mittag-Leffler function can be found e.g., in ^{[12,13,14,15]}. We prompt that Mittag-Leffler function $\mathbf{E}_{\alpha, \beta }(z)$ is not a member of $\mathcal{A}.$ Consequently, it is probable to cogitate the following normalization of $\mathbf{E}_{\alpha, \beta}(z)$ as below due to Bansal and Prajapat ^[13] :

it grasps for complex parameters $\; z\in\mathbb{C}, \; \alpha, \beta\in\mathbb{C}, \; \rm{with}\; \operatorname{Re}\alpha > 0, \; \operatorname{Re}\beta > 0.$

In our present study, we shall confine our attention to the case of real-valued $\alpha, \beta \rm{ and}\; z \in \mathbb{D}$ and hence define a new linear operator based on convolution (or hadamard) product as below:

For real parameters $\alpha, \beta, \gamma, \delta\; \; (\alpha, \beta, \gamma, \delta\notin\left\{0, -1, -2, \dots\right\})$, we let

and define the convolution operator $\Lambda\left(f \right)$ by

Inclusion relations between different subclasses of analytic and univalent functions by using hypergeometric functions (see for example, ^{[16,17,18,19,20,21]}) and by the recent investigations related with distribution series (see for example, ^[22,23,24] and references cited there in) were exclusively studied in the literature. Lately, there has been triggering interest to investigate mapping properties and inclusion results for the families of harmonic univalent functions including various linear and nonlinear operators, (see ^{[25,26,27,28]}) and references cited there in) and also by Bessel functions studied by Porwal ^[29]. Motivated by aforementioned works and recent study in ^[30], in our current paper, we launch connections among the classes $\mathrm{K}^{0}_\mathrm{H}$, $\mathrm{S}^{*, 0}_\mathrm{H}$, $\mathrm{C}^{0}_\mathrm{H},$ $\mathcal{S}_\mathrm{H}(F, G)$, and $\mathcal{R}_\mathrm{H}(F, G)$ by applying the convolution operator $\Lambda$ related with Mittag-Leffler function.

2.   Mapping properties of the class $\mathrm{H},$ related with Mittag-Leffler function

In order to establish our main results, throughout this paper we let the following:

and

Now by using Lemma 1, we establish connection between harmonic convex functions and $\mathcal{S}_\mathrm{H}(F, G)$:

Theorem 2.1. Let $\alpha, \beta, \gamma, \delta, \; (\alpha, \beta, \gamma, \delta \notin\left\{0, -1, -2, \dots\right\})$ are real. If for some $-G \leq F < G \leq 1$ and the inequality

is satisfied then $\Lambda\left(\mathrm{K}^{0}_\mathrm{H}\right)\subset \mathcal{S}^*_\mathrm{H}(F, G).$

Proof. Let $f\in \mathrm{K}_\mathrm{H}^0$ be as assumed in (1.2) with $\mathrm{B}_1 = 0.$ We requisite to show that $\Lambda \left(f \right) = \mathcal{F}(z)\in \mathcal{S}^*_\mathrm{H}(F, G)$, which is given by (1.17) with $\mathrm{B}_1 = 0.$ In view of Lemma 3, we need to assert that

In view of Lemma 1, we have $\mathrm{A}_j = \frac{j+1}{2}$ and $\mathrm{B}_j = \frac{j-1}{2},$ thus

Expressing ${j^2} = {j(j-1)}+{j}$, we get

Now by using (2.1)–(2.3), we get

but $\Psi_1$ is bounded above by $G-F,$ if and only if (2.5) holds.

As in Theorem 2.1, using Lemma 2, we determine the analogous results for the classes $\mathrm{ST}^{\ast, 0}_\mathrm{H}$, $\mathrm{C}^{0}_\mathrm{H}$ with $\mathcal{S}^*_\mathrm{H}(F, G).$

Theorem 2.2. Let $\alpha, \beta, \gamma, \delta, \; (\alpha, \beta, \gamma, \delta \notin\left\{0, -1, -2, \dots\right\})$ are real. If for some $-G \leq F < G \leq 1$ and the inequality

is satisfied, then $\Lambda(\mathrm{S}^{\ast, 0}_\mathrm{H})\subset \mathcal{S}^*_\mathrm{H}(F, G)\; \mathit{\rm{and}}\; \Lambda(\mathrm{C}^{0}_\mathrm{H})\subset \mathcal{S}^*_\mathrm{H}(F, G).$

Proof. Let $f\in S^{\ast, 0}_\mathrm{H}$ $(or\; C^{0}_\mathrm{H})$ be as assumed in (1.2) with $\mathrm{B}_1 = 0.$ It is adequate to verify that $\Lambda \left(f \right) = \mathcal{F} (z)\in \mathcal{S}^*_\mathrm{H}(F, G).$ In sight of Theorem 3, it is profuse to show that

By Lemma 2, we have $\mathrm{A}_j = \frac{(2j+1)(j+1)}{6}$ and $\mathrm{B}_j = \frac{(2j-1)(j-1)}{6},$ thus

Writing ${j^3} = j(j-1)(j-2)+3j(j-1)+j$ and ${j^2} = {j(j-1)}+{j}$, we have

Now by using (2.1)–(2.4), we get

But $\Psi_2$ confined above by $G-F,$ if and only if (2.6) holds.

Theorem 2.3. Let $\alpha, \beta, \gamma, \delta, \; (\alpha, \beta, \gamma, \delta \notin\left\{0, -1, -2, \dots\right\})$ are real. If for some $-\mathbf{G} \leq \mathbf{F} < \mathbf{G} \leq 1$ and $-G \leq F < G \leq 1$ the inequality

is satisfied then $\Lambda\left(\mathcal{RT}_\mathrm{H}(\mathbf{G}, \mathbf{F})\right)\subset \mathcal{S}^*_\mathrm{H}(F, G).$

Proof. Let $f\in \mathcal{RT}_\mathrm{H}(\mathbf{G}, \mathbf{F})$ be as assumed in (1.3). By virtue of Lemma 3, it is enough to show that $\Psi_3\le G-F$, where

By Remark 3, we have

and $\Psi_3$ is bounded above by $G-F,$ if and only if (2.7) holds.

In next theorem, we establish association between $\Lambda(\mathcal{S}^*_\mathrm{H}(F, G))$ and $\mathcal{S}^*_\mathrm{H}(F, G)$.

Theorem 2.4. Let $\alpha, \beta, \gamma, \delta, \; (\alpha, \beta, \gamma, \delta \notin\left\{0, -1, -2, \dots\right\})$ are real. If for some $-G \leq F < G \leq 1$ the inequality

is satisfied, then $\Lambda \left(\mathcal{S}^*_\mathrm{H}(F, G)\right)\subset \mathcal{S}^*_\mathrm{H}(F, G).$

Proof. In sight of Lemma 3, it is profuse to show that

By Remark 2, it follows that

But $\Psi_4$ is bounded above by $G-F,$ if and only if (2.8) holds.

Now we attain a depiction for $\Lambda$ which maps $\mathcal{ST}_\mathrm{H}(F, G)$ on to itself.

Theorem 2.5. Let $\alpha, \beta, \gamma, \delta, \; (\alpha, \beta, \gamma, \delta \notin\left\{0, -1, -2, \dots\right\})$ are real and $-G \leq F < G \leq 1$. Then

if and only if,

Proof. The proof follows in lines similar to the proof of Theorem 2.4, so we overlook the details.

3.   Conclusions

In this investigation we obtained sufficient conditions and inclusion results for functions $f\in\mathcal{A}$ to be in the classes $\mathcal{S}^*_\mathrm{H}(F, G)$ and information regarding the images of functions by applying convolution operator with Mittag-Leffler function. By specializing the parameter $G = 1$ and $F = 2\varrho-1(0\leq\varrho < 1)$ we can easily derive the inclusion results and mapping properties for the function classes $f\in\mathcal{S}_\mathrm{H}(\varrho)$ and $f\in\mathcal{R}_\mathrm{H}(\varrho)$ in association with Mittag-Leffler function. By using the Alexander theorem $f\in\mathcal{K}_H \Leftrightarrow zf'\in \mathcal{S}_H$ we can define $\mathcal{K}_\mathrm{H}(F, G)$ whose members are given by (1.2), satisfying the condition

By using the result given in Lemma 5, one can easily discuss the above results for $f\in\mathcal{K}_\mathrm{H}(F, G)$ on lines similar to above theorems, we left this as exercise for interested readers.

Acknowledgments

The authors would like to acknowledge the financial support of Taif University Researchers Supporting Project number (TURSP-2020/162), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declares no conflicts of interest in this paper.