Applications of a differential operator to a class of harmonic mappings defined by Mittag-leffer functions

1.   Introduction

1.1.   Background of the research

The study of analytic functions has been the core interest of various prominent researchers in the last decade. Much emphasis has been on the aspect of introduction of various concepts in this field. Uralegaddi ^[1], in 1994, introduced the subclasses of starlike, convex and close-to-convex functions with positive coefficients and opened a new side of Geometric Function Theory. Motivated by his work Dixit and Chandra ^[2] introduced new subclass of analytic functions with positive coefficients. Continuing the trend Dixit et al. ^[3], Porwal and Dixit ^[4] and Porwal et al. ^[5] made a substantial amount of important theory which illuminated various new directions of this field. One such area is Harmonic Analysis which has vastly influenced and nurtured the branch of Geometric Function Theory. Dixit and Porwal ^[6] defined and investigated the class of harmonic univalent functions with positive coefficients. With the introduction of this work many mathematicians generalized various important results with the help of some operators, the work of Pathak et al. ^[7], Porwal and Aouf ^[8] and Porwal et al. ^[9] are worth mentioning here. More recently new subclasses of harmonic starlike and convex functions are introduced and studied by Porwal and Dixit ^[10], see also ^[5].

Recently attention has been drawn to Mittag-Leffer functions as these functions can be widely applied across the fields of engineering, chemical, biological, physical sciences as will as in various other applied sciences. Various factors in applying such functions are evident within chaotic, stochastic, dynamic systems, fractional differential equations and distribution of statistics. The geometric characteristics such as convexity, close-to-convexity and starlikeness of the functions investigated here has been broadly examined by many authors. Direct applications of these functions can be seen in a number of fractional calculus tools which includes significant work by ^{[11,12,13,14,15,16,17,18,19,20]}.

1.2.   Preliminaries

Before we go into details about our new work we give some basics which will be helpful in understanding the concepts of this research.

A real-valued function $u\left(x, y\right)$ is said to be harmonic in a domain $\mathbb{D}\subset \mathbb{C}$ if it has continuous second partial derivative and satisfy the Laplace's equation

and complex-valued function $f = u+iv$ is said to be harmonic in a domain $\mathbb{D}$ if and only if $u$ and $v$ are both real harmonic functions in domain $\mathbb{D}.$ Every complex-valued harmonic function $f$ which is harmonic in $\mathbb{D}$, containing the origin, can be represented in the canonical form as

where $h$ and $g$ are analytic functions in $\mathbb{D}$ with $g\left(0\right) = 0.$Then functions $h$ and $g$ are known as analytic and co-analytic parts of $f$ respectively. The Jacobian of $f = u+iv$ is given by

which can be represented in terms of derivatives with respect to $z$ and $\overline{z}$ as

It can be noted that if $f$ is analytic in $\mathbb{D},$ then $f_{\overline{z }} = 0$ and $f_{z}\left(z\right) = f^{^{\prime }}\left(z\right).$ A well- known result for analytic functions state that an analytic function $f$ is locally univalent at a point $z_{0}$ if and only if $J_{f}\left(z\right) \neq 0$ in $\mathbb{D}$ (see for example ^[21]). In ^[22], Lewy proved the converse of this theorem which is also true for harmonic mappings. Therefore, $f$ is sense-preserving and locally univalent if and only if

Let $\mathcal{H}$ denote the class of functions $f$ which are harmonic in the unit disc $\mathfrak{A}: = \mathfrak{A}\left(1\right)$, where $\mathfrak{ A}\left(r\right) : = \left \{ z\in \mathbb{C} :\left \vert z\right \vert < r\right \}.$ Also, let $\mathcal{H}_{0}$ denote the class of functions $f\in \mathcal{H}$ which satisfy the normalization conditions

Therefore the analytic functions $h$ and $g$ given by $\left(1.1 \right)$ can be written in the form

and

Let

It is clear that $\mathcal{S}_{\mathcal{H}}$ reduces to the class $\mathcal{S },$ and by $\mathcal{A}$ whenever the co-analytic part of $f$ vanishes, i.e., $g\left(z\right) = 0$ in $\mathfrak{A}$. Clunie and Sheil-Small ^[23] and Sheil-Small ^[24] studied $\mathcal{S}_{\mathcal{H}}$ together with some of its geometric subclasses. We say that a function $f\in \mathcal{ H}_{0}$ is said to be harmonic starlike in $\mathfrak{A}$ if it satisfy

where

A function $f\left(z\right)$ is subordinated to a function $g\left(z\right)$ denoted by $f\left(z\right) \prec g\left(z\right)$, if there is complex-valued function $w(z)$ with $\left \vert w(z)\right \vert \leq 1$ and $g\left(0\right) = 0$ such that

Also, if $g\left(z\right)$ is univalent in $\mathfrak{A}$, we have equivalence condition

Convolution or Hadamard product of two function $f_{1}$ and $f_{2}$ is denoted by $f_{1}$ $\ast$ $f_{2}$ and is defined by

In 1973, Janowski ^[25] introduced the idea of circular domain by introducing Janowski functions as;

A function $k\left(z\right),$ analytic in $\mathfrak{A}$ with $k\left(0\right) = 1,$ is said to be in class $T\left[A, B\right]$ if for $-1\leq B < A\leq 1$

Janowski showed that the function $k$ maps $\mathfrak{A}$ onto the domain $\Delta \left(A, B\right)$ with centre on real axis and $D_{1} = \frac{1-A}{1-B }$ and $D_{2} = \frac{1+A}{1+B}$ are diameter end points with $0 < D_{1} < 1 < D_{2}.$

The Mittag-Leffer function is defined as

The initial two parametric generalizations for the function shown in $\left(1.5\right)$ were given by Wiman ^[26,27]. It is defined in the following way

where $\alpha, \beta \in \mathbb{C},$ $\text{Re}\left(\alpha \right) > 0$, $\text{Re}\left(\beta \right) > 0$ and $\Gamma \left(z\right)$ is gamma function.

Now the function $\text{Q}_{\alpha, \beta }$ is defined by

Using the function $\text{Q}_{\alpha, \beta }$ Elhaddad et al. ^[28] defined the differential operator for the class of analytic functions as $\mathcal{D}_{\delta }^{m}\left(\alpha, \beta \right) :\mathcal{ A\rightarrow A}$ as illustrated below :

where $m = \mathbb{N} _{0} = \left \{ 0, 1, 2, ...\right \},$ $\delta > 0.$

Where the operator $\mathcal{D}_{\delta }^{m}\left(\alpha, \beta \right)$ for a function $f\in \mathcal{H}$ given by $\left(1.1\right)$ can be defined as below:

where

for $m\in \mathbb{N} _{0}.$

Motivated by ^[29,30] and using the operator $\mathcal{D}_{\delta }^{m}\left(\alpha, \beta \right) f\left(z\right),$ we introduced the class of harmonic univalent functions as:

Definition Let $-B\leq A < B\leq 1,$ $0\leq a < 1\ $ and $\mathcal{S}_{\mathcal{H} }^{\alpha, \beta }\left(m, \delta, A, B\right)$ denote the class of functions $f\in \mathcal{S}_{\mathcal{H}}$ such that

with

Note that,

1. $\mathcal{S}_{\mathcal{H}}^{0, 1}\left(0, \delta, A, B\right) = \mathcal{S }_{\mathcal{H}}^{\ast }\left(A, B\right),$ which was studied by Deziok ^[29].

2. $\mathcal{S}_{\mathcal{H}}^{0, 1}\left(0, \delta, 2a-1, 1\right) = \mathcal{S}_{\mathcal{H}}^{\ast }\left(a\right),$ defined by Jahangiri in ^[31]

3. $\mathcal{S}_{\mathcal{H}}^{0, 1}\left(1, 1, 2a-1, 1\right) = \mathcal{S}_{ \mathcal{H}}^{c}\left(a\right),$ introduced by Jahangiri, see ^[32] for details.

Let $\mathcal{V}\subset \mathcal{H}_{0},$ $\mathfrak{A}_{0} = \mathfrak{ A\diagdown }\left \{ 0\right \}.$ Using Ruscheweyh's approach in ^[33] we define the dual set of $\mathcal{V}$ by

2.   Main criteria

In this section we prove some important results beginning with necessary and sufficient condition. Then some inequality regarding the coefficients of the functions in their series form are evaluated along with examples for justifications.

2.1.   Theorem

Let $f\in \mathcal{H}_{0}$ and is given by $\left(1.3 \right)$ is in the class $\mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right)$ if and only if

where

Proof. Let $f\in \mathcal{H}_{0}$, then $f\in \mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right)$ if and only if the following holds

Now as

and

thus

Thus $f\in \mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right)$ if and only if $f\left(z\right) \ast \mathcal{D}_{\delta }^{m}\left(\alpha, \beta \right) \varphi _{\xi }\left(z\right) \neq 0$ for $z\in \mathfrak{A}_{0},$ $\left \vert \xi \right \vert = 1$ i.e. $\mathcal{S} _{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right) = \left \{ \mathcal{D}_{\delta }^{m}\left(\alpha, \beta \right) \varphi _{\xi }\left(z\right); \text{ }\left \vert \xi \right \vert = 1\right \} ^{\ast }$.

A sufficient coefficient bound for the class $\mathcal{S}_{\mathcal{H} }^{\alpha, \beta }\left(m, \delta, A, B\right)$ is provided in the following.

2.2.   Theorem

Let $f\in \mathcal{H}_{0}$ be of the form $\left(1.3 \right)$ and satisfies the condition

with

then $f\in \mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right).$

Proof. Obviously the theorem is true for $f\left(z\right) = z$. Suppose $f\in \mathcal{H}_{0}$ given by $\left(1.3\right)$ and let there exist $n\geq 2$ such that $a_{n}\neq 0$ or $b_{n}\neq 0$. Since

by $\left(2.1\right)$ we have

and

Therefore$\left \vert h^{^{\prime }}\left(z\right) \right \vert > \left \vert g^{^{\prime }}\left(z\right) \right \vert$ which shows that $f$ is locally univalent and sense-preserving in $\mathfrak{A}$. Moreover if$z_{1}, z_{2}\in \mathfrak{A}$ and $z_{1}\neq z_{2}$ then

Hence by $\left(2.4\right)$ we have

This shows that $f$ is univalent, i.e. $f\in \mathcal{S}_{\mathcal{H}}$. Therefore $f\in \mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right)$ if and only if there exists a complex-valued function $\omega$, $\omega \left(0\right) = 0,$ $\left \vert \omega \left(z\right) \right \vert < 1$ $\left(z\in \mathfrak{A}\right),$ such that

or equivalently

Thus, it is sufficient to prove that

where $z\in \mathfrak{A}\backslash \left \{ 0\right \},$ now by putting $\left \vert z\right \vert = r,$ $\ r\in \left(0, 1\right)$ we get

hence $f\in \mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right).$

2.3.   Example

For function

such that $\sum_{n = 2}^{\infty }\left(\left \vert p_{n}\right \vert +\left \vert q_{n}\right \vert \right) = 1,$ we have

Thus $f\in \mathcal{S}_{\mathcal{H}}^{\alpha, \beta }\left(m, \delta, A, B\right)$ and above inequality $\left(2.1\right)$ is sharp for this function.

Motivated from Silverman ^[34], we introduce the class $\mathfrak{\tau }$ for functions $f\in \mathcal{H}_{0}$ of the form $\left(1.3\right)$ such that $a_{n} = -\left \vert a_{n}\right \vert,$ $b_{n} = \left \vert b_{n}\right \vert$ $\left(n = 2, 3, ...\right),$ i.e.

Further, let us define

Where $\alpha = 0,$ $\beta = 1$ and $m = 0$ the class is studied by Dziok see ^[29].

2.4.   Theorem

Let $f\in \mathfrak{\tau }$ and of the form $\left(2.6 \right).$ Then $f$ $\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ if and only if condition $\left(2.1 \right)$ holds true.

Proof. In Theorem 2.2 we need only to show that each function $f$ $\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ satisfies coefficient inequality $\left(2.1\right).$ If $f$ $\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ then it is of the form $\left(2.6\right)$ and satisfies $\left(2.5\right)$ or equivalently

where $z\in \mathfrak{A, }$ therefore by putting $z = r$ $,$ $r\in \left[0, 1\right)$, we get

It is clear that the denominator of the left hand side cannot vanishes for $r\in \left(0, 1\right)$. Moreover, it is positive for $r = 0$, and in consequence for $r\in \left(0, 1\right)$. Thus, by $\left(2.7 \right)$ we have

The sequence of partial sums $\left \{ S_{n}\right \}$ associated with the series $\sum_{n = 2}^{\infty }\left(\lambda _{n}\left \vert a_{n}\right \vert +\sigma _{n}\left \vert b_{n}\right \vert \right)$ is a non-decreasing sequence. Moreover, by $\left(2.8\right)$ it is bounded by $B-A$. Hence, the sequence $\left \{ S_{n}\right \}$ is convergent and

which yields assertion $\left(2.1\right).$

2.5.   Example

For function

such that $\sum_{n = 2}^{\infty }\left(\left \vert c_{n}\right \vert +\left \vert d_{n}\right \vert \right) = 1,$ we have

Thus $f\in \mathcal{S}_{\mathcal{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$.

3.   Topological properties

we consider the usual topology on $\mathcal{H}$ in which a sequence $\left \{ f_{n}\right \}$ in $\mathcal{H}$ converges to $f$ if and only if it converges to $f$ uniformly on each compact subset of $\mathfrak{A}$. The metric induces the usual topology on $\mathcal{H}.$ It is to verify that the obtained topological space is complete.

Let $\mathcal{F}$ be a subclass of the class $\mathcal{H}$. A function $f\in \mathcal{F}$ is called an extreme point of $\mathcal{F}$ if the condition

implies $f_{1} = f_{2} = f$. We shall use the notation $E\mathcal{F}$ to denote the set of all extreme points of $\mathcal{F}$. It is clear that $E\mathcal{F }\subset \mathcal{F}$.

We say that $\mathcal{F}$ is locally uniformly bounded if for each $r$, $0 < r < 1$, there is a real constant $M = M\left(r\right)$ so that

We say that a class $\mathcal{F}$ is convex if

Moreover, we define the closed convex hull of $\mathcal{F}$ as the intersection of all closed convex subsets of $\mathcal{H}$ that contain $\mathcal{F}.$ We denote the closed convex hull of $\mathcal{F}$ by $\overline{co}\mathcal{F}$.

A real-valued function $\mathfrak{J}:\mathcal{H}\rightarrow \mathbb{R}$ is called convex on a convex class $\mathcal{F}\subset \mathcal{H}$ if

The Krein-Milman theorem (see ^[35]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.

3.1.   Lemma

Let $\mathcal{F}$ be a non-empty convex compact subclass of the class $\mathcal{H}$ and let $\mathfrak{J}:\mathcal{H}\rightarrow \mathbb{R}$ be a real-valued, continuous and convex function on $\mathcal{F}$. Then

3.2.   Lemma

A class $\mathcal{F}\subset \mathcal{H}$ is compact if and only if $\mathcal{F}$ is closed and locally uniformly bounded.

Since $\mathcal{H}$ is complete metric space, Montel's theorem (see ^[36]) implies the following lemma.

3.3.   Lemma

Let $\mathcal{F}$ be a non-empty compact subclass of the class $\mathcal{H}$, then $E\mathcal{F}$ is non-empty and $\overline{co}E\mathcal{F} = \overline{co}\mathcal{F}$.

3.4.   Theorem

The class $\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ is a convex and compact subset of $\mathcal{H}.$

Proof. Let $f_{l}\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ be a functions of the form

and $0\leq \gamma \leq 1.$ Since

and by Theorem 2.4, we have

the function $\Psi = \gamma f_{1}+\left(1-\gamma \right) f_{2}\in \mathcal{S} _{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$. Hence, the class is convex. Furthermore, for $f\in \mathcal{S}_{\mathfrak{\tau } }^{\alpha, \beta }\left(m, \delta, A, B\right)$, $\left \vert z\right \vert \leq r$ $r\in \left(0, 1\right)$, we have

Thus, we conclude that the class $\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ is locally uniformly bounded. By Lemma 3.2, we need only to show that it is closed, i.e. if $f_{l}\rightarrow f$, then $f\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$. Let $f_{l}$ and $f$ be given by $\left(3.1 \right)$ and $\left(2.6\right)$, respectively. Using Theorem 2.4, we have

Since $f_{i}\rightarrow f$, we conclude that $\ \left \vert a_{i, n}\right \vert \rightarrow \left \vert a_{n}\right \vert$ and $\left \vert b_{i, n}\right \vert \rightarrow \left \vert b_{n}\right \vert$ as $i\rightarrow \infty$ $\left(n\in \mathbb{N} \right).$ The sequence of partial sums $\left \{ S_{n}\right \}$ associated with the series $\sum_{n = 2}^{\infty }\left(\alpha _{n}\left \vert a_{i, n}\right \vert +\beta _{n}\left \vert b_{i, n}\right \vert \right)$ is non-decreasing sequence. Moreover, by $\left(3.3\right)$ it is bounded by $B-A$. Therefore, the sequence $\left \{ S_{n}\right \}$ is convergent and

This gives condition $\left(2.1\right)$ and in consequence, $f\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$, which complete the proof.

3.5.   Theorem

We have

where

Proof. Suppose that $0 < \gamma < 1$ and

where $f_{1}$, $f_{2}\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ are functions of the form $\left(3.1 \right)$. Then, by $\left(2.1\right)$ we have

and, in consequence, $a_{1, i} = a_{2, i} = 0$ for $i\in \left \{ 2, 3, ...\right \}$ and $b_{1, i} = b_{2, i} = 0$ for $i\in \left \{ 2, 3, ...\right \} \backslash \left \{ n\right \}.$ It follows that $g_{n}^{\ast } = f_{1} = f_{2}$, and consequently $g_{n}^{\ast }\in E\mathcal{S}_{\mathfrak{ \tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$. Similarly, we verify that the functions $h_{n}^{\ast }$ of the form $\left(3.4\right)$ are extreme points of the class $\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$. Now, suppose that $f\in E\mathcal{S}_{ \mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ and $f$ is not of the form $\left(3.4\right)$. Then there exists $i\in \left \{ 2, 3, ...\right \}$ such that

If $0 < \left \vert a_{i}\right \vert < \frac{B-A}{\lambda _{i}},$ then putting

we have that $0 < \gamma < 1$, $h_{i}^{\ast }$, $\Phi \in$ $\mathcal{S}_{ \mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right),$ $h_{i}^{\ast }\neq \Phi$ and

Thus, $f\notin E\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$. Similarly, if $0 < \left \vert b_{i}\right \vert < \frac{B-A}{\sigma _{i}},$ then putting

we have that $0 < \gamma < 1$, $g_{i}^{\ast }$, $\Phi \in \mathcal{S}_{ \mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right),$ $g_{i}^{\ast }\neq \Psi$ and

It follows that $f\notin E\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right),$ and this completes the proof.

4.   Radii of starlikeness and convexity

A function $f\in \mathcal{H}_{0}$ is said to be starlike of order $\alpha$ in $\mathfrak{A}\left(r\right)$ if

Also, A function $f\in \mathcal{H}_{0}$ is said to be convex of order $\alpha$ in $\mathfrak{A}\left(r\right)$ if

It easy to verify that for function $f\in \mathcal{\tau }$ the condition $\left(4.1\right)$ is equivalent to the following

or equivalently

Let $\mathcal{B}$ be a subclass of the class $\mathcal{H}_{0}.$ We define the radius of starlikeness and convexity

In simple word these show the subregion of the open unit disc where the functions would behave starlike and convex of order $\alpha.$

4.1.   Theorem

The radii of starlikeness of order $\alpha$ for the class $\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ is given by

where $\lambda _{n}$ and $\sigma _{n}$ are define in $\left(2.2 \right)$ and $\left(2.3\right)$ respectively.

Proof. Let $f\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ be of the form $\left(2.6\right).$ Then, for $\left \vert z\right \vert = r < 1$ we have

Thus the condition $\left(4.2\right)$ is true if and only if

By Theorem 2.2, we have

where $\lambda _{n}$ and $\sigma _{n}$ are defined by $\left(2.2 \right)$ and $\left(2.3\right)$ respectively. Thus the conditions $\left(4.4\right)$ is true if

i.e.,

It follows that the function $f$ is starlike of order $\alpha$ in the disc $\mathfrak{A}\left(r^{\ast }\right),$ where $r^{\ast }$

The functions $h_{n}^{\ast }$ and $g_{n}^{\ast }$ are define by $\left(3.4\right)$ realize equality in $\left(4.5\right),$ and the radius $r^{\ast }$ cannot be larger, thus we have $\left(4.3 \right).$

The following theorem may be proved in much same fashion as Theorem 4.1..

4.2.   Theorem

The radii of convexity of order $\alpha$ for the class $\mathcal{S}_{ \mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ is given by

where $\lambda _{n}$ and $\sigma _{n}$ are define in $\left(2.2 \right)$ and $\left(2.3\right)$ respectively.

5.   Applications

In this section we give some applications of the work discussed in this article in the form of some results and examples. It is clear that if the class

is locally uniformly bounded, then

5.1.   Corollary

where $h_{n}$ and $g_{n}$ are defined by Eq $\left(3.4 \right).$

Proof. By Theorem 3.4 and Lemma 3.3 we have

Thus, by Theorem 3.5 and by $\left(5.1\right)$ we have Eq $\left(5.2\right).$

We observe, that for each $n\in \mathbb{N}$, $z\in \mathfrak{A}$, the following real-valued functionals are continuous and convex on $\mathcal{H}:$

and

Therefore, using Lemma 3.1 and Theorem 3.5 we have the following corollaries.

5.2.   Corollary

Let $f\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$ be a function of the form $\left(2.6\right).$ Then

where $\lambda _{n}$ and $\sigma _{n}$ are defined by $\left(2.2 \right)$ and $\left(2.3\right)$ respectively. The result is sharp. The function $h_{n}^{\ast }$ and $g_{n}^{\ast }$ of the form $\left(3.4\right)$ are extremal functions.

Proof. Since for the extremal functions $h_{n}^{\ast }$ and $g_{n}^{\ast }$ we have $\left \vert a_{n}\right \vert = \frac{B-A}{\lambda _{n}}$ and $\left \vert b_{n}\right \vert = \frac{B-A}{\sigma _{n}}.$ Thus, by Lemma 3.1 we have Eq $\left(5.3\right).$

5.3.   Example

Since $\frac{B-A+2}{\lambda _{2}} > \frac{B-A}{\lambda _{2}}$ the polynomial

by Corollary 5.2, clearly $k\left(z\right)$ does not belong to $\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right).$

5.4.   Corollary

Let $f\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right)$, $\left \vert z\right \vert = r < 1.$ Then

Due to Littlewood ^[37] we consider the integral means inequalities for functions from the class $\mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right).$

5.5.   Lemma

Let $f, g\in \mathcal{A}.$ If $f\prec g,$ then

5.6.   Theorem

Let $0 < r < 1,$ $\gamma > 0.$ Then

and

where $h_{n}^{\ast }$ and $g_{n}^{\ast }$is defined by Eq $\left(3.4\right).$

Proof. Let $h_{n}^{\ast }$ and $g_{n}^{\ast }$ are define by Eq $\left(3.4\right)$ and let $\widetilde{g_{n}}\left(z\right) = z+\frac{B-A}{ \sigma _{n}}z^{n}$ $\left(n = 2, 3, ...\right).$ Since $\frac{h_{n}^{\ast }}{z} \prec \frac{h_{2}^{\ast }}{z}$ and $\frac{\widetilde{g_{n}}}{z}\prec \frac{ h_{2}^{\ast }}{z},$ by Lemma 5.5 we have

and

which complete the proof.

5.7.   Corollary

If $f\in \mathcal{S}_{\mathfrak{\tau }}^{\alpha, \beta }\left(m, \delta, A, B\right) \,$ then

and

where $\gamma \geq 1,$ $0 < r < 1$ and $h_{2}^{\ast }$ is the function defined by Eq $\left(3.4\right).$

6.   Conclusions

With the use of Mittag-Leffer functions, we introduced a new subclass of harmonic mappings in Janowski domain. We studied some useful results, like necessary and sufficient conditions, coefficient inequality, topological properties, radii problems, distortion bounds and integral mean of inequality for newly defined classes of functions. It can be seen that our defined class not only generalizes various well known classes and their respective results but also give new direction to this field by the introduction of Mittag-Leffer functions here. Further using the concepts of Mittag-Leffer functions these problems can be studied for classes of meromorphic harmonic functions, Bazilevi'c harmonic functions and for p-valent harmonic functions as well.

Conflict of interest

The authors declare that they have no competing interests.