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

1 Department of Mathematics, Abdul Wali Khan University Mardan, 23200 Mardan, Pakistan 2 Department of Mathematics, Govt. Degree College Mardan, 23200 Mardan, Pakistan 3 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 4 Department of Medical research, China Medical University Taiwan, Taichung, Taiwan 5 Department of Computer Science and information Engineering, Asia University, Taichung, Taiwan


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].

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 (x, y) is said to be harmonic in a domain D ⊂ 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 D if and only if u and v are both real harmonic functions in domain D. Every complex-valued harmonic function f which is harmonic in D, containing the origin, can be represented in the canonical form as Let H denote the class of functions f which are harmonic in the unit disc A := A (1), where A (r) := {z ∈ C : |z| < r} . Also, let H 0 denote the class of functions f ∈ H which satisfy the normalization conditions Therefore the analytic functions h and g given by (1.1) can be written in the form It is clear that S H reduces to the class S, and by A whenever the co-analytic part of f vanishes, i.e., g (z) = 0 in A. Clunie and Sheil-Small [23] and Sheil-Small [24] studied S H together with some of its geometric subclasses. We say that a function f ∈ H 0 is said to be harmonic starlike in A if it satisfy Re where .
A function f (z) is subordinated to a function g (z) denoted by f (z) ≺ g (z), if there is complexvalued function w(z) with |w(z)| ≤ 1 and g (0) = 0 such that Convolution or Hadamard product of two function f 1 and f 2 is denoted by f 1 * f 2 and is defined by (1.4) In 1973, Janowski [25] introduced the idea of circular domain by introducing Janowski functions as; Janowski showed that the function k maps A onto the domain ∆ (A, B) with centre on real axis and D 1 = 1−A 1−B and D 2 = 1+A 1+B are diameter end points with 0 < D 1 < 1 < D 2 .
The Mittag-Leffer function is defined as . (1.5) The initial two parametric generalizations for the function shown in (1.5) were given by Wiman [26,27]. It is defined in the following way where α, β ∈ C, Re (α) > 0, Re (β) > 0 and Γ (z) is gamma function. Now the function Q α,β is defined by Using the function Q α,β Elhaddad et al. [28] defined the differential operator for the class of analytic functions as D m δ (α, β) : A → A as illustrated below : Where the operator D m δ (α, β) for a function f ∈ H given by (1.1) can be defined as below: for m ∈ N 0 . Motivated by [29,30] and using the operator D m δ (α, β) f (z) , we introduced the class of harmonic univalent functions as: with , which was studied by Deziok [29].
, introduced by Jahangiri, see [32] for details. [33] we define the dual set of V by

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.

Theorem
Let f ∈ H 0 and is given by

Now as
A sufficient coefficient bound for the class S α,β H (m, δ, A, B) is provided in the following.

Theorem
Let f ∈ H 0 be of the form (1.3) and satisfies the condition 3) Obviously the theorem is true for f (z) = z. Suppose f ∈ H 0 given by (1.3) and let there exist n ≥ 2 such that a n 0 or b n 0. Since Therefore h (z) > g (z) which shows that f is locally univalent and sense-preserving in A. Moreover ifz 1 , z 2 ∈ A and z 1 z 2 then Hence by (2.4) we have This Thus, it is sufficient to prove that where z ∈ A\ {0} , now by putting |z| = r, r ∈ (0, 1) we get δ, A, B) .
Proof. In Theorem 2.2 we need only to show that each function f ∈ S α,β τ (m, δ, A, B) satisfies coefficient inequality (2.1) . If f ∈ S α,β τ (m, δ, A, B) then it is of the form (2.6) and satisfies (2.5) or equivalently It is clear that the denominator of the left hand side cannot vanishes for r ∈ (0, 1). Moreover, it is positive for r = 0, and in consequence for r ∈ (0, 1). Thus, by (2.7) we have ∞ n=2 (λ n |a n | + σ n |b n |) r n−1 ≤ B − A r ∈ [0, 1) . (2.8) The sequence of partial sums {S n } associated with the series ∞ n=2 (λ n |a n | + σ n |b n |) is a non-decreasing sequence. Moreover, by (2.8) it is bounded by B − A. Hence, the sequence {S n } is convergent and ∞ n=2 (λ n |a n | + σ n |b n |) r n−1 = lim n→∞ S n ≤ B − A, which yields assertion (2.1) .

Topological properties
we consider the usual topology on H in which a sequence { f n } in H converges to f if and only if it converges to f uniformly on each compact subset of A. The metric induces the usual topology on H. It is to verify that the obtained topological space is complete.
Let F be a subclass of the class H. A function f ∈ F is called an extreme point of F if the condition implies f 1 = f 2 = f . We shall use the notation EF to denote the set of all extreme points of F . It is clear that EF ⊂ F . We say that F is locally uniformly bounded if for each r, 0 < r < 1, there is a real constant We say that a class F is convex if Moreover, we define the closed convex hull of F as the intersection of all closed convex subsets of H that contain F . We denote the closed convex hull of F by coF . A real-valued function J : H → R is called convex on a convex class F ⊂ H if The Krein-Milman theorem (see [35]) is fundamental in the theory of extreme points. In particular, it implies the following lemma.

Lemma
Let F be a non-empty convex compact subclass of the class H and let J : H → R be a real-valued, continuous and convex function on F . Then

A class F ⊂ H is compact if and only if F is closed and locally uniformly bounded.
Since H is complete metric space, Montel's theorem (see [36]) implies the following lemma.

Lemma
Let F be a non-empty compact subclass of the class H, then EF is non-empty and coEF = coF .

Theorem
The class S α,β τ (m, δ, A, B) is a convex and compact subset of H.

(3.2)
Thus, we conclude that the class S α,β τ (m, δ, A, B) is locally uniformly bounded. By Lemma 3.2, we need only to show that it is closed, i.e. if f l → f , then f ∈ S α,β τ (m, δ, A, B). Let f l and f be given by (3.1) and (2.6), respectively. Using Theorem 2.4, we have ∞ n=2 α n a i,n + β n b i,n ≤ B − A (i ∈ N) . (3. 3) Since f i → f , we conclude that a i,n → |a n | and b i,n → |b n | as i → ∞ (n ∈ N) . The sequence of partial sums {S n } associated with the series ∞ n=2 α n a i,n + β n b i,n is non-decreasing sequence. Moreover, by (3.3) it is bounded by B − A. Therefore, the sequence {S n } is convergent and ∞ n=2 α n a i,n + β n b i,n = lim This gives condition (2.1) and in consequence, f ∈ S α,β τ (m, δ, A, B), which complete the proof.

Theorem
We have Proof. Suppose that 0 < γ < 1 and It follows that f ES α,β τ (m, δ, A, B) , and this completes the proof.

Radii of starlikeness and convexity
A function f ∈ H 0 is said to be starlike of order α in A (r) if It easy to verify that for function f ∈ τ the condition (4.1) is equivalent to the following Re Let B be a subclass of the class 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 α.

Theorem
The radii of starlikeness of order α for the class S α,β τ (m, δ, A, B) is given by where λ n and σ n are define in (2.2) and (2.3) respectively.
Proof. Let f ∈ S α,β τ (m, δ, A, B) be of the form (2.6) . Then, for |z| = r < 1 we have Thus the condition (4.2) is true if and only if where λ n and σ n are defined by (2.2) i.e., It follows that the function f is starlike of order α in the disc A (r * ) , where r * The functions h * n and g * n are define by (3.4) realize equality in (4.5) , and the radius r * cannot be larger, thus we have (4.3) .
The following theorem may be proved in much same fashion as Theorem 4.1..

Theorem
The radii of convexity of order α for the class S α,β τ (m, δ, A, B) is given by where λ n and σ n are define in (2.2) and (2.3) respectively.

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 (γ n h n + δ n g n ) : ∞ n=2 (γ n + δ n ) = 1, δ 1 = 0, γ n , δ n ≥ 0 (n ∈ N) where h n and g n are defined by Eq (3.4) .

Proof. By Theorem 3.4 and Lemma 3.3 we have
Thus, by Theorem 3.5 and by (5.1) we have Eq (5.2) .
We observe, that for each n ∈ N, z ∈ A, the following real-valued functionals are continuous and convex on H : ( f ∈ H, γ ≥ 1, 0 < r < 1) .
Therefore, using Lemma 3.1 and Theorem 3.5 we have the following corollaries.

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.