On the Generalization of a Class of Harmonic Univalent Functions Defined by Differential Operator

In this article, a new class of harmonic univalent functions, defined by the differential operator, is introduced. Some geometric properties, like, coefficient estimates, extreme points, convex combination and convolution (Hadamard product) are obtained.


Introduction
A continuous function f = u + iv is a complex-valued harmonic function in a complex domain C if both u and v are real harmonic.In any simply connected domain B ⊂ C, we can write f = h + g, where h and g are analytic in B. We call h and g are analytic part and co-analytic part of f respectively.Clunie and Sheil-Small [1] observed that a necessary and sufficient condition for the harmonic functions f = h + g to be locally univalent and sense-preserving in B is that |h (z)| > |g (z)|, (z ∈ B).
Denote by S H the family of harmonic functions f = h + g, which are univalent and sense-preserving in the open unit disc U = {z ∈ C : |z| < 1} where h and g are analytic in B and f is normalized by f (0) = h(0) = f z (0) − 1 = 0. Then for f = h + g ∈ S H , we may express the analytic functions h and g as Note that S H reduces to the class of normalized analytic univalent functions if the co-analytic part of its members equals to zero.
Also, denote by S H the subclass of S H consisting of all functions f k (z) = h(z) + g k (z), where h and g are given by In 1984 Clunie and Sheil-Small [1] investigated the class S H , as well as its geometric subclass and obtained some coefficient bounds.Many authors have studied the family of harmonic univalent function (see References [2][3][4][5][6][7]).
In 2016 Makinde [8] introduced the differential operator F k such that where Thus, it implies that F k f (z) is identically the same as f (z) when k = 0. Also, it reduced the first differential coefficient of the Salagean differential operator when k = 1.
For f = h + g given by Equation ( 1), Sharma and Ravindar [9] considered the differential operator which defined by Equation (3) of f as where In this paper, motivated by study in [9], a new class introduced and studied.Furthermore, coefficient conditions, distortion bounds, extreme points, convex combination and radii of convexity for this class are obtained.A H (k, α, γ) Definition 1.Let f (z) = h(z) + g(z) be a harmonic f unction, where h(z) and g(z) are given by Equation (1).Then f (z)
Here, we give a sufficient condition for a function f to be in the class A H (k, α, γ).
Theorem 1.Let f (z) = h(z) + g(z) where h(z) and g(z) were given by (1). where then f (z) is harmonic univalent and sense-preserving in U and f (z we have by inequality so that According to the condition of Equation ( 5), we only need to show that if Equation ( 6) holds, then Now, substituting for A(z) and B(z Substituting for Equations ( 8) and ( 9) in the inequality we obtain Therefore, we have The harmonic univalent function where k ∈ N 0 and shows that the coefficient bound given by Equation ( 6) is sharp.Since Now, we show that the condition of Equation ( 6) is also necessary for functions f k = h + g k , where h and g n are given by Equation (6).Theorem 2. Let f k = h + g k be given by Equation (6).Then f k (z) ∈ A H (k, α, γ) if and only if the coefficient in condition of Equation ( 6) holds.
Proof.We only need to prove the "only if" part of the theorem because of A H (k, α, γ) ⊂ A H (k, α, γ).Then by Equation ( 5), we have We observe that the above-required condition of Equation ( 11) must behold for all values of z in U.If we choose z to be real and z → 1 − , we get If the condition ( 6) does not hold, then the numerator in Equation ( 12) is negative for r sufficiently closed to 1. Hence there exist z 0 = r 0 in (0, 1) for which the quotient in Equation ( 12) is negative, therefore there is a contradicts the required condition for f k ∈ A H (k, α, γ).

Extreme Points
Here, we determine the extreme points of the closed convex hull of A H (k, α, γ), denoted by clcoA H (k, α, γ).Theorem 3. Let f k given by (1.2).Then f k ∈ A H (k, α, γ) i f and only i f In particular the extreme points of A H (k, α, γ) are {h n } and {g kn }. Proof.Suppose The required representation is obtained as

Convex Combination
Here, we show that the class A H (k, α, γ) is closed under convex combination of its members.Let the function f k,i (z) be defined, for i = 1, 2, . . ., m by Theorem 4. Let the functions f k,i (z), defined by Equation (13) be in the class A H (k, α, γ), for every i = 1, 2, . . ., m.Then the functions c i (z) defined by Proof.According to the definition of c i (z), we can write Further, since f k,i (z) are in A H (k, α, γ) for every i = 1, 2, . . ., m, then by Theorem 2, we obtain which is required coefficient condition.

Convolution (Hadamard Product) Property
Here, we show that the class A H (k, α, γ) is closed under convolution.The convolution of two harmonic functions and is defined as Using Equations ( 12)-( 14), we prove the following theorem.
Then the convolution f n * Q n is given by Equation ( 16), we want to show that the coefficients of f n * Q n satisfy the required condition given in Theorem 1. For Since 0 ≤ µ ≤ α < 1 and f n ∈ A H (k, α, γ).Therefore f n * Q n ∈ A H (k, α, γ) ⊂ A H (k, µ, γ).

Integral Operator
Here, we examine the closure property of the class A H (k, α, γ) under the generalized Bernardi-Libera-Livingston integral operator (see References [10,11]) L u ( f ) which is defined by, by Theorem 2. Therefore, we have L u ( f k (z)) ∈ A H (k, α, γ).