On a Harmonic Univalent Subclass of Functions Involving a Generalized Linear Operator

: In this paper, a subclass of complex-valued harmonic univalent functions deﬁned by a generalized linear operator is introduced. Some interesting results such as coe ﬃ cient bounds, compactness, and other properties of this class are obtained.


Introduction
Let H represent the continuous harmonic functions which are harmonic in the open unit disk U = {z : z ∈ C, |z| < 1} and let A be a subclass of H which represents the functions which are analytic in U. A harmonic function in U could be expressed as f = h + g, where h and g are in A, h is the analytic part of f , g is the co-analytic part of f and h (z) > g (z) is a necessary and sufficient condition for f to be locally univalent and sense-preserving in U (see Clunie and Sheil-Small [1]). Now we write, h(z) = z + ∞ n=2 a n z n , g(z) = ∞ n=2 b n z n . (1) Let SH represents the functions of the form f = h + g which are harmonic and univalent in U, which normalized by the condition f (0) = f z (0) − 1 = 0. The subclass SH 0 of SH consists of all functions in SH which have the additional property f z (0) = 0. The class SH was investigated by Clunie and Sheil-Smallas [1]. Since then, many researchers have studied the class SH and even investigated some subclasses of it. Also, we observe that the class SH reduces to the class S of normalized analytic univalent functions in U, if the co-analytic part of f is equal to zero. For f ∈ S, the Salagean differential operator D n (n ∈ N 0 = N ∪ {0}) was defined by Salagean [2]. For f = h + g given by (1), Jahangiri et al. [3] defined the modified Salagean operator of f as n m a n z n , D m g(z) = ∞ n=2 n m b n z n .
A function f : U → C is subordinate to the function g : U → C denoted by f (z) ≺ g(z), if there exists an analytic function w : U → U with w(0) = 0 such that f (z) = g(w(z)), (z ∈ U).
In this paper we use the same techniques that have been used earlier by Dziok [21] and Dziok et al. [22], to investigate coefficient bound, distortion bounds, and some other properties for the class SH 0 (δ, µ, λ, ς, τ, m, A, B).

Coefficient Bounds
In this section we find the coefficient bound for the class SH 0 (δ, µ, λ, ς, τ, m, A, B).

Theorem 1.
Let the function f (z) = h + g be defined by (1). where and Proof. Let a n 0 or b n 0 for n ≥ 2. Since C n , D n ≥ n(B − A) by (6), we obtain Therefore, f is univalent in U. To ensure the univalence condition, consider z 1 , z 2 ∈ U so that So, we have On the other hand, f ∈ SH 0 (δ, µ, λ, ς, τ, m, A, B) if and only if there exists a function w; with w(0) = 0, and w(z) < 1(z ∈ U) such that The above inequality (9) holds, since for |z| = r (0 < r < 1) we obtain µ+λ a n z n Therefore, f ∈ SH 0 (δ, µ, λ, ς, τ, m, A, B), and so the proof is completed. Next we show that the condition (6) is also necessary for the functions f ∈ H to be in the class SH 0 T (δ, µ, λ, ς, τ, m, A, B) = T m ∩ SH 0 (δ, µ, λ, ς, τ, m, A, B) where T m is the class of functions f = h + g ∈ SH 0 so that Theorem 2. Let f = h + g be defined by (10). Then f ∈ SH 0 T (δ, µ, λ, ς, τ, m, A, B) if and only if the condition (6) holds.
Therefore, SH 0 T (δ, µ, λ, ς, τ, m, A, B) is uniformly bounded. Let also, let f = h + g where h and g are given by (1). Then by Theorem 2 we get If we assume f t → f , then we get that a t,n → |a n | and b t,n → |b n | as n → +∞ (t ∈ N). Let ρ n be the sequence of partial sums of the series ∞ n=2 C n a t,n + D n b t,n . Then ρ n is a non-decreasing sequence and by (13) it is bounded above by B − A. Thus, it is convergent and ∞ n=2 C n a t,n + D n b t,n = lim Therefore, f ∈ SH 0 T (δ, µ, λ, ς, τ, m, A, B) and therefore the class SH 0 T (δ, µ, λ, ς, τ, m, A, B) is closed. As a result, the class is closed, and the class SH 0 T (δ, µ, λ, ς, τ, m, A, B) is also compact subset of SH, which completes the proof. Lemma 1 [23]. Let f = h + g be so that h and g are given by (1). Furthermore, let where 0 ≤ α < 1. Then f is harmonic, orientation preserving, univalent in U and f is starlike of order α.

Theorem 4.
Let 0 ≤ α < 1, C n and D n be defined by (7) and (8). Then where r * α is the radius of starlikeness of order α.
So, the function f is starlike of order α in the disk U * r α where So, the radius r * α cannot be larger. Then we get (14).

Extreme Points
In this section we find the extreme points for the class SH 0 (δ, µ, λ, ς, τ, m, A, B).
It follows that f is not in the family of extreme points of the class SH 0 T (δ, µ, λ, ς, τ, m, A, B) and so the proof is completed.