On Harmonic Functions Defined by Derivative Operator

Abstract

Let S H denote the class of functions f h-g that are harmonic univalent and sense-preserving in the unit disk U {z : |z| < 1}, where h z z ∞ k 2 a k z k , g z ∞ k 1 b k z k |b 1 | < 1. In this paper, we introduce the class M H n, λ, α of functions f h-g which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class M-H n, λ, α if f n z h-g n ∈ M H n, λ, α , where h z z − ∞ k 2 |a k |z k , g n z −1 n ∞ k 1 |b k |z k and n ∈ N 0. Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M-H n, λ, α , are obtained.

Some properties on a class of harmonic univalent functions defined by differential operator

“…The most recent ones, can have a look at Al-Shaqsi and Darus [10], AlShaqsi and Darus [11]. In this paper, we extend some well-known results to the subclasses , ( , , , , ) 1 2 m b S H and , ( , , , , ).…”

Certain class of harmonic multivalent functions

“…Theorem 2. Let f m = h + g m be given by (6). Also, suppose that λ < p,λ (q, ℓ, α) ⊂ SH m,n p,λ (q, ℓ, α), we only need to prove the necessary part of the theorem.…”

“…A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h(z)| > |g(z)| for all z in D (see [1]). Many researcher introduced and studied certain classes of harmonic univalent functions (see [2][3][4][5][6][7][8]). For p 1, n ∈ N , denote by SH(n, p) the class of functions of the form (1) that are harmonic multivalent and sense-preserving in the unit disk U = {z : |z| < 1}, where h and g are defined by…”

On an Application of Neighborhood for a Class of Harmonic Univalent Functions

“…This class includes a variety of well-known subclasses of S H (n). For example we can reobtain several classes introduced earlier by (Jahangiri, 1998), (Shaqsi & Darus, 2008), (Ahuja & Jahangiri, 2003). Further we will give an application of the class mentioned above.…”