On Harmonic Functions Defined by Derivative Operator

A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the classM –– H n, λ, α if fn z h –– gn∈ MH n, λ, α , where h z z− ∑∞ k 2|ak|z, gn z −1 n ∑∞ k 1|bk|z and n ∈ N0. Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M –– H n, λ, α , 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 C. In any simply connected domain D ⊂ C, we can write f h g, where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h z | > |g z | in D; see 2 .
Denote by S H the class of functions f h g that are harmonic, univalent, and sensepreserving in the unit disk U {z : |z| < 1} for which 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

Journal of Inequalities and Applications
Observe that S H reduces to S, the class of normalized univalent analytic functions, if the coanalytic part of f is zero. Also, denote by S * H the subclasses of S H consisting of functions f that map U onto starlike domain.
For f h g given by 1.1 , we define the derivative operator introduced by authors see 1 of f as We let M H n, λ, α denote the family of harmonic functions f of the form 1.1 such that where D n λ f is defined by 1.2 . If the coanalytic part of f h g is identically zero, then the class M H n, λ, α turns out to be the class R n λ α introduced by Al-Shaqsi and Darus 1 for the analytic case. Let M H n, λ, α denote that the subclass of M H n, λ, α consists of harmonic functions f n h g n such that h and g n are of the form It is clear that the class M H n, λ, α includes a variety of well-known subclasses of S H . For example, M H 0, 0, α ≡ S * H α is the class of sense-preserving, harmonic, univalent functions f which are starlike of order α in U, that is, ∂/∂θ arg f re iθ > α, and M H 1, 0, α ≡ M H 0, 1, α ≡ HK α is the class of sense-preserving, harmonic, univalent functions f which are convex of order α in U, that is, ∂/∂θ arg ∂/∂θ f re iθ > α. Note that the classes S * H and HK α were introduced and studied by Jahangiri 3 . Also we notice that the class M H n, 0, α is the class of Salagean-type harmonic univalent functions introduced by Jahangiri et al. 4 ; and M H 0, λ, α is the class of Ruscheweyh-type harmonic univalent functions studied by Murugusundaramoorthy and Vijaya 5 .
In 1984, Clunie and Sheil-Small 2 investigated the class S H as well as its geometric subclasses and obtained some coefficient bounds. Since then, there has been several related papers on S H and its subclasses such that Silverman 6 , Silverman and Silvia 7 , and Jahangiri 3, 8 studied the harmonic univalent functions. Jahangiri and Silverman 9 prove the following theorem.
then f is sense-preserving, harmonic, and univalent in U and f ∈ S * H consists of functions in S H which are starlike in U.
The condition 1.5 is also necessary if f ∈ TH ≡ M H 0, 0, 0 . In this paper, we will give sufficient condition for functions f h g, where h and g are given by 1.1 to be in the class M H n, λ, α ; and it is shown that this coefficient condition is also necessary for functions in the class M H n, λ, α . Also, we obtain distortion theorems and characterize the extreme points for functions in M H n, λ, α . Closure theorems and application of neighborhood are also obtained.

Coefficient bounds
We begin with a sufficient coefficient condition for functions in M H n, λ, α .
which proves univalence. Note that f is sense-preserving in U. This is because

2.3
Using the fact that Rew > α if and only if |1 − α w| ≥ |1 α − w|, it suffices to show that Substituting D n λ f z in 2.4 yields, by 2.1 , we obtain

2.5
This last expression is nonnegative by 2.1 , and so the proof is complete.
The harmonic function where n, λ ∈ N 0 and ∞ k 2 |x k | ∞ k 1 |y k | 1 show that the coefficient bound given by 2.1 is sharp. The functions of the form 2.6 are in M H n, λ, α because In the following theorem, it is shown that the condition 2.1 is also necessary for functions f n h g n , where h and g n are of the form 1.4 .

Proof.
Since M H n, λ, α ⊂ M H n, λ, α , we only need to prove the "if and only if" part of the theorem. To this end, for functions f n of the form 1.4 , we notice that the condition 1.3 is equivalent to K. Al-Shaqsi and M. Darus

5
The above required condition 2.9 must hold for all values of z in U. Upon choosing the values of z on the positive real axis, where 0 ≤ z r < 1, we must have If the condition 2.8 does not hold, then the numerator in 2.10 is negative for r sufficiently close to 1. Hence there exist z 0 r 0 in 0, 1 for which the quotient in 2.8 is negative. This contradicts the required condition for f n ∈ M H n, λ, α and so the proof is complete.

Distortion bounds
In this section, we will obtain distortion bounds for functions in M H n, λ, α .

3.1
Proof. We only prove the left-hand inequality. The proof for the right-hand inequality is similar and will be omitted. Let f n ∈ M H n, λ, α . Taking the absolute value of f n , we obtain

6
Journal of Inequalities and Applications The following covering result follows from the left-hand inequality in Theorem 3.1.

Convolution, convex combination, and extreme points
In this section, we show that the class M H n, λ, α is invariant under convolution and convex combination of its member.
For harmonic functions f n z z − ∞ k 2 a k z k −1 n ∞ k 1 b k z k and F n z z − ∞ k 2 A k z k −1 n ∞ k 1 B k z k , the convolution of f n and F n is given by