On Some Subclasses of Harmonic Functions Defined by Fractional Calculus

The purpose of this paper is to study subclasses of normalized harmonic functions with positive real part using fractional derivative. Sharp estimates for coefficients and distortion theorems are given.


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 orientationpreserving in D is that |g z | < |h z | in D, see 1 .
Denote by H the class of functions f h g which are harmonic univalent and orientation-preserving in the open unit disk U {z : |z| < 1} so that f h g is normalized by f 0 h 0 f z 0 − 1 0. Therefore, for f h g ∈ H, we can express h and g by the following power series expansion: Observe that H reduces S, the class of normalized univalent analytic functions, if the coanalytic part of f is zero.
For f h g given by 1.1 and n > −1, Murugusundaramoorthy 2 defined the Ruscheweyh derivative of the harmonic function f h g in H by where the Ruscheweh derivative of a power series f z z n n 2 a n z n is given by The operator * stands for the Hadamard product or convolution of two power series In 3 , Owa introduced the following definition.
Definition 1.1.Let the function f z be analytic in a simply connected domain of the z-plane containing the origin and let 0 ≤ λ < 1.The fractional derivative of f of order λ is defined by where the multiplicity of z − ζ −λ is removed by requiring log z − ζ to be real when z − ζ > 0.
In 4 , Owa gave the relation between the fractional derivative and Ruscheweyh operator for the function f z z ∞ n 2 a n z n as Using 1.2 and the relation between the fractional derivative and Ruscheweyh operator, we define the fractional derivative of order λ, 0 ≤ λ < 1, for the harmonic function f h g as

1.8
Since D λ f D λ h D λ g, it was proved in 1 that the harmonic function D λ f is starlike of order 1/2 if and only if the analytic function D λ h − D λ g is starlike of order 1/2, and it was shown in 4, Theorem 3 that D λ h − D λ g is starlike of order 1/2 if and only if Re{D Recently, Owa and Srivastava 5 studied the linear Ω λ defined by operator where f is normalized and analytic function on U.
It is easily seen that Analogously, we studied the linear operator Ω λ defined on the harmonic function f h g by where We will define subclasses of normalized harmonic functions obtained by the Hadamard product and using the fractional derivative.

Main results
Let h and g be analytic in U. Let P H stand for harmonic functions f h g so that Re f > 0, z ∈ U and f 0 1.If the function f z f z h g belongs to P H for the analytic and normalized functions then the class of functions f h g is denoted by P 0 H 6 .

International Journal of Mathematics and Mathematical Sciences
The function is analytic on U when α is a complex number different from −1, − 1/2 , − 1/3 , . . . .For Ω λ f ∈ P 0 H , we denote by P λ,0 H α the class of functions defined by Therefore, 4 , with a n , b n being the coefficients of Ω λ f ∈ P 0 H , then Ω λ F P λ,0 H α .Note that P 0,0 H α ≡ P 0 H α 7 and P 0,0 H 0 ≡ P 0 H .

2.9
where Therefore, and

2.14
Conversely, if the function Ω λ F Ω λ H Ω λ G of the form 2.4 satisfies 2.10 , then by Theorem 2.1 Ω λ h Ω λ g ∈ P H and the following function holds:

2.17
Since P 0 H is compact, see 6 ,

2.19
Equality is obtained for the function 2.3 where

2.21
Since by 5, Proposition 2.2 this completes the proof.

2.31
The required results are obtained.
On the other hand, from 2.10 , it is known 6, Corollary 2.5 that

2.32
Then we get the coefficient inequalities for P λ,0 0 α .

Positive order
We say that the harmonic function f h g of the form 2.
1 is in the class P H β , 0 ≤ β < 1 for |z| r if Re f > β and f 0 1.If the function f z f z h g belongs to P H β for the analytic and normalized functions h and g of the form 2.1 , then the class of functions f h g is denoted by P 0 H β .Denote by P λ,0 H β, α the class of functions defined By 2.3 where Ω λ f ∈ P 0 H β .Many of our results can be rewritten for functions in the class P λ,0 H β, α .For instance, see the following theorems., then there exists Ω λ f ∈ P 0 H β so thatα z Ω λ F z z z Ω λ F z z 1 − α Ω λ F z Ω λ f z .3.1Conversely, for any function f such that Ω λ f ∈ P 0 H β , there exists Ω λ F ∈ P λ,0 H β, α satisfying 3.1 .Theorem 3.2.A function Ω λ F belongs to P λ,0 H β, α if and only if If Ω λ F ∈ P λ,0 H β, α and Re α > 0, then there exists Ω λ f ∈ P 0 H β so that λ f zζ dζ, z ∈ U.3.3Theorem 3.4.If Re α > 0, then P λ,0 H β, α ⊂ P 0 H β .