A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.


Introduction
A continuous complex-valued function f u iv defined in a simply connected domain D, is said to be harmonic in D if both u and v are harmonic in D. In any simply connected domain D we can write f h g, where h and g are analytic in D. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h z | > |g z |, z ∈ D. See Clunie and Sheil-Small 1 .
Denote by S H the class of functions f h g that are harmonic univalent and sense preserving in the unit disk U {z : |z| < 1} for which f 0 f z 0 − 1 0. Then for f h g ∈ S H , we may express the analytic function h and g as Note that S H , is reduced to S the class of normalized analytic univalent functions if the coanalytic part of f h g is identically zero.

1.3
Corresponding to the function The Dziok-Srivastava operator 6, 7 where * stands for convolution of two power series.
To make the notation simple, we write Special cases of the Dziok-Srivastava operator includes the Hohlov operator 8 , the Carlson-Shaffer operator L a, c 9 , the Ruscheweyh derivative operator D n 10 , and the Srivastava-Owa fractional derivative operators 11-13 .
We define the Dziok-Srivastava operator of the harmonic functions f h g given by 1.1 as Recently, Porwal 14, Chapter 5 defined the subclass M H β ⊂ S H consisting of harmonic univalent functions f z satisfying the following condition: International Journal of Mathematics and Mathematical Sciences 3 He proved that if f h g is given by 1.1 and if 1.9 For g ≡ 0 the class of M H β is reduced to the class M β studied by Uralegaddi et al. 15 .
Generalizing the class M H β , we let M H α 1 , β denote the family of functions f h g of form 1.1 which satisfy the condition In this paper, we give a sufficient condition for f h g, given by 1.1 to be in M H α 1 , β , and it is shown that this condition is also necessary for functions in M H α 1 , β .We then obtain distortion theorem, extreme points, convolution conditions, and convex combinations and discuss a class preserving integral operator for functions in M H α 1 , β .

Main Results
First, we give a sufficient coefficient bound for the class We have where But 2.6 is true by hypothesis.Hence and the theorem is proved.
International Journal of Mathematics and Mathematical Sciences 5 In the following theorem, it is proved that the condition 2.1 is also necessary for functions f h g ∈ M H α 1 , β that are given by 1.11 .
Proof.Since M H α 1 , β ⊂ M H α 1 , β , we only need to prove the "only if" part of the theorem.
For this we show that f / ∈ M H α 1 , β if the condition 2.8 does not hold.Note that a necessary and sufficient condition for f h g given by 1.9 is in The above condition must hold for all values of z, |z| r < 1. Upon choosing the values of z on the positive real axis where 0 ≤ z r < 1, we must have

2.11
If the condition 2.8 does not hold then the numerator of 2.11 is negative for r and sufficiently close to 1. Thus there exists a z 0 r 0 in 0,1 for which the quotient in 2.11 is negative.This contradicts the required condition for f ∈ M H α 1 , β and so the proof is complete.
Next, we determine the extreme points of the closed convex hulls of where

2.13
x k ≥ 0, and y k ≥ 0. In particular the extreme points of M H α 1 , β are {h k } and {g k }.
Proof.For functions f of the form 2.12 , we have

2.16
International Journal of Mathematics and Mathematical Sciences 7 Proof.We only prove the right hand inequality.The proof for left hand inequality is similar and will be omitted.Let f ∈ M H α 1 , β .Taking the absolute value of f, we have

2.17
For our next theorem, we need to define the convolution of two harmonic functions.For harmonic functions of the form

2.18
We define the convolution of two harmonic functions f and F as

2.19
Using this definition, we show that the class M H α 1 , β is closed under convolution.
Then the convolution f * F is given by 2.19 .We wish to show that the coefficients of f * F satisfy the required condition given in Theorem 2.2.For F z ∈ M H α 1 , β we note that |A k | ≤ 1 and |B k | ≤ 1.Now, for the convolution function f * F, we obtain

2.23
For ∞ i−1 t i 1, 0 ≤ t i ≤ 1, the convex combination of f i may be written as Then by 2.23 ,

2.25
This is the condition required by Theorem 2.2 and so ∞ i−1 t i f i z ∈ M H α 1 , β .

A Family of Class Preserving Integral Operator
Let f z h z g z be defined by 1.1 ; then F z defined by the relation By definition of F z , we have Now we have

Theorem 3 . 1 .∞ k 2
Let f z h z g z ∈ S H be given by 1.11 andf ∈ M H α 1 , β , then F z is defined by 3.1 also belong to M H α 1 , β .Proof.Let f z z |a k |z k − ∞ k 1 |b k |z k be in M H α 1 , β ,then by Theorem 2.2, we have