On a New Subclass of Harmonic Univalent Functions Defined by Multiplier Transformation

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by using Multliplier transformation. Coefficient estimates, distortion bounds, extreme points, convolution condition and convex combination for functions belonging to this class are determined. The results obtained for the class reduce to the corresponding several known results are briefly indicated.


Introduction
A continuous complex-valued function f = u + iv defined in a simplyconnected domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simply-connected domain D, we can write f = h + g, where h and g are analytic in D. We call h the analytic part and g the co-analytic 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)|, z ∈ D. See Clunie and Sheil-Small [2], for more basic results on harmonic functions one may refer to the following standard text book by Duren [8], (see also [1] and [11]).
Denote by S H the class of functions f = h + g which are harmonic univalent and sense-preserving in the open 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 functions h and g as (1) h Note that S H reduces to the class S of normalized analytic univalent functions if the co-analytic part of its member is zero. For this class the function f (z) may be expressed as (2) f (z) = z + ∞ k=2 a k z k .
Cho and Srivastava [3], (see also [4]), introduced the operator I n λ : A → A defined as (3) I n λ f (z) = z + ∞ k=2 k + λ 1 + λ n a k z k , where A denote the class of functions of the form (2) which are analytic in the open unit disc U . For λ = 1, the operator I n λ ≡ I n was studied by Uralegaddi and Somanatha [19] and for λ = 0 the operator I n λ reduce to well-known Salagean operator introduced by Salagean [18].
Motivated with the definition of modified Salagean operator introduced by Jahangiri et al. [9], we define the modified multiplier transformation for f = h + g given by (1) as Now, for 1 < β ≤ 4 3 , −1 < λ ≤ 1, n ∈ N 0 and z ∈ U , suppose that S H (n, λ, β) denote the family of harmonic functions f of the form (1) such that where I n λ f is defined by (4). Further, let the subclass V S H (n, λ, β) consisting of harmonic functions f n = h + g n in S H (n, λ, β) so that h and g n are of the form In 1994 Uralegaddi et al. [20] introduced an analogous subclass of starlike, convex and close-to-convex functions with positive coefficients and opened up a new and interesting direction of research in Geometric Function Theory. In fact, they considered the functions of the form (2) where the coefficients are positive. Motivated by the work of Uralegaddi et al. [20], several researchers (e.g. Dixit and Chandra [5], Dixit et al. [7], Porwal and Dixit [15], Porwal et al. [17]) introduce new subclasses of analytic functions with positive coefficients. Recently, analogues to these results, Dixit and Porwal [6] introduced the class R H (β) of harmonic univalent functions with positive coefficients and opened up a new direction of research in the theory of harmonic univalent functions. After the appearence of this paper several researchers (e.g. see Pathak et al. [10], Porwal and Aouf [12], Porwal et al. [16]) generalized the result of [6] by using certain operator. Very recently Porwal and Dixit [13] (see also [14]) investigated new subclasses of harmonic starlike and convex functions. In the present paper, analogues to these results, we study the subclass of harmonic univalent functions defined by Multliplier transformation in which h has positive coefficients.
Assigning specific values to n, λ and β in the subclass S H (n, λ, β), we obtain the following known subclasses studied earlier by various researchers.
In the present paper, we extend the above results to the families S H (n, λ, β) and V S H (n, λ, β). We also obtain extreme points, distortion bounds, convolution conditions and convex combinations of functions belonging to the class V S H (n, λ, β).

Main Results
We first prove a sufficient coefficient condition for function in S H (n, λ, β).
Theorem 2.1. Let f = h + g be such that h and g are given by (1). Furthermore, let .
To prove that f ∈ S H (n, λ, β). Using the fact that ω < β, if and only if, |ω − 1| < |ω + 1 − 2β|, it suffies to show that We have which is bounded above by 1 by using (7) and so the proof is complete.
The harmonic univalent functions |y k | = 1 show that the coefficient bound given by (7) is sharp.
In the following theorem, it is shown that the condition (7) is also necessary for functions f n = h + g n where h and g n are of the form (6).
Theorem 2.2. Let f n = h + g n be given by (6).
Then f n ∈ V S H (n, λ, β), if and only if Proof. Since V S H (n, λ, β) ⊂ S H (n, λ, β), we only need to prove the "only if" part of the theorem. To this end, for functions f n of the form (6), we notice that the condition The above required condition (10) 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 (9) does not hold, then the numerator in (11) is negative for r sufficiently close to 1. Hence there exists z 0 = r 0 in (0,1) for which the quotient in (11) is negative. This contradicts the required condition for f n ∈ V S H (n, λ, β) and so the proof is complete.
Next, we determine the extreme points of closed convex hulls of V S H (n, λ, β) denoted by clco V S H (n, λ, β). Theorem 2.3. Let f n be given by (6). Then f n ∈ V S H (n, λ, β), if and only if In particular, the extreme points of V S H (n, λ, β) are {h k } and {g n k }.
Proof. For functions f of form (12), we have On a New Subclass of Harmonic Univalent Functions. . .
and so f n ∈ V S H (n, λ, β).

Set
y k and note that, by Theo- as required.
The following theorem gives the distortion bounds for functions in V S H (n, λ, β) which yields a covering result for the class.
Theorem 2.4. Let f n ∈ V S H (n, λ, β). Then for |z| = r < 1, we have Proof. We only prove the right hand inequality. The proof for left hand inequality is similar and will be omitted. Let f n (z) ∈ V S H (n, λ, β). Taking the absolute value of f , we have The following covering result follows from the left hand inequality in Theorem 2.4. Corollary 2.1. Let f n of the form (6) be so that f n ∈ V S H (n, λ, β). Then For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form We define their convolution (13) Using this definition, we show that the class V S H (n, λ, β) is closed under convolution.
Proof. Let Then the convolution (f n * F n )(z) is given by (13). We wish to show that the coefficients of (f n * F n )(z) satisfy the required condition given in Theorem 2.2. For F n (z) ∈ V S H (n, λ, α), we note that |A k | ≤ 1 and |B k | ≤ 1. Now, for the convolution function (f n * F n )(z), we have .
Next, we show that the class V S H (n, λ, β) is closed under convex combinations of its members.
Theorem 2.6. The class V S H (n, λ, β) is closed under convex combination.

A Family of Class Preserving Integral Operator
Let f (z) = h(z) + g(z) be defined by (1)  Theorem 3.1. Let f n (z) = h(z) + g n (z) ∈ S H be given by (6) and f n (z) ∈ V S H (n, λ, β) then F (z) be defined by (14) also belong to V S H (n, λ, β).
Proof. From the representation of (14) of F (z), it follows that c + 1 c + k |b k |z k .