Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator

Making use of a Salagean operator, we introduce a new class of complex valued harmonic functionswhich are orientation preserving and univalent in the open unit disc. Among the results presented in this paper including the coeffcient bounds, distortion inequality, and covering property, extreme points, certain inclusion results, convolution properties, and partial sums for this generalized class of functions are discussed.


Introduction and Preliminaries
A continuous function  =  + V is a complex-valued harmonic function in a complex domain G if both  and V are real and harmonic in G.In any simply connected domain D ⊂ G, we can write  = ℎ + , where ℎ and  are analytic in D. We call ℎ the analytic part and  the coanalytic part of .A necessary and sufficient condition for  to be locally univalent and orientation preserving in D is that |ℎ  ()| > |  ()| in D (see [1]).
Denote by H the family of functions which are harmonic, univalent, and orientation preserving in the open unit disc U = { : || < 1} so that  is normalized by (0) =   (0)−1 = 0. Thus, for  = ℎ+ ∈ H, the functions ℎ and  are analytic in U and can be expressed in the following forms: and () is then given by (0 ≤      1     < 1) .
We note that the family H of orientation preserving, normalized harmonic univalent functions reduces to the well-known class S of normalized univalent functions if the coanalytic part of  is identically zero; that is,  ≡ 0.
For functions  ∈ H, Jahangiri et al. [2] defined Salagean operator on harmonic functions given by  ℓ  () =  ℓ ℎ () + (−1) ℓ  ℓ  (), where In 1975, Silverman [3] introduced a new class T of analytic functions of the form () =  − ∑ ∞ =2 |  |  and opened up a new direction of studies in the theory of univalent functions as well as in harmonic functions with negative coefficients [4].Uralegaddi et al. [5] introduced analogous subclasses of star-like, convex functions with positive coefficients and opened up a new and interesting direction of research.In fact, they considered the functions where the coefficients are positive rather than negative real numbers.Motivated by the initial work of Uralegaddi et al. [5], many researchers (see [6][7][8][9]) introduced and studied various new subclasses of analytic functions with positive coefficients but analogues results on harmonic univalent 2 International Journal of Analysis functions have not been explored in the literature.Very recently, Dixit and Porwal [10] attempted to fill this gap by introducing a new subclass of harmonic univalent functions with positive coefficients.
Denote by V H the subfamily of H consisting of harmonic functions  = ℎ +  of the form Motivated by the earlier works of [11][12][13][14] on the subject of harmonic functions, in this paper an attempt has been made to study the class of functions  ∈ V H associated with Salagean operator on harmonic functions.Further, we obtain a sufficient coefficient condition for functions  ∈ H given by (3) and also show that this coefficient condition is necessary for functions  ∈ V H , the class of harmonic functions with positive coefficients.Distortion results and extreme points, inclusion relations, and convolution properties and results on partial sums are discussed extensively.

Coefficient Bounds
In our first theorem, we obtain a sufficient coefficient condition for harmonic functions in P ℓ H (, ).
Proof.Since V ℓ H (, ) ⊂ P ℓ H (, ), we only need to prove the "only if " part of the theorem.To this end, for functions  of the form (6), we notice that the condition Equivalently, The above required condition must hold for all values of  in U. Upon choosing the values of  on the positive real axis where 0 ≤  =  < 1, we must have If condition (15) does not hold, then the numerator in ( 18) is negative for  sufficiently close to 1. Hence, there exists  0 =  0 in (0, 1) for which the quotient of ( 18) is negative.This contradicts the required condition for  ∈ V ℓ H (, ).This completes the proof of the theorem.

Distortion Bounds and Extreme Points
By routine procedure (see [10][11][12][13]), we can easily prove the following results; hence we state the following theorems without proof for functions in V ℓ H (, ).

Convolution Properties
For functions  ∈ H given by ( 3) and  ∈ H given by we recall the Hadamard product (or convolution) of  and  by Let   () ∈ V ℓ H (, ) ( = 1, 2, 3, . . ., ) be given by then the convolution is defined by ,        . (28) Proof.We use the principle of mathematical induction in our proof.Let  1 ∈ V ℓ H (,  1 ), and  2 ∈ V ℓ H (,  2 ).By using Theorem 2, we have Thus, by applying Cauchy-Schwarz inequality, we have Then, we get that is, if then Hence we get Consequently, if International Journal of Analysis That is, if then Since () for  ≥ 2 and () for  ≥ 1 are increasing, and also then ( 1 *  2 )() ∈ V ℓ H (, ) where Next, we suppose that (49)

Partial Sums Results
In 1985, Silvia [16] studied the partial sums of convex functions of order  (0 ≤  < 1).Later on, Silverman [17] and several researchers studied and generalized the results on partial sums for various classes of analytic functions only but analogues results on harmonic functions have not been explored in the literature.Very recently, Porwal [18] and Porwal and Dixit [19] filled this gap by investigating interesting results on the partial sums of star-like harmonic univalent functions.Now in this section we discussed the partial sums results for the class of harmonic functions with positive coefficients based on Salagean operator of order  (1 <  ≤ 4/3) on lines similar to Porwal [18].Let P ℓ H (  /,   /) denote the subclass of H consisting of functions  = ℎ +  of the form (3) which satisfy the inequality where and  =  − 1, unless otherwise stated.
International Journal of Analysis 7 Now, we discuss the ratio of a function of the form (6) with  1 = 0 being We first obtain the sharp bounds for R{()/  ()}.