Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator

Abstract

Making use of a Salagean operator, we introduce a new class of complex valued harmonic functions which 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.

1. Introduction and Preliminaries

A continuous function is a complex-valued harmonic function in a complex domain if both and are real and harmonic in . In any simply connected domain , we can write , where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and orientation preserving in is that in (see [1]).

Denote by the family of functions which are harmonic, univalent, and orientation preserving in the open unit disc so that is normalized by . Thus, for , the functions and are analytic in and can be expressed in the following forms: and is then given by We note that the family of orientation preserving, normalized harmonic univalent functions reduces to the well-known class of normalized univalent functions if the coanalytic part of is identically zero; that is, .

For functions , Jahangiri et al. [2] defined Salagean operator on harmonic functions given by where

In 1975, Silverman [3] introduced a new class of analytic functions of the form 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 [6–9]) introduced and studied various new subclasses of analytic functions with positive coefficients but analogues results on harmonic univalent 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 the subfamily of consisting of harmonic functions of the form

Motivated by the earlier works of [11–14] on the subject of harmonic functions, in this paper an attempt has been made to study the class of functions associated with Salagean operator on harmonic functions. Further, we obtain a sufficient coefficient condition for functions given by (3) and also show that this coefficient condition is necessary for functions , 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.

For , , we let be a new subclass of , consisting of all functions of the form (3) satisfying the condition where is given by (4) (see [2]). Also let .

2. Coefficient Bounds

In our first theorem, we obtain a sufficient coefficient condition for harmonic functions in .

Theorem 1. Let be given by (3). If where and , then .

Proof. We let (8) hold for the coefficients of . It suffices to show that where Substituting for and in (9), we get
The above expression is bounded above by 1 if which is equivalent to But (8) is true by hypothesis. Hence, , , and the theorem is proved for and is given by (10) and (11), respectively.

Theorem 2. For and , if and only if

Proof. Since , 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 . Upon choosing the values of on the positive real axis where , we must have If condition (15) does not hold, then the numerator in (18) is negative for sufficiently close to 1. Hence, there exists in for which the quotient of (18) is negative. This contradicts the required condition for . This completes the proof of the theorem.

3. Distortion Bounds and Extreme Points

By routine procedure (see [10–13]),  we can easily prove the following results; hence we state the following theorems without proof for functions in .

Theorem 3 (distortion bounds). Let . Then for , we have

Corollary 4 (covering result). If , then

Next we state the extreme points of closed convex hulls of denoted by .

Theorem 5. A function if and only if where ,  , , and , also , and . In particular, the extreme points of are and .

Theorem 6. The family is closed under convex combinations.

4. Inclusion Results

Now, we will examine the closure properties of the class under the generalized Bernardi-Libera-Livingston integral operator which is defined by ,  .

Theorem 7. Let . Then .

Lemma 8 (see [15]). Let be given by (3). If where and , then .

Theorem 9. Let be given by (6). Then .

Proof. Since , then by Theorem 1 we must have To show that , by virtue of Lemma 8 we have to show that where . For this, it is sufficient to prove that or equivalently , and , which is true and the theorem is proved.

Corollary 10. .

5. Convolution Properties

For functions given by (3) and given by we recall the Hadamard product (or convolution) of and by Let be given by then the convolution is defined by

Theorem 11. Let , then , where

Proof. We use the principle of mathematical induction in our proof. Let , and . By using Theorem 2, we have then Thus, by applying Cauchy-Schwarz inequality, we have Then, we get Therefore, if that is, if then . By (30) we have Hence we get Consequently, if That is, if then . Then we see that Since for and for are increasing, and also then where Next, we suppose that , where
We can show that , where Since we have

Corollary 12. Let , then , where

6. Partial Sums Results

In 1985, Silvia [16] studied the partial sums of convex functions of order . 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 on lines similar to Porwal [18].

Let denote the subclass of consisting of functions of the form (3) which satisfy the inequality where and , unless otherwise stated.

Now, we discuss the ratio of a function of the form (6) with being

We first obtain the sharp bounds for .

Theorem 13. If of the form (6) with satisfies the condition (50), then where The result (53) is sharp with the function given by

Proof. Define the function by It suffices to show that . Now, from (56), we can write Hence we obtain Now if From condition (50), it is sufficient to show that which is equivalently to To see that the function given by (55) gives the sharp result, we observe that for

We next determine bounds for .

Theorem 14. If of the form (6) with satisfies condition (50), then where The result (63) is sharp with the function given by

Proof. Define the function by Hence we obtain The last inequality is equivalent to Making use of (50) and the condition (64), we obtain (61). Finally, equality holds in (63) for the extremal function given by (65).

We next turns to ratios for and .

Theorem 15. If of the form (6) with satisfies condition (50), then where The result (69) is sharp with the function given by .

Proof. Define the function by The result (69) follows by using the techniques as used in Theorem 13.

Proceeding exactly as in the proof of Theorem 14, we can prove the following theorem.

Theorem 16. If of the form (6) with satisfies the condition (50), then The result is sharp with the function given by .

We next determine bounds for and .

Theorem 17. If of the form (6) with satisfies condition (50), then where The result (73) is sharp with the function given by .

Theorem 18. If of the form (6) with satisfies condition (50), then where The result (75) is sharp with the function given by .

Proof. Define the function by We omit the details of proof, because it runs parallel to that from Theorem 14.

Theorem 19. If of the form (6) with satisfies condition (50), then where The result (78) is sharp with the function given by .

Theorem 20. If of the form (6) with satisfies condition (50), then where

Theorem 21. If of the form (6) with satisfies condition (50), then

Theorem 22. If of the form (6) with satisfies condition (50), then The result (83) is sharp with the function given by .

Theorem 23. If of the form (6) with satisfies condition (50), then where The result (84) is sharp with the function given by .

Theorem 24. If of the form (6) with satisfies condition (50), then The result (86) is sharp with the function given by .

Concluding Remarks. By choosing (or ) and (or ), the various results presented in this paper would provide interesting extensions and generalizations of the subclasses of harmonic star-like functions with positive coefficients of order based on Salagean operator and similarly for convex functions. The details involved in the derivations of such specializations of the results presented in this paper are fairly straight-forward and hence omitted.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

We record our sincere thanks to the referees for their valuable suggestions.