New Class of Close-to-Convex Harmonic Functions Defined by a Fourth-Order Differential Inequality

Abstract

In the recent past, various new subclasses of normalized harmonic functions have been defined in open unit disk U which satisfy second-order and third-order differential inequalities. Here, in this study, we define a new class of normalized harmonic functions in open unit disk U which is satisfying a fourth-order differential inequality. We investigate some useful results such as close-to-convexity, coefficient bounds, growth estimates, sufficient coefficient condition, and convolution for the functions belonging to this new class of harmonic functions. In addition, under convex combination and convolution of its members, we prove that this new class is closed, and we also give some lemmas to prove our main results.

1. Introduction and Definitions

Let represent the class of all harmonic functions in open unit disk , and all these harmonic functions are normalized byand can be expressed as , where is the analytic part and is the coanalytic part of in , and also, these functions have a series of the form

Harmonic functions is locally univalent and sense-preserving in , if it satisfies a necessary sufficient condition [1, 2]. If coanalytic part of is zero, then class of complex valued harmonic functions reduces to class of normalized analytic functions.

Let denote the family of analytic univalent and normalized functions in and also which are defined as

Also, let class define aswhere class represent the class of functions which are harmonic, univalent, and sense-preserving in open unit disk .

We can see that class is compact and normal, but class is only normal. Let , and are the subclasses of which map open unit disk onto starlike, convex, and close-to-convex domains for harmonic functions, respectively. We can observe that

These subclasses , , and map open unit disk onto their respective domains.

For [2], Ponnusamy et al. defined the class of harmonic functions which satisfy the condition

In this class, they studied about close-to-convexity of harmonic functions. After that, Li and Ponnusamy [3, 4] discussed univalency and convexity of the abovementioned class. A class of harmonic functions defined by Nagpal and Ravichandran in [5] and the functions in this class satisfy the condition

Note thatand members of are fully starlike in .

Recently, Ghosh and Vasudevarao [6] for defined a new class for harmonic functions satisfying the condition

Rajbala and Prajapat [7] for , , defined a new class of harmonic functions which satisfy the following inequality:

For this class, authors used Gaussian hypergeometric function and created harmonic polynomials for the class .

Very recently, for the functions , Yaşar and Yalçın [8] introduced the class which satisfy the following inequality:for . For further study about harmonic functions, refer [2, 5, 9–11].

By taking the inspiration from the abovementioned work, we define new class of harmonic functions in which satisfy the fourth-order differential inequality.

Definition 1. For , let denote the class of functions and satisfy the condition

Definition 2. For , let denote a class of functions if it satisfies the inequalityNote thatSpecial cases are(1), defined by Yaşar and Yalçın [8].(2), discussed by Nagpal and Ravichandran [5].(3), defined by Ghosh and Vasudevarao [6].(4), defined by Rajbala and Prajapat [7].In this section, we prove that all the members of the class are close-to-convex. We will derive coefficient bounds, growth estimates, and sufficient coefficient condition for the class . Furthermore, we investigate example of harmonic polynomial belonging to .

2. Main Results

Lemma 1 (see [1]). Let and are analytic functions in open unit disk along with and is close-to-convex for each . Then,

Theorem 1. Let if and only if for each .

Proof. Suppose . For each ,Thus, for each . Conversely, let , then By choosing , we getHence, .

Lemma 2 (see [12]). Let , , with . Let defined bybe analytic in , such that

Then, for , , such thator

Then,

Lemma 3. If with , thenand hence,

Proof. Let , and we haveThen,Let be an analytic function in withWe have to show that , . Then,Since is analytic in ,such thatThen, by using Lemma 2, we may writeFor all , we getwhich opposes the hypothesis. Hence, there is no , such thatHence, for all . So, we get

Theorem 2. A function is close-to-convex in .

Proof. According to Lemma 3, we derive that are close-to-convex in , . Therefore, in the light of Theorem 1 and Lemma 1, we get are close-to-convex in .

Theorem 3. Let . Then,

The equality is satisfied for

Proof. Let . Applying the series of , we getTaking , we get required result.

Theorem 4. Let for , . Then,

The equality holds in each case for the function

Proof. LetThen, from Theorem 1, we havefor each . Thus, for each , we havefor , whereThen, there exists an analytic function of the formwithsuch thatwhereComparing coefficients on both sides of (47), we yieldSince for and is arbitrary, first part of Theorem 4 is complete. Similarly, we can prove (2) and (3).
Now, we investigate sufficient condition for .

Theorem 5. Let withThen, .

Proof. Let . Then by using (50), we haveHence, .

Corollary 1. Let . Ifthen with .

Example 1. By taking and , in the light of Theorem 5, then harmonic polynomialsbelong to .

Example 2. By taking and , in the light of Theorem 5, then harmonic polynomialsbelong to .

Remark 1. The results which we obtained above lead to the results of the classes and which are defined and studied in [2–4, 6], respectively.

Remark 2. The class with is a special case of the class defined in [13] and also our results lead the results of [13].
Now, we investigate convex combinations and convolutions for the class .

Theorem 6. The class of harmonic functions is closed under convex combinations.

Proof. Let, for and ,and convex combination of functions can be defined aswhereand and are the analytic in withNow,showing that .
A sequence of nonnegative real numbers is said to be a convex null sequence, if as andWe need Lemma 4 and Lemma [14] to prove results for convolution.

Lemma 4 (see [15]). If is a convex null sequence, thenis analytic and

Lemma 5. Let . Then, .

Proof. Suppose be given by . Then,which is equivalent towhere Now, for , we consider a sequence defined bySince is a convex null sequence and using Lemma 4, the functionis analytic and in . Writingand using Lemma [14], we get .

Lemma 6. Let , for . Then, .

Proof. Suppose and . Then, the convolution of and is defined bySinceThen, we have Since and using Lemma 5, in .
Now applying Lemma [14] on (71) yieldsin . Thus, .
Now, by using Lemma 6, let us prove that the class is closed under convolutions of its members.

Theorem 7. Let , for . Then,

Proof. LetThen, the convolution of and is defined byin order to show thatWe have to prove thatfor each . By Lemma 6, the class is closed under convolutions for each , for . Then, both and given bybelong to . Since is closed under convex combinations, thenbelongs to Hence, is closed under convolution.
Goodloe [16] defined the Hadamard product of a harmonic function as follows:where and . By considering this Hadamard product of a harmonic function, we investigate following result.

Theorem 8. Let and be such that , for . Then, .

Proof. SupposeThen,for each . Using Theorem 1 and in order to show that , we need to show thatfor each . Write , and Since andin . Lemma [14] proves that .

Corollary 2. Let and . Then, .

Proof. Suppose . Then, for . Theorem 8 concludes that .

3. Conclusion

Various new subclasses of normalized harmonic functions have been defined in open unit disk , satisfying second-order and third-order differential inequalities. In this study, we defined a new class of normalized harmonic functions in open unit disk , satisfying a fourth-order differential inequality. We gave some useful results such that close-to-convexity, coefficient bounds, growth estimates, sufficient coefficient condition, and convolution for the functions belong to this new class of harmonic functions. Further using the concepts of fourth-order differential inequality, all these problems can be studied for classes of meromorphic harmonic functions, Bazilevic harmonic functions, and for valent harmonic functions as well.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally contributed to this study.

Acknowledgments

This study was supported by UKM (GUP-2019-032).