Abstract
We preface and examine classes of (p, q)-convex harmonic locally univalent functions associated with subordination. We acquired a coefficient characterization of (p, q)-convex harmonic univalent functions. We give necessary and sufficient convolution terms for the functions we will introduce.
1. Introduction
First, let us give the basic definitions and notations that we will use in our article. In order not to spoil the generality of this study, let us denote the continuous complex valued harmonic functions class with which are harmonic in the open unit disk and let be the subclass of which consists of functions that are analytic in A function harmonic in can be written in the form where and are analytic in Here, the analytic part of the function is and the co-analytical part is . The necessary and sufficient condition for to be locally univalent and the sense-preserving in is that ([1]). In the light of this information, we can write without losing generality as follows:
Let us denote the class of functions satisfying which are harmonic, univalent, and sense-preserving in for which conditions with . From this point of view, we can easily say that if there is a sense-preserving feature.
In 1984 Clunie and Sheil-Small [1] defined and analyzed characteristic features of the class . Over the years, many articles on the class of and its subclasses have been made by many researchers by referring to this article.
Many studies have been done on quantum calculus. As the importance of this subject can be understood from its multidisciplinary nature, it is known to be innovative and important in many fields. The quantum calculus is also known as -calculus. We can roughly define this calculus as the traditional infinitesimal calculus. In fact, Euler and Jacobi first started to study the subject of -calculus, they are also people who find many attractive implementations in several fields of mathematics and other sciences.
In the last study by Sahai and Yadav [2], the quantum calculus was based on two parameters . In fact, this two-parameter definition is the postquantum calculus denoted by -calculus, which is the generalization of -calculus. We will use the definition of -calculus in this article as the basis of the article published by Chakrabarti and Jagannathan in 1991 [3]. Let , for any nonnegative integer , the -integer number , denoted by is
It can be seen that this twin-basic number defined above is a generalization of the -number defined as follows:
In like manner, the -differential operator of a function , analytic in is defined by
It may be easily shown that as and . We can easily see that
For more information and details on -calculus and -calculus, [2, 4] can be used as references. Apart from these, different studies have also been carried out [5–7].
Ismail et al. [8] and Ahuja et al. and Ahuja and Ҫetinkaya [6, 9]z) used - calculus in the theory of analytic univalent functions. The -difference operator’s definition is where and are of the form which given in [4] and we get the following result for same and
For , let
We say that an analytic function is subordinate to an analytic function and write , if there exists a complex valued function which maps into oneself with , such that
Additionally, if is univalent in , then we can give the following result:
Denote by the subclass of consisting of functions of the form (1) that satisfy the condition where is and are defined by (8) and (9).
By suitably specializing the parameters, the classes reduce to the various subclasses of harmonic univalent functions. That is, by assigning special values instead of ,, , and , we are saying that they become classes that used to be studied. This is an indication that this article is a general subclass that includes other classes in harmonic functions. Such as (i) for ([10])(ii) for and ([11]),(iii) for ([12]),(iv) for and ([6, 13]),(v) for , and ([14, 15]),(vi) for , and ([16]).
Using the method that used by Dziok et al. [11, 17–19] we find necessary and sufficient conditions for the above defined class .
2. Main Results
For functions and of the form we define the Hadamard product of and by
In the first theorem, we introduce a sufficient coefficient bound for harmonic functions in
Theorem 1. Let us first assume that Then, if and only if where
Proof. Let be of the form (1). Then if and only if it satisfies (11) or equivalently where and Since the inequality (17) yields
Theorem 2. Let be given by (1). If where , then,
Proof. if and only if there exists a complex valued function ; such that or equivalently The above inequality (20) holds, since for , we obtain The harmonic function where shows that the coefficient bound given by (20) is sharp. The functions of the form are in because
Denote by the subclass of consisting of functions of the form (1) that satisfy the inequality (20). It is clear that .
Theorem 3. The class is closed under convex combination.
Proof. For , suppose that where Then, by Theorem 2., For we can write the convex combination of as follows: Then, by (9), hence
3. Conclusions
As a result, a general subclass has been defined in this article. Thus, with this study, which will be a good reference for the new results to be obtained, a subclass study has been made for harmonic functions using the derivative, which is still popular today.
Data Availability
There is no data available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.