Certain Properties of Harmonic Functions Deﬁned by a Second-Order Differential Inequality

: The Theory of Complex Functions has been studied by many scientists and its application area has become a very wide subject. Harmonic functions play a crucial role in various ﬁelds of mathematics, physics, engineering, and other scientiﬁc disciplines. Of course, the main reason for maintaining this popularity is that it has an interdisciplinary ﬁeld of application. This makes this subject important not only for those who work in pure mathematics, but also in ﬁelds with a deep-rooted history, such as engineering, physics, and software development. In this study, we will examine a subclass of Harmonic functions in the Theory of Geometric Functions. We will give some deﬁnitions necessary for this. Then, we will deﬁne a new subclass of complex-valued harmonic functions, and their coefﬁcient relations, growth estimates, radius of univalency, radius of starlikeness and radius of convexity of this class are investigated. In addition, it is shown that this class is closed under convolution of its members.


Introduction
In this section, some definitions and necessary information that we will use in this paper will be given.After these definitions, we will give some important properties about harmonic functions and introduce the representations of a few subclasses.
When |_| < 1 for _ ∈ C, notation D is called open unit disk.Here, C is complex number set and $ and are analytic in D. Let H be the class of complex-valued harmonic functions 3 = $ + in the open unit disk D, normalized so that $(0) = 0 = (0), $ (0) = 1.From this point of view, we can give the definition of H 0 = 3 = $ + ∈ H : (0) = 0 for the H 0 class.Every function 3 ∈ H 0 has the canonical representation of a harmonic function 3 = $ + in the open unit disk D as the sum of an analytic function $ and the conjugate of an analytic function .The power series expansions of $ and functions be defined as follows: In this case, the functions in Relation (1) are analytic on the open unit disk.SH 0 , just as the class A of analytic and normalized functions in the open unit disk D is a subclass of H 0 .
Let K, S * and CK be the subclasses of S mapping D onto convex, starlike and closeto-convex domains, respectively, just as KH 0 , SH * ,0 and CKH 0 are the subclasses of SH 0 mapping the open unit disk to their respective domains.
The classes introduced above have been studied and developed by many researchers.One of these researchers, Ponussamy et al. [3], introduced the following class in 2013: PH 0 = 3 ∈ H 0 : Re $ (_) > (_) for _ ∈ D and they proved that functions in PH 0 are close-to-convex.After this study, the following subclass definition has been made using this class and some important features such as coefficient bounds, growth estimates, etc., are examined by Ghosh and Vasudevarao [4]: Nagpal and Ravichandran [5] studied a special version of subclass WH 0 of functions which is a harmonic analogue of the class W defined by Chichra [6] consisting of functions In 1977, Chichra [6] studied the class G(α) for some α ≥ 0, where G(α) consists of Recently, Liu and Yang [7] defined a class where α ≥ 0, 0 < r ≤ 1.Also, Rajbala and Prajapat [8] studied such properties of the class WH 0 (δ, λ) of func- tions 3 ∈ H 0 satisfies the following inequality: Apart from all these past studies, there are many ongoing studies today.For important studies that we can use as references in this article, References [5,[9][10][11][12] can be consulted.

The Sharp Coefficient Estimates and Growth Theorems of WH WH WH 0 (α, β)
In this section, we will examine the WH 0 (α, β) class.
The first theorem is about the conditions under which a given function will belong to the WH 0 (α, β) class, and what properties a function in the WH 0 (α, β) class has.

Now, let us examine the coefficient relation of the co-analytical part of a function
The result is sharp and equality applies to the function Proof.Let us assume that the function 3 defined in type (1) belongs to class WH 0 (α, β).
Using the series representation of (ρe iϕ ), 0 ≤ ρ < 1 and ϕ ∈ R, we derive Allowing ρ → 1 − , we prove the inequality (4).Moreover, the equality is achieved for The following theorem, which allows us to understand the relationship between the coefficients of the function 3, also allows us to solve the problem of finding an upper bound for the coefficients of the functions in the WH 0 (α, β) class.Theorem 3. Let 3 be a function of type 3 = $ + in WH 0 (α, β) class.Then, for 9 ≥ 2, we have All boundaries are sharp here.Conditions of equality for all boundaries are satisfied if Proof.(i) Let us assume that the function 3 defined in type (1) belongs to class WH 0 (α, β).
Then, from Theorem 1, σ = $ + σ ∈ W 0 (α, β) for each σ (|σ| = 1).Thus, for each From here, we see that there exists an analytical function p of the type If we equate the coefficients in Equation ( 5), we obtain the following relation According to Caratheodory (for detailed information, see [20]), since |p 9 | ≤ 2 for 9 ≥ 1, and σ (|σ| = 1) is arbitrary, the proof of the first inequality is thus completed.The proof can be completed by using the method used in the first proof in other parts of the theorem.In all cases, the state of equality is provided by the function Proof.Let us assume that the function 3 defined in type (1) belongs to class H 0 .Then, using (6), The following theorem determines the lower and upper bounds for the modulus of the function 3.

Theorem 5. Let 3 be a function of type
The result is sharp and equalities apply to the function Proof.Let 3 be a function of type 3 = $ + in class WH 0 (α, β).Then, using Theorem 1, σ = $ + σ ∈ W 0 (α, β) , and for each |σ| = 1, we have Re{p(_)} > 0, where If we then apply the method used by Rosihan et al. (Theorem 2.1 [16]), we get the following result where u and v be two nonnegative real constants satisfying Thus, _s u is written instead of ω and, after a few operations, is obtained.We say that an analytic function f is subordinate to an analytic function g, and write f ≺ g, if there exists a complex valued function which maps D into oneself with (0) = 0, such that f (z) = g( (z)) (z ∈ D).Where ≺ shows subordination symbol, on the other hand, since Re{p(_)} > 0, then p(_) and . Using Equality (7), we obtain especially, we obtain

Geometric Properties of Harmonic Mappings in WH WH WH 0 (α, β)
In this section, we will examine the geometric properties of the functions in the WH 0 (α, β) class.We shall provide the radius of univalency, starlikeness and convexity for functions belonging to the class WH 0 (α, β).Let us consider and remember the three lemmas that will guide us in the theorems given in this section and shed light on the proofs.Lemma 1 (Corollary 2.2 [21]).Let 3 = $ + be a sense-preserving harmonic mapping in the open unit disk.If for all σ (|σ| = 1), the analytic functions σ = $ + σ are univalent in D, then 3 is univalent in D.

Convex Combinations and Convolutions
In this section, we investigate that the class WH 0 (α, β) is convolutions and closed under convex combinations of its members.Theorem 9.The class WH 0 (α, β) is closed under convex combinations.
A sequence {λ 9 } ∞ 9=0 of non-negative real numbers is said to be a convex null sequence, if λ 9 → 0 as 9 → ∞, and The following lemmas are needed to complete the proof.

Discussion
In this research, we examine some specific properties for harmonic functions defined by a second-order differential inequality.First, we gave the necessary definitions and preliminary information.Then, we define and prove the coefficient relations and growth theorems for the WH 0 (α, β) class.Then, we examined the geometric properties of the harmonic mappings belonging to the WH 0 (α, β) class.Finally, we proved the theorems about convex combinations and convolutions.Today, it is known that harmonic functions have a very wide field of study.Moreover, it is known that application areas are used by different disciplines.With this study, we aim to shed light on studies in other disciplines.We think that the results of this study, which will be used by many researchers in the future, will connect with different disciplines.In addition to all these, this study will act as a bridge between the articles written in the past and the articles to be written in the future.

Figure 1 .
Figure 1.Image of the unit disk under the 3 function.

Figure 2 .
Figure 2. Image of the unit disk under the 3 function.

Lemma 6 .
Let the function p be analytic in the open unit disk D with p(0) = 1 and Re[p(_)] > 1/2 in the open unit disk D.Then, for any analytic function in D, the function p * takes values in the convex hull of the image of D under .Let ∈ W 0 (α, β), then Re (_)