1. Introduction
A complex-valued harmonic function
⨍
in the open unit disk
is given by
where
and
are analytic in
. We call
h the analytic part and
g is co-analytic part of
. A necessary and sufficient condition for
to be locally univalent and sense-preserving in
is
for all
(see [
1]). Let
,
denote the class of functions
of the form
which are harmonic in
We next denote by
the subclass of functions
which are
m-valent and sense-preserving in
. Then, we say that
is a
m-valently harmonic function in
. The functions in
are harmonic and sense-preserving in
if
in
. The class
of the harmonic univalent and sense-preserving functions was studied in detail by Clunie and Sheil-Small [
1]. Furthermore, note that
reduces to the class
of normalized analytic univalent functions in
, if
Let
and
be the subclasses of
mapping
onto convex and starlike domains, respectively.
In [
2], Owa et al. investigated the class
for
and
and
which consisted of analytic functions
such that
The research described in the present paper is motivated by this work and focuses on analyzing the new class of multivalent harmonic functions that is introduced in the following definition:
Definition 1. Denote by γ
the class of functions and satisfyfor some and where Remark 1. If and , then the class is defined byfor some and This class was studied by Çakmak et al. (see [3]). It is evident that, for and . Here, the following classes are obtained when special values are given to the variables:
([
4]),
([
5,
6]),
([
7]),
([
7]),
([
8]),
([
9]),
([
10]),
([
11]),
for
([
2]) and
([
12]),
([
13]),
([
12]),
([
12]),
([
12]),
([
12,
14]),
([
12]),
([
15]),
([
16]).
In order to better understand the importance of this subclass analysis, the important subclasses of the multivalent harmonic functions are mentioned first. The subclasses of multivalent harmonic functions are used in a wide range of mathematical and physical contexts, and each subclass has its own unique properties and applications. By studying these subclasses, researchers can gain insights into the behavior and structure of multivalent harmonic functions, and develop new techniques and tools for analyzing complex systems. Some important subclasses are quasiconformal maps, modular forms, harmonic morphisms, multidimensional harmonic functions, elliptic functions. For more information about these classes, see [
17,
18,
19,
20,
21,
22,
23,
24,
25].
In
Section 2, the necessary and sufficient conditions are specified for certain functions to belong to the class
. For the functions of the class, coefficient bounds and growth predictions are also given. The starlikeness and convexity characteristics of the class
are examined in
Section 3. In
Section 4, convolution properties involving the functions of the class
are examined.
2. The Sharp Coefficient Estimates and Growth Theorems of
The first theorem of this section provides the necessary and sufficient conditions for the functions to be part of the class. The following theorems concern coefficient estimations and distortion limits regarding this class.
Theorem 1. The mapping if and only if for each
Proof. Suppose that
For each
Thus,
for each
Conversely, let
then
With the convenient choice of
we have
and hence
□
Theorem 2. Let then for where This result is sharp, and equality holds for the following function Proof. Suppose that
Using the series representation of
and
we derive
where
Allowing
we prove the result (
3). Moreover, it can be easily seen that the equality is achieved for the function given in (
4). □
Theorem 3. Let Then, for we havewhere All the results given in this theorem are certain and the equations are provided for the following function Proof. Suppose that
then, from Theorem 1,
for each
Thus, for each
we have
for
Therefore, there exists an analytic function of the form
with a positive real part in
such that
where
Comparing coefficients on both sides of (
5), we obtain
According to Caratheodory [
26], since
for
and
is arbitrary, the proof of (i) is complete. By following the methods in proof (i), proof (ii), and proof (iii) are obtained. The function
provides equations. □
We will now give a sufficient condition for a function to be of class
Theorem 4. Let and where withthen Proof. Suppose that
Then, using (
6),
where
Hence,
□
Theorem 5. Let for and where ThenInequalities are sharp for the function Proof. Let
Then, using Theorem 1,
for each
Thus, there exists an analytic function
with
and
in
, such that
where
. Simplifying (
8), we obtain
where
and
. Therefore, using the Schwarz lemma, we have
and
Similarly, we have
and
Since
is arbitrary, we have (
7). □
3. Geometric Properties of Harmonic Mappings in
In this section, the radius of the m-valently starlikeness and convexity for functions in the class will be provided.
The two following lemmas are used in order to demonstrate the key findings:
Lemma 1 ([
27]).
Let . If , then is m-valently starlike in . Lemma 2 ([
28]).
Let . If , then is m-valently convex in . The starlikeness result is first presented in the next theorem.
Theorem 6. Let be a sense-preserving harmonic mapping in Then, is m-valently starlike in where Proof. Let
and let
be fixed. Then
and
According to Lemma 1, it suffices to show that
for
. Using Theorem 3 (i) gives that
Furthermore, considering that
we know that the inequality (
9) can be written by
Thus, if
for all
, then
□
The radius of convexity for the class is determined in the next theorem.
Theorem 7. Let be a sense-preserving harmonic mapping in Then is m-valently convex in where Proof. Let
and let
be fixed. Then
and
According to Lemma 2, it suffices to show that
for
. Using Theorem 3 (i); gives that
Furthermore, considering that
we know that the inequality (
10) can be written by
Thus, if
for all
, then
□
4. Convex Combinations and Convolutions
In this section, we show that the class is closed under convex combinations and convolutions of its members.
Theorem 8. The class is closed under convex combinations.
Proof. Suppose
for
and
The convex combination of functions
may be written as
where
Then, both
h and
g are analytic in
with
and
showing that
. □
A sequence of non-negative real numbers is said to be a convex null sequence, if as , and We shall require the following Lemmas 3 and 4 to prove the results of the convolution.
Lemma 3 ([
29]).
If be a convex null sequence, then functionis analytic and in Lemma 4 ([
14]).
Let the function ϕ be analytic in with and in Then, for any analytic function, ψ in the function takes the values in the convex hull of the image of under Lemma 5. Let then
Proof. Suppose that
will be given by
then
which is equivalent to
in
where
Now consider a sequence
defined by
It is easy to see that the sequence
is a convex null sequence. Using Lemma 3, this explains the following function
is analytic and
in
Writing
and using Lemma 4 gives that
for
□
Lemma 6. Suppose for Then, ∈
Proof. Let
and
Then, the convolution of
and
is defined by
Since
and
then we have
Since
and using Lemma 5,
in
Now applying Lemma 4 to (
11) yields
in
Thus,
∈
□
Now, using Lemma 6, we prove that the class is closed under the convolutions of its members.
Theorem 9. Let for Then, ∈
Proof. Suppose
. Then, the convolution of
and
is defined as
In order to show that
∈
we need to show that
for each
By Lemma 6, the class
is closed under convolutions for each
for
Then, both
and
given by
are members of
. Since
is closed under convex combinations, then the function
is a member of
. Thus,
is closed under convolution. □
In [
30], Goodloe defined the Hadamard product of a harmonic function
with an analytic function
in
as follows:
Theorem 10. Suppose that and with for then ∈
Proof. Suppose that
then
for each
By Theorem 1, in order to prove that
∈
we need to prove that
∈
for each
Write
as
and
Since and in Lemma 4 yields . □
5. Discussion and Conclusions
As stated in the information provided in the introduction, multivalent harmonic functions are becoming more and more significant. A new class of these functions is defined in this paper and is denoted by in Definition 1. By assigning various values to the variables in the class defined in this study, it is established that numerous subclasses that have already been investigated by many different researchers can be obtained. The coefficient relations, growth theorems, and geometric features of the recently introduced class are examined in the four sections of this research paper.
We believe that, as this article thoroughly covers the topic of multivalent harmonic functions from the past to the present, it will be relevant to numerous future studies. We suggest future studies regarding the theories of differential subordination and superordination involving functions from the class
(γ
as seen for harmonic complex-valued functions in recent papers such as [
31,
32]. Quantum calculus aspects can be introduced in this study, as presented in recent investigations [
33,
34,
35].
Author Contributions
Conceptualization, G.I.O., S.Y. and H.B.; methodology, G.I.O., S.Y. and H.B.; software, G.I.O., S.Y. and H.B.; validation, G.I.O., S.Y. and H.B.; formal analysis, G.I.O., S.Y. and H.B.; investigation, G.I.O., S.Y. and H.B.; resources, G.I.O., S.Y. and H.B.; data curation, G.I.O., S.Y. and H.B.; writing—original draft preparation, S.Y. and H.B.; writing—review and editing, G.I.O., S.Y. and H.B.; visualization, G.I.O., S.Y. and H.B.; supervision, S.Y.; project administration, G.I.O.; funding acquisition, G.I.O. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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