Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction ()
1. Introduction
In-depth research into the properties of harmonic mappings is beneficial for addressing various problems encountered in the field of engineering. Generally, the linear combinations of harmonic mappings often fail to preserve the original characteristics. For instance, the linear combinations of two convex harmonic mappings may not necessarily be convex, in some cases, may not even be univalent. One can refer to the survey paper by Campbell in [1] . Therefore, a thorough investigation into the univalency and convexity properties of the combinations of some harmonic mappings becomes crucial.
A continuous complex-valued function
is said to be harmonic in the open unit disk
if u and v are real-value harmonic function in
. Such harmonic mappings can be written as
where h, g are both analytic in
. The equation
is called the dilatation of a harmonic mapping
. Lewy [2] has proved that the harmonic mapping
is locally univalent and sense-preserving in
if
for all
.
Let
denote the class of all locally univalent and sense-preserving harmonic mappings, and
be the subclass of
which normalized by the conditions
.
A domain
is said to be convex in the direction
(
) if for all
, the set
is either connected or empty. Specifically, a domain is convex in the vertical direction if for every line parallel to the imaginary axis has a connected intersection with Ω. A function
is called convex in the vertical direction if it maps
onto a domain Ω convex in the imaginary axis.
A common way to construct a new function is to take the linear combination of two functions with a real coefficient
. Let
and
be two harmonic mappings, then
(
) is called the linear combination of
and
.
The following Lemma 1.1 called shear construction which was proposed by Clunie and Shile-Small in [3] , can help us to verify the vertical convexity of harmonic mappings.
Lemma 1.1. A locally univalent and sense-preserving harmonic mapping
on
is univalent and maps
onto a domain convex in the direction of
(
) if and only if the analytic function
is univalent and maps
onto a domain convex in the direction of
.
In particular, when the harmonic mapping maps the unit disk to convex along the horizontal direction, then the parameter
, then it can be seen that both the analytic function
and the harmonic mapping
are convex in the horizontal direction. Similarly, for the case that is convex in vertical direction
, then it can be obtained that the analytic function
and the harmonic mapping
are both convex in the vertical direction. This lemma will help us construct harmonic mappings that are convex along some special directions, and extend them to arbitrary directions.
In order to judge the univalency of some harmonic mappings, we need to use the relevant Lemma 1.2 proposed by Rahman, Q. I. in [4] , which is usually called Cohn’s Rule. Converting the judgment of the univalency of the harmonic mappings into the problem of analyzing the distribution of the zeros of a polynomial function.
Lemma 1.2. (Cohn’s Rule) Given a polynomial
of degree n, and let
satisfy the following equation
(1)
Assuming that
has r and s zeros located inside and on the unit disk
, respectively. If the coefficients in the polynomial satisfy
, then we can construct a function
as follows
(2)
Which is of degree
and has
number of zeros inside and on the unit disk
.
The following result due to Hengarther [5] is useful in checking whether an analytic function is convex in the vertical direction.
Lemma 1.3. Suppose
is an analytic function and non-constant in
. Then
(3)
if and only if
1) It is univalent in
;
2) It is convex in the vertical direction;
3) There exist sequences
and
converging to
and
, respectively, such that
(4)
Linear combination of two functions is an important way to construct new harmonic mapping. However, some geometry properties may not exist after their combination [1] . In 2013, Wang, Z. G. [6] et al. derived several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the horizontal direction. In 2016, Kumar, R. [7] et al. proposed a new family of univalent harmonic mappings which map the unit disk onto domains convex in the vertical direction, and he also identifies the conditions under which linear combinations of mappings from this family remain univalency and convex in the vertical direction. In 2018, Long, B. Y. [8] et al. considered the linear combination of two vertical strip mappings with various dilatation, and proved it is convex in the vertical direction. Zireh, A. [9] et al. proved the sufficient conditions for the linear combinations of two slanted half-plane harmonic mappings to be univalent and convex in an arbitrary direction of
. In recent years, Beig, S. [10] et al. have demonstrated that the linear combination of two different kinds of harmonic mappings is univalent and convex in a special direction
, they have further generalized this result to more common cases by setting certain conditions.
In this paper, inspired by the research conducted in [6] [7] [8] [9] [10] , we investigate the linear combinations
, where
represents right half-plane mappings and
represents vertical strip mappings. These harmonic mappings are sheared by the analytic functions respectively given by
(5)
In particular, we demonstrate that
is univalent and convex in the vertical direction under certain conditions. Furthermore, by setting some coefficients such as
be the dilatation of
and
, we establish the sufficient conditions for their combination to be locally univalent and convex in the vertical direction.
2. Main Results
Theorem 2.1. For
, let
and satisfy Equation (5). Then
is convex in the vertical direction for a real constant
if it is locally univalent and sense-preserving.
Proof. Since
(6)
We set
(7)
By setting
and differentiate the Equation (8), we get
(8)
Substituting
into
, we can get
(9)
It is obviously that
for all
, and we can easy to verify that
,
for
. Therefore, by using the minimum principle for harmonic mappings with all
, we get
.
By applying Lemma 1.3, we can determine that analytic function
is convex in the direction of
. According to Lemma 1.1, the harmonic mapping
also convex in the vertical direction.
Theorem 2.2. For
, let
and satisfy Equation (5). Let
be the dilatation of these two harmonic mappings
. Then the linear combination
is convex in the vertical direction for a real constant
if one of the following conditions is met.
1)
.
2)
and
.
Proof. In views of (6), we set
where
,
. Let
be the dilatation of
, we get
(10)
Since
, the above equations give
(11)
Substituting
and
into
, we get
(12)
Let
, we can simplify the Equation (12) to obtain
. This means that
and
is locally univalent and sense-preserving. According to Theorem 2.1, we can conclude that
is convex in the vertical direction. The proof of first condition is complete.
Next, we examine the second condition with
, where
is defined by Equation (2) in Theorem 2.2. Substituting
into the equation
, we have
(13)
After simplify, we can get
(14)
Combining formulas (14) and (12), we can obtain
(15)
Therefore,
, which indicate that
is locally univalent and sense-preserving. By applying Theorem 2.1, we know that
is convex in the vertical direction. This concludes the proof of the second condition.
Theorem 2.3. For
, let
and satisfy Equation (5). Let
and
be the dilatation of
and
respectively. Then, for a real constant
, (
), the linear combination
is convex in the vertical direction.
Proof. Let
be the dilatation of
which satisfy the Equation (10). By setting
,
, and substituting them into Equation (11), we obtain
(16)
Then, by substituting these into
from Equation (10), we can derive
(17)
where
(18)
And
, which satisfy Lemma 1.2. Therefore, let
be a zero of
and
, implying
is a zero of
. With this, we can rewrite (17) as follow
(19)
It is evident that
for
. Thus, we can apply Cohn’s Rule to
, and conclude that all zeros of
lie inside or on
. Consequently, we have
(20)
where
(21)
By simple calculate, we get
(22)
which satisfies the condition of Lemma 1.2. Therefore, we can apply Cohn’s Rule again to reduce the quadratic polynomial to once and judge the size of the function root.
By setting
, substituting it into the Equation (2), we have
(23)
where
(24)
It is obviously that the coefficients given above satisfy
. Therefore, the unique root of
from Equation (23) lies on the unit circle. According to Lemma 1.2, we know that the dilatation of harmonic mapping
satisfies
. Moreover, by Theorem 2.1, we obtain that the linear combination of harmonic mapping
is locally univalent and convex in the vertical direction. Thus, the proof of Theorem 2.3 is completed.
Theorem 2.4. For
, let
and satisfy Equation (5). Considering some special dilatations of two harmonic mappings
and
, by setting
and
respectively. Then, for a real constant
, the linear combination
is convex in the vertical direction if the following inequation holds true
(24)
Proof. Let
be the dilatation of
which satisfies the Equation (10). By setting
,
, and substituting them into Equation (11), we obtain
(25)
Then, by substituting these into
from Equation (10), we can derive
(26)
where
(27)
Similarly, the above Equation (26) satisfies
and the conditions of Lemma 1.2 mentioned earlier. Therefore, let
be the zero of
, and
be the zero of
, and we express
in form of education (19) again. By simple calculations, we have
(28)
which satisfies the condition of Lemma 1.2. Thus, we apply the Cohn’s Rule and get
(29)
where
(30)
Simplify the above equation, we have
(31)
Since the condition
is met, we can use Cohn’s Rule again and get
(32)
where
(33)
For
, we have
(34)
Thus, the unique zero of the above equation will lie inside or on the unit disk
if the inequality (24) is met. According to Lemma 1.2 and Theorem 2.1, we know that harmonic mapping
is locally univalent and convex in the vertical direction. Thus, the proof of Theorem 2.4 is complete.
In recent years, Long B Y [8] et al. introduced a variable dilatation
for harmonic mappings. Therefore, following their setup, in the theorem below, we similarly make a transformation to the dilatation of harmonic mappings.
Theorem 2.5. For
, let
and satisfy Equation (5). Let
and
be the dilatations of two harmonic mappings
and
respectively. Then, for a real constant
, the linear combination
is convex in the vertical direction if
(35)
where
(36)
and
(37)
Proof. Let
be the dilatation of
which satisfies the Equation (10). By setting
,
, substituting them into Equation (11), we have
(38)
where
(39)
It can be easily calculated that the following inequality
(40)
consistently holds for
. Therefore, we can use the Cohn’s Rule and get
(41)
where
(42)
Assuming the inequality
holds, we can use the Cohn’s Rule again and get
(43)
After simplification, we have
(44)
where
(45)
Since
, the zeros of
and
both lie inside or on the unit disk. Therefore,
. By Theorem 2.1, we know that the harmonic mapping
is locally univalent and convex in the vertical direction. Thus, all the proofs in this article have been completed.
3. Examples
In this section, we give several examples to illustrate our main results.
Example 1. Let
and satisfy (5), we consider the linear combination
with
,
and
. Taking
, by shearing we obtain
(46)
Also, taking
,
, we have
(47)
Since
satisfy the condition of Theorem 2.1, the linear combination of harmonic mappings
is convex in the vertical direction. The images of
under
and
are shown in Figure 1, respectively.
Figure 1. Images of
under
and
with
.
Figure 2. Images of
under
and
with
.
Example 2. Let
be the harmonic mappings considered in Theorem 2.3 with
,
,
and
. Taking
, by shearing we obtain
Also, taking
, we have
Let
and
, the harmonic mapping
is convex in the vertical direction. The images of
under
and
are shown in Figure 2, respectively.
4. Conclusions
In the main results, we demonstrate that the combinations of right half-plane mappings and vertical strip mappings are convex in the vertical direction if and only if they are locally univalent. Furthermore, we extend the above theorem to more general cases by imposing two conditions
and
. By considering parameters
and
as the dilatations of these harmonic mappings, respectively. We prove the sufficient conditions that their combinations are locally univalent and convex in the vertical direction.
From the above proofs in this paper, it is evident that the linear combinations of right half-plane mappings and vertical strip mappings are convex in the vertical direction only when specific conditions are met. In future studies, we can set the combination coefficient
to be a complex number, consider the univalency and convexity properties of their combinations. This can extend the content presented by Liu Z. H. and Khurana, D. et al. in [11] [12] . Also, we can use the slanted half-plane harmonic mappings which proposed in [9] , and prove the sufficient conditions that the combinations of slanted half-plane harmonic mappings and vertical-strip mappings are locally univalent and convex in some certain direction. This holds significant implications for the progress of research on harmonic mappings and minimal surfaces.