A proof of a conjecture on convolution of harmonic mappings and some related problems
UDC 517.5 Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for $n=1,2,3$ and $4$. Also, it has been proved only for $\beta=\pi/2$. In this paper, by using of a new...
Gespeichert in:
| Datum: | 2021 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2021
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/94 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507122116067328 |
|---|---|
| author | Yalçın, S. Ebadian , A. Azizi, S. Yalçın, Sibel Yalçın, S. Ebadian , A. Azizi, S. |
| author_facet | Yalçın, S. Ebadian , A. Azizi, S. Yalçın, Sibel Yalçın, S. Ebadian , A. Azizi, S. |
| author_sort | Yalçın, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:28Z |
| description | UDC 517.5
Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for $n=1,2,3$ and $4$. Also, it has been proved only for $\beta=\pi/2$. In this paper, by using of a new method, we settle this conjecture in the affirmative for all $n\in\mathbb{N}$ and $\beta\in(0,\pi)$. Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new method simplifies calculations and shortens the proof of results remarkably. |
| doi_str_mv | 10.37863/umzh.v73i2.94 |
| first_indexed | 2026-03-24T02:04:17Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i2.94
UDC 517.5
S. Yalçın (Bursa Uludag Univ., Turkey),
A. Ebadian (Urmia Univ., Iran),
S. Azizi (Payame Noor Univ., Tehran, Iran)
A PROOF OF A CONJECTURE ON CONVOLUTION OF HARMONIC
MAPPINGS AND SOME RELATED PROBLEMS
ДОВЕДЕННЯ ГIПОТЕЗИ ПРО ЗГОРТКУ
ГАРМОНIЧНИХ ВIДОБРАЖЕНЬ ТА ДЕЯКI ПОВ’ЯЗАНI ЗАДАЧI
Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with
a vertical strip mapping. They have verified the above conjecture for n = 1, 2, 3 and 4. Also, it has been proved only
for \beta = \pi /2. In this paper, by using of a new method, we settle this conjecture in the affirmative for all n \in \BbbN and
\beta \in (0, \pi ). Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new
method simplifies calculations and shortens the proof of results remarkably.
Нещодавно Kumar та iн. запропонували гiпотезу щодо згортки узагальнених вiдображень правої пiвплощини з
вiдображеннями вертикальної смуги. Вони перевiрили цю гiпотезу для n = 1, 2, 3 та 4. Крiм цього, гiпотезу
було доведено тiльки для \beta = \pi /2. Використовуючи новий метод, ми доводимо цю гiпотезу для всiх n \in \BbbN
та \beta \in (0, \pi ). Бiльш того, за допомогою цього методу ми отримали деякi результати щодо згортки гармонiчних
вiдображень. Новий метод спрощує обчислення та значно скорочує доведення результатiв.
1. Introduction. Let \scrH denote the class of all complex-valued harmonic functions f = h + g
in the unit disk \BbbD = \{ z \in \BbbC : | z| < 1\} , where h and g are analytic in \BbbD and normalized by
h(0) = g(0) = 0 = h\prime (0) - 1. We call h and g, the analytic and the co-analytic parts of f,
respectively, and have the following power series representation:
h(z) = z +
\infty \sum
n=2
anz
n and g(z) =
\infty \sum
n=1
bnz
n, z \in \BbbD .
A function f \in \scrH is locally univalent and sense-preserving in \BbbD if Jf (z) > 0 for all z in \BbbD , where
the Jacobian of f = h+ g is given by
Jf (z) = | h\prime (z)| 2 - | g\prime (z)| 2.
Using a result of Lewy [12] and the inverse function theorem, one obtains that Jf (z) > 0 is a
necessary and sufficient condition for f \in \scrH to be locally univalent and sense-preserving in \BbbD .
Consequently, f = h + g \in \scrH is sense-preserving in \BbbD if and only if | w(z)| < 1, where w(z) =
=
g\prime (z)
h\prime (z)
is the analytic dilatation of f = h + g. For many basic results on univalent harmonic
mappings, see [3, 7, 18]. Denote by \scrS H the class of all sense-preserving harmonic univalent map-
pings f = h+ g \in \scrH and by \scrS 0
H the class of functions f \in \scrS H such that fz(0) = 0. We denote by
\scrK 0
H and S\ast 0
H the subclasses of S0
H whose functions map \BbbD onto convex and starlike domains.
A domain \Omega \subset \BbbC is said to be convex in the direction \gamma , 0 \leq \gamma < \pi , if every line parallel to the
the line joining 0 to ei\gamma has a connected intersection with \Omega . In particular, if \gamma = 0, we say that \Omega
is convex in horizontal direction (CHD).
c\bigcirc S. YALÇIN, A. EBADIAN, S. AZIZI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2 283
284 S. YALÇIN, A. EBADIAN, S. AZIZI
Let \{ f\beta \} , where f\beta = h\beta +g\beta , be the collection of those harmonic mappings which are obtained
by shearing of analytic vertical strip mappings
h\beta (z) + g\beta (z) =
1
2i \mathrm{s}\mathrm{i}\mathrm{n}\beta
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1 + zei\beta
1 + ze - i\beta
\biggr)
, 0 < \beta < \pi , (1.1)
with suitable dilatations (see [4]).
If
f(z) = h(z) + g(z) = z +
\infty \sum
n=2
anz
n +
\infty \sum
n=1
bnzn
and
F (z) = H(z) +G(z) = z +
\infty \sum
n=2
Anz
n +
\infty \sum
n=1
Bnzn,
then the \itc \ito \itn \itv \ito \itl \itu \itt \iti \ito \itn f \ast F is defined to be the function
(f \ast F )(z) = (h \ast H)(z) + (g +G)(z) = z +
\infty \sum
n=2
anAnz
n +
\infty \sum
n=1
bnBnzn.
The properties of the harmonic convolutions are not as nice as that of the analytic functions. For
example, harmonic convolution of two mappings from class \scrK 0
H , is not necessarily in \scrK 0
H . In view
of this statement, a good number of papers appeared on this topic (see, for example, [5, 6, 9, 10, 14]).
In particular, the following results were obtained by Dorff [5].
Theorem 1.1 [5]. Let f = h+ g \in \scrK 0
H with h(z) + g(z) = z/(1 - z). Then f \ast f\beta \in \scrS 0
H and
is CHD, provided f \ast f\beta is locally univalent and sense-preserving, where f\beta is given by (1.1).
Theorem 1.2 [5]. Let f1 = h1+g1 \in \scrS 0
H and f2 = h2+g2 \in \scrS 0
H with hi(z)+gi(z) = z/(1 - z)
for i = 1, 2. If f1 \ast f2 is locally univalent and sense-preserving, then f1 \ast f2 \in \scrS 0
H and is convex in
the horizontal direction.
In [9], Kumar et al. defined harmonic right half-plane mappings Fa = Ha + Ga such that
Ha(z) +Ga(z) = z/(1 - z) with dilatations wa(z) = (a - z)/(1 - az), - 1 < a < 1. Clearly, for
a = 0 the mapping Fa reduces to the standard right half-plane mapping.
Recently, Kumar et al. [10] studied the harmonic convolution of mapping f\beta with the mapping
Fa and posed the following conjecture.
Conjecture A. Let f\beta = h\beta + g\beta be the harmonic mappings given by (1.1) with dilatation
w(z) = ei\varphi zn, \varphi \in \BbbR . Then Fa \ast f\beta \in \scrS 0
H and is CHD for all n \in \BbbN provided a \in [(n - 2)/(n+
+ 2), 1).
They proved the above conjecture for n = 1, 2, 3, 4. Also, in [11], it has been verified only for
\beta = \pi /2. In this paper (see Theorem 3.1), by using of a new method, we settle this conjecture in the
affirmative for all n \in \BbbN and any \beta \in (0, \pi ) . Namely, we prove that this conjectue is true for all
for all n \in \BbbN provided a \in [(n - 2)/(n+ 2), 1) and any \beta \in (0, \pi ).
Note that in the most of papers, Cohn rule and Schur – Cohn algorithm play central role to prove
the obtained results on the harmonic convolution (see [6, 9 – 11, 13, 14]). For example, the following
results were proved by using of Cohn rule and Schur – Cohn algorithm.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
A PROOF OF A CONJECTURE ON CONVOLUTION OF HARMONIC MAPPINGS . . . 285
Theorem B [9]. If fn = h+g is the right half-plane mapping given by h(z)+g(z) = z/(1 - z)
with w(z) = ei\varphi zn, \varphi \in \BbbR , n \in \BbbN , then Fa \ast fn \in \scrS 0
H is CHD for a \in [(n - 2)/(n+ 2), 1).
Theorem C [15]. Let f = h + g \in \scrS 0
H with h(z) + g(z) = z/(1 - z) and w(z) = - z(z +
+a)/(1+az), then F0 \ast f \in \scrS 0
H and is convex in the horizontal direction for a = 1 or - 1 \leq a \leq 0.
In this paper, we will use a new method to prove the above results which remarkably simplifies
the calculation and shortens the proof of results compared with Cohn rule and Schur – Cohn algorithm
and this is an advantage.
A more general class of harmonic univalent mappings, Lc = Hc + Gc, c > 0, was defined by
Muir [17]
Lc(z) = Hc(z) +Gc(z) =
1
1 + c
\biggl[
z
1 - z
+
cz
(1 - z)2
\biggr]
+
1
1 + c
\biggl[
z
1 - z
+
cz
(1 - z)2
\biggr]
. (1.2)
Clearly, for c = 1, we obtain the standard right half-plane mapping.
In view of Lemmas 2.1 and 2.2 in [16], similar to the approach used in the proof of Theorem 3.1
(or Conjecture A), we get the following result which solves the problem 4.4 proposed in [16].
Theorem 1.3. Let Lc = Hc + Gc \in \scrK 0
H be a mapping given by (1.2). If f\beta = h\beta + g\beta is
given by (1.1) with dilatation w(z) = ei\varphi zn, \varphi \in \BbbR , n \in \BbbN , then Lc \ast f\beta \in \scrS 0
H and is CHD for
0 < c \leq 2/n.
2. Preliminaries. The following lemmas will be required in the proof of our main results.
Lemma 2.1. For \eta > 1 and n \in \BbbN , we have \eta n > \eta n - 1 + \eta n - 2 + . . .+ \eta + 1.
Proof. By mathematical induction the proof is easy, so we skip the details.
Lemma 2.2 (see [1] and also [8]). Let p(z) = zn +
\sum n - 1
j=0
ajz
j , be a complex polynomial.
Then all the zeros of p(z) lie in the disk
\{ z : | z| < \eta \} \subset \{ z : | z| < 1 +A\} ,
where
A = \mathrm{m}\mathrm{a}\mathrm{x}
0\leq j\leq n - 1
| aj |
and \eta is the unique positive root of the real-coefficient polynomial
Q(x) = xn - | an - 1| xn - 1 - | an - 2| xn - 2 - . . . - | a1| x - | a0| .
In view of Lemma 2.2, we obtain the result stated below which play a central role in proofs of
our results in this paper.
Corollary 2.1. If | a0| \leq 1, | a1| \leq 1, . . . , | an - 1| \leq 1, then all the zeros of complex polynomial
p(z) = zn +
\sum n - 1
j=0
ajz
j lie in the unit disk \{ z : | z| < \eta \leq 1\} .
Proof. In contrary, let \eta > 1, where \eta is the unique positive root of the real-coefficient
polynomial
Q(x) = xn - | an - 1| xn - 1 - | an - 2| xn - 2 - . . . - | a1| x - | a0| .
Namely, \eta > 1 such that Q(\eta ) = 0. So,
\eta n = | an - 1| \eta n - 1 + | an - 2| \eta n - 2 + . . .+ | a1| \eta + | a0| \leq
\leq \eta n - 1 + \eta n - 2 + . . .+ \eta + 1.
This contradicts Lemma 2.1. Then \eta \leq 1 and in view of Lemma 2.2 we get the desired result.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
286 S. YALÇIN, A. EBADIAN, S. AZIZI
Lemma 2.3 (see [10], Lemma 2.1). If f\beta = h\beta + g\beta is the mapping given by with dilatation
w = g\prime \beta /h
\prime
\beta , then \widetilde w, the dilatation of Fa \ast f\beta , is given by
\widetilde w(z) = 2w(1 + w)(a+ az \mathrm{c}\mathrm{o}\mathrm{s}\beta + z \mathrm{c}\mathrm{o}\mathrm{s}\beta + z2) - zw\prime (1 - a)(1 + 2z \mathrm{c}\mathrm{o}\mathrm{s}\beta + z2)
2(1 + z \mathrm{c}\mathrm{o}\mathrm{s}\beta + az \mathrm{c}\mathrm{o}\mathrm{s}\beta + az2)(1 + w) - zw\prime (1 - a)(1 + 2z \mathrm{c}\mathrm{o}\mathrm{s}\beta + z2)
. (2.1)
Lemma 2.4 (see [9], Eq. (4)). If f = h + g \in \scrS 0
H is right half-plane mapping, where h(z) +
+ g(z) = z/(1 - z) with dilatation w = g\prime /h\prime (h\prime (z) \not = 0, z \in \BbbD ), then \widetilde w2, the dilatation of Fa \ast f,
is given by
\widetilde w1(z) =
2(a - z)w(1 + w) + (a - 1)w\prime z(1 - z)
2(1 - az)(1 + w) + (a - 1)w\prime z(1 - z)
. (2.2)
Lemma 2.5 (see [6], Eq. (6)). If f = h+ g \in \scrS 0
H with h(z) + g(z) = z/(1 - z) and dilatation
w = g\prime /h\prime , then the dilatation F0 \ast f is given by
\widetilde w2(z) = - z
w2 +
\biggl[
w - 1
2
w\prime z
\biggr]
+
1
2
w\prime
1 +
\biggl[
w - 1
2
w\prime z
\biggr]
+
1
2
w\prime z2
. (2.3)
3. Main results. In the following result, we prove Conjecture A.
Theorem 3.1. If f\beta = h\beta + g\beta is the harmonic mapping obtained from the relation (1.1) with
dilatation w(z) = ei\varphi zn, \varphi \in \BbbR , n \in \BbbN , then Fa \ast f\beta \in \scrS 0
H and is CHD for a \in
\biggl[
n - 2
n+ 2
, 1
\biggr)
.
Proof. By Theorem 1.1, it suffices to show that Fa \ast f\beta is locally univalent and sense-preserving
or equivalently the dilatation \widetilde w of Fa \ast f\beta satisfies | \widetilde w(z)| < 1 for all z \in \BbbD . Setting w(z) = ei\varphi zn
in (2.1), we obtain
\widetilde w(z) = zne2i\varphi
p(z)
p\ast (z)
,
where
p(z) = zn+2 + (a+ 1) \mathrm{c}\mathrm{o}\mathrm{s}\beta zn+1 + azn +
1
2
(2 + an - n)e - i\theta z2+
+[(a(1 + n) + 1 - n) \mathrm{c}\mathrm{o}\mathrm{s}\beta ]e - i\theta z +
1
2
(2a+ an - n)e - i\theta
and
p\ast (z) =
1
2
(2a+ an - n)ei\theta zn+2 + [(a(1 + n) + 1 - n) \mathrm{c}\mathrm{o}\mathrm{s}\beta ]ei\theta zn+1+
+
1
2
(2 + an - n)e - i\theta zn + az2 + (a+ 1) \mathrm{c}\mathrm{o}\mathrm{s}\beta z + 1
such that p\ast (z) = zn+2p(1/z).
Clearly, if z0, z0 \not = 0, is a zero of p then
1
z0
is a zero of p\ast . Hence, if \alpha 1, \alpha 2, . . . , \alpha n+2 are the
zeros of p (not necessarily distinct), then we can write
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
A PROOF OF A CONJECTURE ON CONVOLUTION OF HARMONIC MAPPINGS . . . 287
\widetilde w(z) = zne2i\varphi
(z - \alpha 1)
(1 - \alpha 1z)
(z - \alpha 2)
(1 - \alpha 2z)
. . .
(z - \alpha n+2)
(1 - \alpha n+2z)
for | \alpha i| \leq 1, since
z - \alpha i
1 - \alpha iz
maps the closed unit disk onto itself. Therefore, to prove that | \widetilde w(z)| < 1
in \BbbD , we will show that all the zeros of polynomial p, i.e., \alpha 1, \alpha 2, . . . , \alpha n+2 lie inside or on the
unit circle | z| = 1 for a \in [(n - 2)/(n + 2), 1). We obviously have a0 =
1
2
(2a + an - n)e - i\theta ,
a1 = [(a(1+ n) + 1 - n) \mathrm{c}\mathrm{o}\mathrm{s}\beta ]e - i\theta , a2 =
1
2
(2+ an - n)e - i\theta , a3 = a4 = . . . = an - 1 = 0, an = a,
an+1 = (a+ 1) \mathrm{c}\mathrm{o}\mathrm{s}\beta and an+2 = 1.
For a \in
\biggl[
n - 2
n+ 2
, 1
\biggr)
, we can easily see that
| a0| \leq 1, | a2| \leq 1, . . . , | an| \leq 1.
Also, if a \in
\biggl(
n - 1
n+ 1
, 1
\biggr)
we observe that | a1| < | a0| \leq 1 and if a \in
\biggl[
n - 2
n+ 2
,
n - 1
n+ 1
\biggr)
, we have
| a1| < | a2| \leq 1
\Bigl(
for a =
n - 1
n+ 1
, separately, it is clear that | a1| < 1
\Bigr)
.
So, Corollary 2.1 implies that all the zeros of polynomial
q(z) = p(z) - an+1z
n+1 =
= zn+2 + azn +
1
2
(2 + an - n)e - i\theta z2 + [(a(1 + n) + 1 - n) \mathrm{c}\mathrm{o}\mathrm{s}\beta ]e - i\theta z +
1
2
(2a+ an - n)e - i\theta
lie inside the unit circle | z| = 1 (namely in the unit disk \BbbD ).
On the other hand, since p(z) = q(z) + an+1z
n+1, then
p(z)
q(z)
= 1 +
an+1z
n+1
q(z)
and this approaches 1 as z goes to infinity. So, there is a sufficiently large number R with\bigm| \bigm| \bigm| \bigm| p(z)q(z)
- 1
\bigm| \bigm| \bigm| \bigm| < 1
for | z| = R, that is, | p(z) - q(z)| < | q(z)| for | z| = R. Now, the application of Rouché’s theorem
(see [2]) allows us to conclude that all the zeros of polynomial p lie in the unit disk \BbbD for a \in
\in [(n - 2)/(n+ 2), 1) and this completes the proof.
Similar to the method used in he proof of the above theorem, by using of Corollary 2.1, Theorem
1.2 and the relation (2.2), we get Theorem B. So, we skip the proof.
Theorem 3.2. Let f = h+g \in \scrS 0
H with h(z)+g(z) = z/(1 - z) and w(z) = - z(z+a)/(1+az).
Then F0 \ast f \in \scrS 0
H and is convex in the horizontal direction for a = 1 or - 1 \leq a \leq 0.
Proof. By Theorem 1.2, we need only to prove that F0 \ast f is locally univalent and sense-
preserving. Setting \widetilde w2(z) = - z(z + a)/(1 + az) in (2.3) and simplifying, we obtain
\widetilde w2(z) = z
z3 +
2 + 3a
2
z2 + (1 + a)z + a/2
1 +
2 + 3a
2
z + (1 + a)z2 + (a/2)z3
= z
\varphi (z)
\varphi \ast (z)
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
288 S. YALÇIN, A. EBADIAN, S. AZIZI
where
\varphi (z) = z3 +
2 + 3a
2
z2 + (1 + a)z + a/2.
To prove that | \widetilde w2(z)| < 1 in \BbbD , it suffices to show that all the zeros of \varphi lie inside | z| = 1 or
on | z| = 1. If a = 1, then \varphi (z) = z3 +
2 + 3a
2
z2 + (1 + a)z + a/2 =
1
2
(1 + z)2(1 + 2z) has all
its zeros in \BbbD . The repeated application of Corollary 2.1 (as in the proof of Theorem 3.1) shows that
this is in fact true, also for - 1 \leq a \leq 0. We skip the details. This completes the proof.
The new method used in the proof of above theorems can be applied to prove many problems in
convolutions of univalent harmonic mappings. For example, we can derive the following theorem.
We skip the details for similarity.
Theorem 3.3 ([15], Theorem 1.2). Let Lc = Hc + Gc \in \scrK 0
H be a mapping given by (1.2). If
Fa = Ha+Ga is the right half-plane mapping, then Lc\ast Fa is univalent and convex in the horizontal
direction for 0 < c \leq 2(1 + a)/(1 - a).
References
1. A. Cauchy, Exercises de mathematique, Oeuvres (2), 9 (1829).
2. J. Conway, Functions of one complex variable, Second Ed., Vol. I, Springer-Verlag (1978).
3. M. Dorff, Anamorphosis, mapping problems, and harmonic univalent functions, Explorat. Complex Anal., 197 – 269
(2012).
4. M. Dorff, Harmonic univalent mappings onto asymmetric vertical strips, Comput. Methods and Funct. Theory,
171 – 175 (1997).
5. M. Dorff, Convolution of planar harmonic convex mappings, Complex Var. Theory and Appl., 45, № 3, 263 – 271
(2001).
6. M. Dorff, M. Nowak, M. Wołoszkiewicz, Convolution of harmonic convex mappings, Complex Var. Elliptic Equat.,
57, № 5, 489 – 503 (2012).
7. P. Duren, Harmonic mappings in the plane, Cambridge Tracts Math., 156, Cambridge Univ. Press, Cambridge (2004).
8. R. B. Gardner, N. K. Govil, Eneström – Kakeya theorem and some of its generalizations, Current Topics in Pure and
Comput. Complex Anal., 171 – 199 (2014).
9. R. Kumar, M. Dorff, S. Gupta, S. Singh, Convolution properties of some harmonic mappings in the right half-plane,
Bull. Malays. Math. Sci. Soc., 39, № 1, 439 – 455 (2016).
10. R. Kumar, S. Gupta, S. Singh, M. Dorff, An application of Cohn’s rule to convolutions of univalent harmonic mappings,
Rocky Mountain J. Math., 46, № 2, 559 – 570 (2016).
11. R. Kumar, S. Gupta, S. Singh, M. Dorff, On harmonic convolutions involving a vertical strip mapping, Bull. Korean
Math. Soc., 52, № 1, 105 – 123 (2015).
12. H. Lewy, On the nonvanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc., 42, 689 – 692
(1936).
13. L. Li, S. Ponnusamy, Convolutions of slanted half-plane harmonic mappings, Analysis (Munich), 33, № 2, 159 – 176
(2013).
14. L. Li, S. Ponnusamy, Solution to an open problem on convolution of harmonic mappings, Complex Var. Elliptic
Equat., 58, № 12, 1647 – 1653 (2013).
15. Y. Li, Z. Liu, Convolution of harmonic right half-plane mappings, Open Math., 14, 789 – 800 (2016).
16. Z. Liu, Y. Jiang, Y. Sun, Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings,
Filomat, 31, № 7, 1843 – 1856 (2017).
17. S. Muir, Weak subordination for convex univalent harmonic functions, J. Math. Anal. and Appl., 348, 689 – 692
(2008).
18. S. Ponnusamy, A. Rasila, Planar harmonic and quasiregular mappings, Topics in Modern Function Theory, Chapter
in CMFT, RMS-Lect. Notes Ser., № 19, 267 – 333 (2013).
Received 18.05.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
|
| id | umjimathkievua-article-94 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:17Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f1/624b02d8d139dc58647023f08bc23ef1.pdf |
| spelling | umjimathkievua-article-942025-03-31T08:48:28Z A proof of a conjecture on convolution of harmonic mappings and some related problems A PROOF OF A CONJECTURE ON CONVOLUTION OF HARMONIC MAPPINGS AND SOME RELATED PROBLEMS A proof of a conjecture on convolution of harmonic mappings and some related problems Yalçın, S. Ebadian , A. Azizi, S. Yalçın, Sibel Yalçın, S. Ebadian , A. Azizi, S. Harmonic convolutions harmonic vertical strip mappings harmonic half-plane mappings Harmonic convolutions harmonic vertical strip mappings harmonic half-plane mappings UDC 517.5 Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for $n=1,2,3$ and $4$. Also, it has been proved only for $\beta=\pi/2$. In this paper, by using of a new method, we settle this conjecture in the affirmative for all $n\in\mathbb{N}$ and $\beta\in(0,\pi)$. Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new method simplifies calculations and shortens the proof of results remarkably. УДК 517.5 Доведення гіпотези про згортку гармонічних відображень та деякі пов'язані задачі Нещодавно Kumar та ін. запропонували гіпотезу щодо згортки узагальнених відображень правої півплощини з відображеннями вертикальної смуги. Вони перевірили цю гіпотезу для $n=1,2,3$ та $4$. Крім цього, гіпотезу було доведено тільки для $\beta=\pi/2$. Використовуючи новий метод, ми доводимо цю гіпотезу для всіх $n\in\mathbb{N}$ та $\beta\in(0,\pi)$. Більш того, за допомогою цього методу ми отримали деякі результати щодо згортки гармонічних відображень. Новий метод спрощує обчислення та значно скорочує доведення результатів. Institute of Mathematics, NAS of Ukraine 2021-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/94 10.37863/umzh.v73i2.94 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 2 (2021); 283 - 288 Український математичний журнал; Том 73 № 2 (2021); 283 - 288 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/94/8969 |
| spellingShingle | Yalçın, S. Ebadian , A. Azizi, S. Yalçın, Sibel Yalçın, S. Ebadian , A. Azizi, S. A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title | A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title_alt | A PROOF OF A CONJECTURE ON CONVOLUTION OF HARMONIC MAPPINGS AND SOME RELATED PROBLEMS A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title_full | A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title_fullStr | A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title_full_unstemmed | A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title_short | A proof of a conjecture on convolution of harmonic mappings and some related problems |
| title_sort | proof of a conjecture on convolution of harmonic mappings and some related problems |
| topic_facet | Harmonic convolutions harmonic vertical strip mappings harmonic half-plane mappings Harmonic convolutions harmonic vertical strip mappings harmonic half-plane mappings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/94 |
| work_keys_str_mv | AT yalcıns aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT ebadiana aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT azizis aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT yalcınsibel aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT yalcıns aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT ebadiana aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT azizis aproofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT yalcıns proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT ebadiana proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT azizis proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT yalcınsibel proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT yalcıns proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT ebadiana proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems AT azizis proofofaconjectureonconvolutionofharmonicmappingsandsomerelatedproblems |