Estimates for $\lambda$-spirallike function of complex order on the boundary
UDC 517.5 We give some results for $\lambda$ -spirallike function of complex order at the boundary of the unit disc $U$. The sharpness of these results is also proved. Furthermore, three examples for our results are considered.  
Gespeichert in:
| Datum: | 2022 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2375 |
| 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_ | 1860508268201246720 |
|---|---|
| author | Akyel, T. . Akyel, T. |
| author_facet | Akyel, T. . Akyel, T. |
| author_sort | Akyel, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-27T15:39:11Z |
| description | UDC 517.5
We give some results for $\lambda$ -spirallike function of complex order at the boundary of the unit disc $U$. The sharpness of these results is also proved. Furthermore, three examples for our results are considered.
  |
| doi_str_mv | 10.37863/umzh.v74i1.2375 |
| first_indexed | 2026-03-24T02:22:30Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i1.2375
UDC 517.5
T. Akyel (Maltepe Univ., Istanbul, Turkey)
ESTIMATES FOR \bfitlambda -SPIRALLIKE FUNCTION OF COMPLEX ORDER
ON THE BOUNDARY
ОЦIНКИ ДЛЯ СПIРАЛЕПОДIБНОЇ \bfitlambda -ФУНКЦIЇ
КОМПЛЕКСНОГО ПОРЯДКУ НА ГРАНИЦI
We give some results for \lambda -spirallike function of complex order at the boundary of the unit disc U . The sharpness of these
results is also proved. Furthermore, three examples for our results are considered.
Наведено деякi результати для спiралеподiбної \lambda -функцiї комплексного порядку на границi одиничного диска U, а
також доведено точнiсть цих результатiв. Крiм того, розглянуто три приклади для iлюстрацiї цих результатiв.
1. Introduction. Let f be a holomorphic function in the unit disc U =
\bigl\{
z : | z| < 1
\bigr\}
, f(0) = 0 and
| f(z)| < 1 for | z| < 1. In accordance with the classical Schwarz lemma, for any point z in the disc
U, we have | f(z)| \leq | z| and | f \prime (0)| \leq 1. Equality in these inequalities (in the first one, for z \not = 0)
occurs only if f(z) = zei\theta , where \theta is a real number [7, p. 329]. The study of generalizations and
variations of Schwarz’s lemma as well as Littlewood’s theorem is of fundamental significance in the
area of geometric function theory and attracts many authors’ interest during the last years (see, for
example, [3, 7, 20] and the references therein). The generalization of the Schwarz lemma as follows:
| f(z)| \leq | z| | z| + | f \prime (0)|
1 + | z| | f \prime (0)|
, z \in U. (1.1)
Inequality (1.1) and its generalizations have important applications in geometric theory of functions
(see, e.g., [7, 16, 19]). Therefore, the interest in such type results has also continued in recent years
(see, e.g., [2, 3, 5, 6, 11, 12, 16 – 19] and references therein).
Let \scrA denote the class of functions
f(z) = z + c2z
2 + c3z
3 + . . .
which are holomorphic in the unit disc U . Let \scrM denote the class of bounded holomorphic functions
h(z) in U, satisfying the condition h(0) = 0 and | h(z)| \leq | z| for z \in U . For a function belonging
to the class \scrA we say that f(z) is \lambda -spirallike function of complex order in U if and only if
\Re
\biggl(
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
zf \prime (z)
f(z)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr] \biggr)
> 0, (1.2)
for some real \lambda , | \lambda | < \pi
2
, b \not = 0, complex.We denote this class by \scrN (\lambda ). It was introduced and
studied by Al-Oboudi and Haidan [1].
It is easy to show that f(z) \in \scrN (\lambda ) if and only if there is an h \in \scrM such that
\mathrm{s}\mathrm{e}\mathrm{c}\lambda ei\lambda
zf \prime (z)
f(z)
- i \mathrm{t}\mathrm{a}\mathrm{n}\lambda =
1 + (2b - 1)h(z)
1 - h(z)
(1.3)
for z \in U and for some \lambda , | \lambda | < \pi
2
.
c\bigcirc T. AKYEL, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1 3
4 T. AKYEL
Therefore, from (1.3), we take
h(z) = ei\lambda
zf \prime (z)
f(z)
- 1
ei\lambda
\biggl(
zf \prime (z)
f(z)
- 1
\biggr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
.
Since h(z) \in \scrM , from Schwarz lemma, we obtain
h(z) = ei\lambda
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
ei\lambda
\bigl(
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
\bigr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
,
h(z)
z
= ei\lambda
c2 +
\bigl(
2c3 - c22
\bigr)
z + . . .
ei\lambda
\bigl(
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
\bigr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
,
| h\prime (0)| = | c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\leq 1
and
| c2| \leq 2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (1.4)
Moreover, the equality in (1.4) occurs for the function
f(z) =
z
(1 - z)2be - i\lambda cos\lambda
.
It is an elementary consequence of Schwarz lemma that if f extends continuously to some
boundary point c with | c| = 1, and if | f(c)| = 1 and f \prime (c) exists, then | f \prime (c)| \geq 1, which is known
as the Schwarz lemma on the boundary. Passing to the angular limit in (1.1), we obtain the boundary
Schwarz lemma [16]
| f \prime (c)| \geq 2
1 + | f \prime (0)|
. (1.5)
For c = 1, the equality in (1.5) occurs for the function
f(z) = z
z + a
1 + az
,
where 0 \leq a \leq 1. It follow that
| f \prime (c)| \geq 1 (1.6)
with equality only if f is of the form f(z) = zei\theta , \theta is real.
V. N. Dubinin has continued this line and has made a refinement on the boundary Schwarz lemma
by considering the function f(z) = cpz
p + cp+1z
p+1 + . . . with a zero set \{ zk\} (see [5]).
S. G. Krantz and D. M. Burns [10] and D. Chelst [4] has studied the uniqueness part of the
Schwarz lemma. P. R. Mercer [11] has proved a version of the Schwarz lemma where the images of
two points are known. Also, he has considered some Schwarz and Carathéodory inequalities at the
boundary, as consequences of a lemma due to Rogosinski [12]. For more general results and related
estimates, we refer to the papers [13 – 15].
Also, M. Jeong [9] has obtained some inequalities at a boundary point for different form of holo-
morphic functions and has found the condition for equality and also in [8], has defined a holomorphic
self map on the closed unit disc with fixed points only on the boundary of the unit disc.
In the proofs of our main results, we will resort to the following lemma due to Julia – Wolff [19].
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
ESTIMATES FOR \lambda -SPIRALLIKE FUNCTION OF COMPLEX ORDER ON THE BOUNDARY 5
Lemma 1 (Julia – Wolff lemma). Let f be a holomorphic function in U, f(0) = 0 and f(U) \subset
\subset U. If , in addition, the function f has an angular limit f(c) at c \in \partial U, | f(c)| = 1, then the
angular derivative f \prime (c) exists and 1 \leq | f \prime (c)| \leq \infty .
Corollary 1. The holomorphic function f has a finite angular derivative f \prime (c) if and only if f \prime
has the finite angular limit f \prime (c) at c \in \partial U.
2. Main results. In this section, for holomorphic function f(z) = z + c2z
2 + c3z
3 + . . .
belonging to the class of \scrN (\lambda ), the modulus of the angular derivative of the function
zf \prime (z)
f(z)
will be
estimated from below on the boundary point of the unit disc. The sharpness of these results is also
proved. Furthermore, examples will be presented for the inequalities obtained.
Theorem 1. Let f(z) \in \scrN (\lambda ). Assume that, for some c \in \partial U, f has angular limit f(c) at c
and
cf \prime (c)
f(c)
= 1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
. Then we have the inequality
\bigm| \bigm| \bigm| \bigm| \biggl( zf \prime (z)
f(z)
\biggr) \prime
z=c
\bigm| \bigm| \bigm| \bigm| \geq | b|
2
\mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.1)
The equality in (2.1) occurs for the function
f(z) =
z
(1 - z)2be - i\lambda cos\lambda
.
Proof. Since f(z) \in \scrN (\lambda ), we write
\mathrm{s}\mathrm{e}\mathrm{c}\lambda ei\lambda
zf \prime (z)
f(z)
- i \mathrm{t}\mathrm{a}\mathrm{n}\lambda =
1 + (2b - 1)h(z)
1 - h(z)
for z \in U and for some \lambda , | \lambda | < \pi
2
and h(z) \in \scrM . Thus, we get
h(z) = ei\lambda
zf \prime (z)
f(z)
- 1
ei\lambda
\biggl(
zf \prime (z)
f(z)
- 1
\biggr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
;
h(z) is a holomorphic function in the unit disc U, | h(z)| < 1 for | z| < 1 and h(0) = 0. For c \in \partial U
and
cf \prime (c)
f(c)
= 1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
, we take
| h(c)| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| e
i\lambda
cf \prime (c)
f(c)
- 1
ei\lambda
\biggl(
cf \prime (c)
f(c)
- 1
\biggr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
- 1
ei\lambda
\biggl(
1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
- 1
\biggr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| - b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
- b \mathrm{c}\mathrm{o}\mathrm{s}\lambda + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| b \mathrm{c}\mathrm{o}\mathrm{s}\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\bigm| \bigm| \bigm| \bigm| = 1.
Let
g(z) =
zf \prime (z)
f(z)
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
6 T. AKYEL
So,
h(z) = ei\lambda
g(z) - 1
ei\lambda (g(z) - 1) + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
.
From (1.6), we obtain
1 \leq | h\prime (c)| =
\bigm| \bigm| \bigm| \bigm| \bigm| ei\lambda 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (c)
(ei\lambda (g(c) - 1) + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda )
2
\bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (c)\biggl(
ei\lambda
\biggl(
1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
- 1
\biggr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr) 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (c)
( - b \mathrm{c}\mathrm{o}\mathrm{s}\lambda + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda )2
\bigm| \bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (c)(b \mathrm{c}\mathrm{o}\mathrm{s}\lambda )2
\bigm| \bigm| \bigm| \bigm| = 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
and
| g\prime (c)| \geq | b|
2
\mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Now, we shall show that inequality (2.1) is sharp. Let
f(z) =
z
(1 - z)2be - i\lambda cos\lambda
. (2.2)
Differentiating (2.2) logarithmically, we have
\mathrm{l}\mathrm{n} f(z) = \mathrm{l}\mathrm{n}
z
(1 - z)2be - i\lambda cos\lambda
,
f \prime (z)
f(z)
=
1
z
+
2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
1 - z
and
g(z) =
zf \prime (z)
f(z)
= 1 + 2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z
1 - z
.
Therefore, we take
g\prime (z) =
1
(1 - z)2
2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
and \bigm| \bigm| g\prime ( - 1)
\bigm| \bigm| = | b|
2
\mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Theorem 1 is proved.
Inequality (2.1) can be strengthened as below by taking into account c2 which is the second
coefficient in the expansion of the function f(z).
Theorem 2. Under the same assumptions as in Theorem 1, we have the inequality\bigm| \bigm| \bigm| \bigm| \biggl( zf \prime (z)
f(z)
\biggr) \prime
z=c
\bigm| \bigm| \bigm| \bigm| \geq 2| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda + | c2|
. (2.3)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
ESTIMATES FOR \lambda -SPIRALLIKE FUNCTION OF COMPLEX ORDER ON THE BOUNDARY 7
Inequality (2.3) is sharp with equality for the function
f(z) =
z\bigl(
z2 + 2dz + 1
\bigr) e - i\lambda b cos\lambda
,
where d =
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
is an arbitrary number on [0, 1] (see (1.4)).
Proof. Let h(z) be as in the proof of Theorem 1. By using inequality (1.5) for the function
h(z), we obtain
2
1 + | h\prime (0)|
\leq | h\prime (c)| =
\bigm| \bigm| \bigm| \bigm| \bigm| ei\lambda 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (c)
(ei\lambda (g(c) - 1) + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda )
2
\bigm| \bigm| \bigm| \bigm| \bigm| = 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
,
where g(z) =
\biggl(
zf \prime (z)
f(z)
\biggr)
.
Since
h\prime (z) = ei\lambda
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (z)
(ei\lambda (g(z) - 1) + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda )
2 ,
h\prime (0) = ei\lambda
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda g\prime (0)
(ei\lambda (g(0) - 1) + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda )
2
and \bigm| \bigm| h\prime (0)\bigm| \bigm| = | c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
,
we take
2
1 +
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\leq 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
.
Thus, we obtain inequality (2.3).
Now, we shall show that inequality (2.3) is sharp. Choose arbitrary d \in [0, 1]. Let
f(z) =
z\bigl(
z2 + 2dz + 1
\bigr) e - i\lambda b cos\lambda
. (2.4)
Differentiating (2.4) logarithmically, we get
\mathrm{l}\mathrm{n} f(z) = \mathrm{l}\mathrm{n}
z\bigl(
z2 + 2dz + 1
\bigr) e - i\lambda b cos\lambda
,
f \prime (z)
f(z)
=
1
z
- 2e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z + d
z2 + 2dz + 1
and
g(z) =
zf \prime (z)
f(z)
= 1 - 2e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z2 + dz
z2 + 2dz + 1
.
Thus, since d =
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
, we obtain
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
8 T. AKYEL
| g\prime (1)| = 2| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda + | c2|
.
Theorem 2 is proved.
Now, inequality (2.3) can be strengthened as below by taking into account c3 which is the third
coefficient in the expansion of the function f(z).
Theorem 3. Let f(z) belong to \scrN (\lambda ). Assume that, for some c \in \partial U, f has angular limit f(c)
at c and
cf \prime (c)
f(c)
= 1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
. Then we have the inequality
\bigm| \bigm| \bigm| \bigm| \biggl( zf \prime (z)
f(z)
\biggr) \prime
z=c
\bigm| \bigm| \bigm| \bigm| \geq | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
2
\Biggl(
1 +
2
\bigl(
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda - | c2|
\bigr) 2
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda - | c2| 2 + | (2c3 - c22)2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22|
\Biggr)
. (2.5)
Inequality (2.5) is sharp with equality for the function
f(z) =
z
(1 - z2)e - i\lambda b cos\lambda
.
Proof. Let h(z) be as in the proof of Theorem 1 and \eta (z) = z. By the maximum principle for
each z \in U, we have | h(z)| \leq | \eta (z)| . So,
p(z) =
h(z)
\eta (z)
is a holomorphic function in U and | p(z)| < 1 for | z| < 1.
Since
h(z) = ei\lambda
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
ei\lambda
\bigl(
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
\bigr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
and
p(z) =
h(z)
z
= ei\lambda
z
\bigl(
c2 +
\bigl(
2c3 - c22
\bigr)
z + . . .
\bigr) \bigl(
ei\lambda
\bigl(
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
\bigr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\bigr)
z
=
= ei\lambda
\bigl(
c2 +
\bigl(
2c3 - c22
\bigr)
z + . . .
\bigr) \bigl(
ei\lambda
\bigl(
c2z +
\bigl(
2c3 - c22
\bigr)
z2 + . . .
\bigr)
+ 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\bigr) ,
in particular, we have
| p(0)| = | c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\leq 1 (2.6)
and
| p\prime (0)| =
\bigm| \bigm| (2c3 - c22)2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22
\bigm| \bigm|
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda
.
Moreover, it can be seen that
ch\prime (c)
h(c)
= | h\prime (c)| \geq | \eta \prime (c)| = c\eta \prime (c)
\eta (c)
.
The function
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
ESTIMATES FOR \lambda -SPIRALLIKE FUNCTION OF COMPLEX ORDER ON THE BOUNDARY 9
\Gamma (z) =
p(z) - p(0)
1 - p(0)p(z)
is holomorphic in the unit disc U, | \Gamma (z)| < 1 for | z| < 1, \Gamma (0) = 0 and | \Gamma (c)| = 1 for c \in \partial U.
From (1.5), we obtain
2
1 + | \Gamma \prime (0)|
\leq
\bigm| \bigm| \Gamma \prime (c)
\bigm| \bigm| = 1 - | p(0)| 2
| 1 - p(0)p(c)| 2
| p\prime (c)| \leq 1 + | p(0)|
1 - | p(0)|
| p\prime (c)| =
=
1 + | p(0)|
1 - | p(0)|
\bigl\{
| h\prime (c)| - 1
\bigr\}
.
Since
\Gamma \prime (z) =
1 - | p(0)| 2\Bigl(
1 - p(0)p(z)
\Bigr) 2 p\prime (z),
| \Gamma \prime (0)| = | p\prime (0)|
1 - | p(0)| 2
=
\bigm| \bigm| \bigl( 2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22
\bigm| \bigm|
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda
1 -
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr) 2 =
\bigm| \bigm| \bigl( 2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22
\bigm| \bigm|
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda - | c2| 2
,
we take
2
1 +
\bigm| \bigm| \bigl( 2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22
\bigm| \bigm|
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda - | c2| 2
\leq
1 +
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
1 - | c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl\{
2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
- 1
\biggr\}
=
=
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda + | c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda - | c2|
\biggl\{
2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
- 1
\biggr\}
.
Therefore, we get
1 +
2
\bigl(
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda - | c2| 2
\bigr)
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda - | c2| 2 + |
\bigl(
2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda - | c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda + | c2|
\leq 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
and
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
2
\Biggl(
1 +
2
\bigl(
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda - | c2|
\bigr) 2
4| b| 2 \mathrm{c}\mathrm{o}\mathrm{s}2 \lambda - | c2| 2 +
\bigm| \bigm| \bigl( 2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22
\bigm| \bigm|
\Biggr)
\leq
\bigm| \bigm| g\prime (c)\bigm| \bigm| .
So, we obtain inequality (2.5).
To show that inequality (2.5) is sharp, take the holomorphic function
f(z) =
z
(1 - z2)e - i\lambda b cos\lambda
. (2.7)
Differentiating (2.7) logarithmically, we obtain
\mathrm{l}\mathrm{n} f(z) = \mathrm{l}\mathrm{n}
z
(1 - z2)e - i\lambda b cos\lambda
,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
10 T. AKYEL
f \prime (z)
f(z)
=
1
z
+ e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
2z
1 - z2
and
g(z) =
zf \prime (z)
f(z)
= 1 + e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
2z2
1 - z2
.
Therefore, we take
g\prime (z) = e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
4z
(1 - z2)2
and
| g\prime (i)| = | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Since | c2| = 0 and | c3| = | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda , (2.5) is satisfied with equality.
Theorem 3 is proved.
If f(z) - z has no zeros different from z = 0 in Theorem 3, inequality (2.5) can be further
strengthened. This is given by the following theorem.
Theorem 4. Let f(z) \in \scrN (\lambda ), f(z) - z has no zeros in U except z = 0 and c2 > 0. Suppose
that, for some c \in \partial U, f has angular limit f(c) at c and
cf \prime (c)
f(c)
= 1 - b
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
ei\lambda
. Then we have the
inequality
\bigm| \bigm| \bigm| \bigm| \biggl( zf \prime (z)
f(z)
\biggr) \prime
z=c
\bigm| \bigm| \bigm| \bigm| \geq | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
2
\left( 1 -
4| c2| | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda \mathrm{l}\mathrm{n}2
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
4| c2| | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda \mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
- |
\bigl(
2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22|
\right) .
(2.8)
Proof. Let c2 > 0 in the expression of the function f(z). Having in mind inequality (2.6) and
the function f(z) - z has no zeros in U except z = 0, we denote by \mathrm{l}\mathrm{n} p(z) the holomorphic branch
of the logarithm normed by the condition
\mathrm{l}\mathrm{n} p(0) = \mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
< 0.
The auxiliary function
\Upsilon (z) =
\mathrm{l}\mathrm{n} p(z) - \mathrm{l}\mathrm{n} p(0)
\mathrm{l}\mathrm{n} p(z) + \mathrm{l}\mathrm{n} p(0)
is holomorphic in the unit disc U, | \Upsilon (z)| < 1, \Upsilon (0) = 0 and | \Upsilon (c)| = 1 for c \in \partial U.
From (1.5), we get
2
1 + | \Upsilon \prime (0)|
\leq | \Upsilon \prime (c)| = | 2 \mathrm{l}\mathrm{n} p(0)|
| \mathrm{l}\mathrm{n} p(c) + \mathrm{l}\mathrm{n} p(0)| 2
\bigm| \bigm| \bigm| \bigm| p\prime (c)p(c)
\bigm| \bigm| \bigm| \bigm| = - 2 \mathrm{l}\mathrm{n} p(0)
\mathrm{l}\mathrm{n}2 p(0) + \mathrm{a}\mathrm{r}\mathrm{g}2 p(c)
\bigl\{
| h\prime (c)| - 1
\bigr\}
.
Replacing \mathrm{a}\mathrm{r}\mathrm{g}2 p(c) by zero, then
1
1 - 1
2 \mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr) |
\bigl(
2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22|
2| c2| | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\leq - 1
\mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr) \biggl\{ 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
- 1
\biggr\}
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
ESTIMATES FOR \lambda -SPIRALLIKE FUNCTION OF COMPLEX ORDER ON THE BOUNDARY 11
and
1 -
4| c2| | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda \mathrm{l}\mathrm{n}2
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
4| c2| | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda \mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
- |
\bigl(
2c3 - c22
\bigr)
2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - ei\lambda c22|
\leq 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
.
Thus, we obtain inequality (2.8).
The following inequality (2.9) is weaker, but is simpler than (2.8) and does not contain the
coefficient c3.
Theorem 5. Under the same assumptions as in Theorem 4, we have the inequality\bigm| \bigm| \bigm| \bigm| \biggl( zf \prime (z)
f(z)
\biggr) \prime
z=c
\bigm| \bigm| \bigm| \bigm| \geq | b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
2
\biggl(
1 - 1
2
\mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr) \biggr)
. (2.9)
Proof. Let c2 > 0. Using inequality (1.6) for the function \Gamma (z), we obtain
1 \leq | \Upsilon \prime (c)| = | 2 \mathrm{l}\mathrm{n} p(0)|
| \mathrm{l}\mathrm{n} p(c) + \mathrm{l}\mathrm{n} p(0)| 2
\bigm| \bigm| \bigm| \bigm| p\prime (c)p(c)
\bigm| \bigm| \bigm| \bigm| = - 2 \mathrm{l}\mathrm{n} p(0)
\mathrm{l}\mathrm{n}2 p(0) + \mathrm{a}\mathrm{r}\mathrm{g}2 p(c)
\bigl\{
| h\prime (c)| - 1
\bigr\}
.
Replacing \mathrm{a}\mathrm{r}\mathrm{g}2 p(c) by zero, then
1 \leq | \Upsilon \prime (c)| \leq - 2
\mathrm{l}\mathrm{n}
\biggl(
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr) \biggl\{ 2| g\prime (c)|
| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
- 1
\biggr\}
.
Therefore, we obtain inequality (2.9).
Theorem 4 is proved.
3. Examples.
Example 1. Let us consider the function f(z) defined by
f(z) =
z
(1 - z)2be - i\lambda cos\lambda
.
From here, we have
zf \prime (z)
f(z)
= 1 + 2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z
1 - z
.
So, we take
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
zf \prime (z)
f(z)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
\biggl(
1 + 2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z
1 - z
\biggr)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z
1 - z
- \mathrm{c}\mathrm{o}\mathrm{s}\lambda + b \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda + 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z
1 - z
- ei\lambda + b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl(
2z
1 - z
+ 1
\biggr) \biggr]
=
1 + z
1 - z
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
12 T. AKYEL
Obviously, the function
k(z) =
1 + z
1 - z
maps the unit disc U to the right half plane and, hence, the real part of the function k(z) is nonne-
gative. Therefore, we obtain
\Re
\biggl(
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
zf \prime (z)
f(z)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr] \biggr)
= \Re
\biggl(
1 + z
1 - z
\biggr)
> 0.
Thus, the function f(z) satisfies condition (1.2) and Theorem 1. That is,
f(z) =
z
(1 - z)2be - i\lambda cos\lambda
and, for some c \in \partial U, we have
( - 1)f \prime ( - 1)
f( - 1)
= 1 + 2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl(
- 1
1 - ( - 1)
\biggr)
= 1 + 2be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl(
- 1
2
\biggr)
and
( - 1)f \prime ( - 1)
f( - 1)
= 1 - be - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Example 2. Let us consider the function f(z) defined by
f(z) =
z\bigl(
z2 + 2dz + 1
\bigr) e - i\lambda b cos\lambda
,
where d =
| c2|
2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
is an arbitrary number from [0, 1] (see (1.4)). Hence, we get
zf \prime (z)
f(z)
= 1 - 2e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z2 + dz
z2 + 2dz + 1
.
Therefore, we obtain
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
zf \prime (z)
f(z)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
\biggl(
1 - e - i\lambda 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z2 + dz
z2 + 2dz + 1
\biggr)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda - 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z2 + dz
z2 + 2dz + 1
- ei\lambda + b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
- 2b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
z2 + dz
z2 + 2dz + 1
+ b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr]
=
=
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl(
- 2
z2 + dz
z2 + 2dz + 1
+ 1
\biggr) \biggr]
=
1 - z2
z2 + 2dz + 1
.
Since d \in [0, 1], we see that
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
ESTIMATES FOR \lambda -SPIRALLIKE FUNCTION OF COMPLEX ORDER ON THE BOUNDARY 13
\Re
\biggl(
1
b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
ei\lambda
zf \prime (z)
f(z)
- (1 - b) \mathrm{c}\mathrm{o}\mathrm{s}\lambda - i \mathrm{s}\mathrm{i}\mathrm{n}\lambda
\biggr] \biggr)
= \Re
\biggl(
1 - z2
z2 + 2dz + 1
\biggr)
> 0.
Thus, the function f(z) satisfies condition (1.2) and Theorem 2. That is,
f(z) =
z\bigl(
z2 + 2dz + 1
\bigr) e - i\lambda b cos\lambda
,
for some c \in \partial U, we have
f \prime (1)
f(1)
= 1 - 2e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
1 + d
1 + 2d+ 1
= 1 - 2e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda
1 + d
2(1 + d)
and
f \prime (1)
f(1)
= 1 - e - i\lambda b \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Example 3. Let us consider the function f(z) given by
f(z) =
z
(1 - z2)e - i\lambda b cos\lambda
.
Similar to other examples, it can be easily shown that the function f(z) provides the properties of
class \scrN (\lambda ).
References
1. F. M. Al-Oboudi, M. M. Haidan, Spirallike functions of complex order, J. Nat. Geom., 19, 53 – 72 (2000).
2. T. A. Azeroğlu, B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Var. and Elliptic Equat., 58,
571 – 577 (2013).
3. H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly, 117, 770 – 785 (2010).
4. D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc., 129, 3275 – 3278 (2001).
5. V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci., 122,
3623 – 3629 (2004).
6. V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci., 207, 825 – 831 (2015).
7. G. M. Golusin, Geometric theory of functions of complex variable (in Russian), 2nd ed., Moscow (1966).
8. M. Jeong, The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B. Pure
and Appl. Math., 21, 275 – 284 (2014).
9. M. Jeong, The Schwarz lemma and boundary fixed points, J. Korean Soc. Math. Educ. Ser. B. Pure and Appl. Math.,
18, 219 – 227 (2011).
10. S. G. Krantz, D. M. Burns, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer.
Math. Soc., 7, 661 – 676 (1994).
11. P. R. Mercer, Sharpened versions of the Schwarz lemma, J. Math. Anal. and Appl., 205, 508 – 511 (1997).
12. P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski’s lemma, J. Class. Anal., 12, 93 – 97 (2018).
13. M. Mateljević, The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings,
Filomat, 29, 221 – 244 (2015).
14. M. Mateljević, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Rev. Roum. Math.
Pures et Appl., 51, 711 – 722 (2006).
15. M. Mateljević, Ahlfors – Schwarz lemma and curvature, Kragujevac J. Math., 25, 155 – 164 (2003).
16. R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 128, 3513 – 3517 (2000).
17. B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc., 50, 2053 – 2059
(2013).
18. B. N. Örnek, T. Akyel, Sharpened forms of the generalized Schwarz inequality on the boundary, Proc. Indian Acad.
Sci. (Math. Sci.), 126, 69 – 78 (2016).
19. Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin (1992).
20. D. Shoikhet, M. Elin, F. Jacobzon, M. Levenshtein, The Schwarz lemma: rigidity and dynamics, harmonic and
complex analysis and its applications, Springer Int. Publ., 135 – 230 (2014).
Received 16.02.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
|
| id | umjimathkievua-article-2375 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:30Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e5/004c702396cfd2026cb160f5731061e5.pdf |
| spelling | umjimathkievua-article-23752022-03-27T15:39:11Z Estimates for $\lambda$-spirallike function of complex order on the boundary Estimates for $\lambda$-spirallike function of complex order on the boundary Akyel, T. . Akyel, T. Schwarz lemma on the boundary Julia-Wolff lemma lambda-spirallikefunction UDC 517.5 We give some results for $\lambda$ -spirallike function of complex order at the boundary of the unit disc $U$. The sharpness of these results is also proved. Furthermore, three examples for our results are considered. &nbsp; УДК 517.5Оцiнки для спiралеподiбної $\lambda$ -функцiї комплексного порядку на границi Наведено деякі результати для спіралеподібної $\lambda$-функції комплексного порядку на границі одиничного диска $U,$ а також доведено точність цих результатів.&nbsp;Крім того, розглянуто три приклади для ілюстрації цих результатів.&nbsp; Institute of Mathematics, NAS of Ukraine 2022-01-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2375 10.37863/umzh.v74i1.2375 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 1 (2022); 3 - 13 Український математичний журнал; Том 74 № 1 (2022); 3 - 13 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2375/9171 Copyright (c) 2022 Tuğba AKYEL |
| spellingShingle | Akyel, T. . Akyel, T. Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title | Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title_alt | Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title_full | Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title_fullStr | Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title_full_unstemmed | Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title_short | Estimates for $\lambda$-spirallike function of complex order on the boundary |
| title_sort | estimates for $\lambda$-spirallike function of complex order on the boundary |
| topic_facet | Schwarz lemma on the boundary Julia-Wolff lemma lambda-spirallikefunction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2375 |
| work_keys_str_mv | AT akyelt estimatesforlambdaspirallikefunctionofcomplexorderontheboundary AT estimatesforlambdaspirallikefunctionofcomplexorderontheboundary AT akyelt estimatesforlambdaspirallikefunctionofcomplexorderontheboundary |