Estimates for analytic functions concerned with Hankel determinant
UDC 517.5 We give an upper bound of Hankel determinant of the first order $(H_{2}(1))$ for the classes of an analytic function. In addition, an evaluation with the Hankel determinant from below will be given for the second angular derivative of $f(z)$ analytic function. For new inequalities, the res...
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2021
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| author | Örnek, B. N. Örnek, B. N. |
| author_facet | Örnek, B. N. Örnek, B. N. |
| author_sort | Örnek, B. N. |
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| description | UDC 517.5
We give an upper bound of Hankel determinant of the first order $(H_{2}(1))$ for the classes of an analytic function. In addition, an evaluation with the Hankel determinant from below will be given for the second angular derivative of $f(z)$ analytic function. For new inequalities, the results of Jack's lemma and Hankel determinant were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.
  |
| doi_str_mv | 10.37863/umzh.v73i9.907 |
| first_indexed | 2026-03-24T02:06:03Z |
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DOI: 10.37863/umzh.v73i9.907
UDC 517.5
B. N. Örnek (Amasya Univ., Turkey)
ESTIMATES FOR ANALYTIC FUNCTIONS
CONCERNED WITH HANKEL DETERMINANT
ОЦIНКИ ДЛЯ АНАЛIТИЧНИХ ФУНКЦIЙ,
ПОВ’ЯЗАНI З ДЕТЕРМIНАНТОМ ГАНКЕЛЯ
We give an upper bound of Hankel determinant of the first order (H2(1)) for the classes of an analytic function. In addition,
an evaluation with the Hankel determinant from below will be given for the second angular derivative of f(z) analytic
function. For new inequalities, the results of Jack’s lemma and Hankel determinant were used. Moreover, in a class of
analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations below
of the modulus of angular derivative have been obtained.
Отримано верхню границю детермiнанта Ганкеля першого порядку (H2(1)) для класiв аналiтичної функцiї. Також
встановлено оцiнку знизу з детермiнантом Ганкеля для другої кутової похiдної аналiтичної функцiї f(z). Для
отримання нових нерiвностей використано лему Джека та детермiнант Ганкеля. Крiм того, для класу аналiтичних
функцiй на одиничному диску за умови iснування кутової границi для межової точки отримано оцiнки знизу для
модуля кутової похiдної.
1. Introduction. The most classical version of the Schwarz lemma examines the behavior of a
bounded, analytic function mapping the origin to the origin in the unit disc U =
\bigl\{
z : | z| < 1
\bigr\}
. It
is possible to see its effectiveness in the proofs of many important theorems. The Schwarz lemma,
which has broad applications and is the direct application of the maximum modulus principle, is
given in the most basic form as follows:
Let U be the unit disc in the complex plane \BbbC . Let f : U \rightarrow U be an analytic function with
f(z) = cpz
p + . . . . Under these conditions,
\bigm| \bigm| f(z)\bigm| \bigm| \leq | z| p for all z \in U and | cp| \leq 1. In addition, if
the equality
\bigm| \bigm| f(z)\bigm| \bigm| = | z| p holds for any z \not = 0 or | cp| = 1, then f is a rotation, that is, f(z) = zpei\theta ,
\theta real [5, p. 329]. Schwarz lemma has several applications in the field of electrical and electronics
engineering. Usage of positive real function and boundary analysis of these functions for circuit
synthesis can be given as an exemplary application of the Schwarz lemma in electrical engineering.
Furthermore, it is also used for analysis of transfer functions in control engineering and multinotch
filter design in signal processing [12, 13].
In order to derive our main results, we have to recall here the following lemma [6].
Lemma 1 (Jack’s lemma). Let f(z) be a nonconstant analytic function in U with f(0) = 0. If\bigm| \bigm| f(z0)\bigm| \bigm| = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
| f(z)| : | z| \leq | z0|
\bigr\}
,
then there exists a real number k \geq 1 such that
z0f
\prime (z0)
f(z0)
= k.
Let \scrA denote the class of functions f(z) = z + c2z
2 + c3z
3 + . . . that are analytic in U. Also,
let \scrM be the subclass of \scrA consisting of all functions f(z) satisfying
c\bigcirc B. N. ÖRNEK, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9 1205
1206 B. N. ÖRNEK
\Re
\left[
\left(
\biggl(
z
f(z)
\biggr) 2
f \prime (z)\biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\right)
\biggl(
2 +
zf \prime \prime (z)
f \prime (z)
- 2
zf \prime (z)
f(z)
\biggr) \right] < 1. (1.1)
The certain analytic functions which is in the class of \scrM on the unit disc U are considered in this
paper. The subject of the present paper is to discuss some properties of the function f(z) which
belongs to the class of \scrM by applying Jack’s lemma.
In this paper, we will give the estimates for the Hankel determinant of the first order for the
class of analytic function f \in \scrA will satisfy the condition (1.1). In particular, upper bounds on
H2(1) will be obtained for the class \scrM . In addition, the relationship between the coefficients of the
Hankel determinant and the angular derivative of the function f, which provides the class \scrM , will
be examined. In this examine, the coefficients c2, c3 and c4 will be used. Let f \in \scrA . The qth
Hankel determinant of f for n \geq 0 and q \geq 1 is stated by Thomas and Noonan [19] as
Hq(n) =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
cn cn+1 . . . cn+q - 1
cn+1 cn+2 . . . cn+q
...
...
...
...
cn+q - 1 cn+q . . . cn+2q - 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
, c1 = 1.
From the Hankel determinant for n = 1 and q = 2, we have
H2(1) =
\bigm| \bigm| \bigm| \bigm| \bigm| c1 c2
c2 c3
\bigm| \bigm| \bigm| \bigm| \bigm| = c3 - c22.
Here, the Hankel determinant H2(1) = c3 - c22 is well-known as Fekete – Szegö functional [18]. In
[19], the authors have obtained the upper bounds of the Hankel determinant
\bigm| \bigm| c2c4 - c23
\bigm| \bigm| . Also, in
[16], the author have obtained the upper bounds the Hankel determinant A(k)
n . Moreover, in [17], the
authors have given bounds for the second Hankel determinant for class \scrM \alpha .
Let f \in \scrM and consider the function
\Psi (z) = 2
\Biggl[ \biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\Biggr]
.
It is an analytic function in U and \Psi (0) = 0. Now, let us show that | \Psi (z)| < 1 in U. If the logarithm
differentiation of both sides is taken in the last equation, we obtain
\mathrm{l}\mathrm{n}
\biggl(
1 +
1
2
\Psi (z)
\biggr)
= \mathrm{l}\mathrm{n}
\Biggl( \biggl(
z
f(z)
\biggr) 2
f \prime (z)
\Biggr)
,
1
2
\Psi \prime (z)
1 +
1
2
\Psi (z)
=
2
z
- 2
f \prime (z)
f(z)
+
f \prime \prime (z)
f \prime (z)
and
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
ESTIMATES FOR ANALYTIC FUNCTIONS CONCERNED WITH HANKEL DETERMINANT 1207
1
2
z\Psi \prime (z)
1 +
1
2
\Psi (z)
= 2 - 2
zf \prime (z)
f(z)
+
zf \prime \prime (z)
f \prime (z)
.
Therefore, we have
\Re
\left[
\left(
\biggl(
z
f(z)
\biggr) 2
f \prime (z)\biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\right)
\biggl(
2 +
zf \prime \prime (z)
f \prime (z)
- 2
zf \prime (z)
f(z)
\biggr) \right] = \Re
\biggl(
z\Psi \prime (z)
\Psi (z)
\biggr)
.
We suppose that there exists a z0 \in U such that
\mathrm{m}\mathrm{a}\mathrm{x}
| z| \leq | z0|
| \Psi (z)| =
\bigm| \bigm| \Psi (z0)
\bigm| \bigm| = 1.
From Jack’s lemma, we get
\Psi (z0) = ei\theta and
z0\Psi
\prime (z0)
\Psi (z0)
= k.
Therefore, we have
\Re
\left[
\left(
\biggl(
z0
f(z0)
\biggr) 2
f \prime (z0)\biggl(
z0
f(z0)
\biggr) 2
f \prime (z0) - 1
\right)
\biggl(
2 +
z0f
\prime \prime (z0)
f \prime (z0)
- 2
z0f
\prime (z0)
f(z0)
\biggr) \right] =
= \Re
\biggl(
z0\Psi
\prime (z0)
\Psi (z0)
\biggr)
= \Re
\biggl(
k\Psi (z0)
\Psi (z0)
\biggr)
= \Re (k) = k \geq 1.
This contradicts the f \in \scrM . This means that there is no point z0 \in U such that
\mathrm{m}\mathrm{a}\mathrm{x}
| z| \leq | z0|
| \Psi (z)| =
\bigm| \bigm| \Psi (z0)
\bigm| \bigm| = 1.
Hence, we take | \Psi (z)| < 1 in U. From the Schwarz lemma, we obtain
\Psi (z) = 2
\Biggl[ \biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\Biggr]
=
= 2
\Biggl[ \biggl(
z
z + c2z2 + c3z3 + . . .
\biggr) 2 \bigl(
1 + 2c2z + 3c3z
2 + . . .
\bigr)
- 1
\Biggr]
=
= 2
\bigl[
(c3 - c22)z
2 + (2c4 - 4c2c3 + 2c32)z
3 + . . .
\bigr]
,
\Psi (z)
z2
= 2
\bigl[
(c3 - c22) + (2c4 - 4c2c3 + 2c32)z + . . .
\bigr]
,
2| c3 - c22| = 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \leq 1
and \bigm| \bigm| H2(1)
\bigm| \bigm| \leq 1
2
.
We thus obtain the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1208 B. N. ÖRNEK
Lemma 2. If f \in \scrM , then we have the inequality\bigm| \bigm| H2(1)
\bigm| \bigm| \leq 1
2
. (1.2)
Consider the product
B(z) =
n\prod
i=1
z - zi
1 - ziz
.
The function B(z) is called a finite Blaschke product, where z1, z2, . . . , zn \in U. Let the function
\Psi (z) satisfy the condition of the Schwarz lemma and also have zeros z1, z2, . . . , zn. Thus, one can
see that the inequality (1.2) can be strengthened by standard methods as follows:
\bigm| \bigm| H2(1)
\bigm| \bigm| \leq 1
2
n\prod
i=1
| zi| .
Since the area of applicability of Schwarz lemma is quite wide, there exist many studies about
it. Some of these studies, which are called the boundary version of Schwarz lemma, are about being
estimated from below the modulus of the derivative of the function at some boundary point of the
unit disc. The boundary version of Schwarz lemma is given as follows:
If f extends continuously to some boundary point b with | b| = 1 and if
\bigm| \bigm| f(b)\bigm| \bigm| = 1 and f \prime (b)
exists, then
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \geq 1, which is known as the Schwarz lemma on the boundary. In addition
to conditions of the boundary Schwarz lemma, if f fixes the point zero, that is, f(z) = cpz
p +
+ cp+1z
p+1 + . . . , then the inequality
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \geq p+
1 - | cp|
1 + | cp|
(1.3)
and \bigm| \bigm| f \prime (b)
\bigm| \bigm| \geq p (1.4)
are obtained [11]. Inequality (1.3) and its generalizations have important applications in geometric
theory of functions and they are still hot topics in the mathematics literature [1 – 4, 7, 9 – 14]. Mercer
[8] proves a version of the Schwarz lemma where the images of two points are known. Also, he
considers some Schwarz and Carathéodory inequalities at the boundary, as consequences of a lemma
due to Rogosinski [9]. In addition, he obtains a new boundary Schwarz lemma, for analytic functions
mapping the unit disk to itself [10].
The following lemma, known as the Julia – Wolff lemma, is needed in the sequel (see [15]).
Lemma 3 (Julia – Wolff lemma). Let f be an analytic function in U, f(0) = 0 and f(U) \subset U.
If , in addition, the function f has an angular limit f(b) at b \in \partial U,
\bigm| \bigm| f(b)\bigm| \bigm| = 1, then the angular
derivative f \prime (b) exists and 1 \leq
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \leq \infty .
Corollary 1. The analytic function f has a finite angular derivative f \prime (b) if and only if f \prime has
the finite angular limit f \prime (b) at b \in \partial U.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
ESTIMATES FOR ANALYTIC FUNCTIONS CONCERNED WITH HANKEL DETERMINANT 1209
2. Main results. In this section, we discuss different versions of the boundary Schwarz lemma
and the Hankel determinant for \scrM class. Assuming the existence of angular limit on a boundary
point, we obtain some estimations from below for the moduli of derivatives of analytic functions
from a certain class. In the inequalities obtained, the relationship between the Hankel determinant
and the second angular derivative of the f(z) function was established.
Theorem 1. Let f \in \scrM . Assume that, for some b \in \partial U, f has an angular limit f(b) at b,
f(b) =
2b
3
and f \prime (b) =
2
3
. Then we have the inequality
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 4
9
. (2.1)
Proof. Let us consider the function
\Psi (z) = 2
\Biggl[ \biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\Biggr]
.
\Psi (z) is an analytic function in U, \Psi (0) = 0 and | \Psi (z)| < 1 for z \in U. Also, since f(b) =
2b
3
and
f \prime (b) =
2
3
, we take
\Psi (b) = 2
\Biggl[ \biggl(
b
f(b)
\biggr) 2
f \prime (b) - 1
\Biggr]
=
= 2
\left[
\left( b
2b
3
\right)
2
2
3
- 1
\right] = 2
\biggl[
9
4
2
3
- 1
\biggr]
= 1
and \bigm| \bigm| \Psi (b)
\bigm| \bigm| = 1.
Therefore, from (1.4) for p = 2, we get
2 \leq | \Psi \prime (b)| = 2
\bigm| \bigm| \bigm| \bigm| \bigm|
\bigl(
2bf \prime (b) + f \prime \prime (b)b2
\bigr) \bigl(
f(b)
\bigr) 2 - 2f(b)f \prime (b)b2f \prime (b)\bigl(
f(b)
\bigr) 4
\bigm| \bigm| \bigm| \bigm| \bigm| =
= 2
\bigm| \bigm| \bigm| \bigm| \bigm| 2bf \prime (b)\bigl(
f(b)
\bigr) 2 +
f \prime \prime (b)b2\bigl(
f(b)
\bigr) 2 - 2b2 (f \prime (b))2\bigl(
f(b)
\bigr) 3
\bigm| \bigm| \bigm| \bigm| \bigm| =
= 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2b
2
3\biggl(
2b
3
\biggr) 2 +
f \prime \prime (b)b2\biggl(
2b
3
\biggr) 2 -
2b2
\biggl(
2
3
\biggr) 2
\biggl(
2b
3
\biggr) 3
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= 2
\bigm| \bigm| \bigm| \bigm| 3b + f \prime \prime (b)
9
4
- 3
b
\bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1210 B. N. ÖRNEK
= 2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| 9
4
=
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm|
and \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 4
9
.
Theorem 1 is proved.
Inequality (2.1) can be strengthened as below by taking into account c2 and c3 which is second
and third coefficients in the expansion of the function f(z) = z + c2z
2 + c3z
3 + . . . .
Theorem 2. Under the same assumptions as in Theorem 1, we have
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 2
9
\Biggl(
1 +
2
1 +
\bigm| \bigm| H2(1)
\bigm| \bigm|
\Biggr)
. (2.2)
Proof. Let \Phi (z) be the same as in the proof of Theorem 1. Therefore, from (1.3) for p = 2, we
obtain
2 +
1 - | a2|
1 + | a2|
\leq | \Psi \prime (b)| = 9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| ,
where | a2| =
\bigm| \bigm| \Psi \prime \prime (0)
\bigm| \bigm|
2!
= 2| c3 - c22| = 2
\bigm| \bigm| H2(1)
\bigm| \bigm| .
Therefore, we take
2 +
1 - 2
\bigm| \bigm| H2(1)
\bigm| \bigm|
1 + 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \leq 9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| ,
1 +
2
1 + 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \leq 9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| ,
and \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 2
9
\Biggl(
1 +
2
1 +
\bigm| \bigm| H2(1)
\bigm| \bigm|
\Biggr)
.
Theorem 2 is proved.
In the following theorem, inequality (2.2) has been strenghened by adding the consecutive term
c4 of f(z) function.
Theorem 3. Let f \in \scrM . Assume that, for some b \in \partial U, f has an angular limit f(b) at b,
f(b) =
2b
3
and f \prime (b) =
2
3
. Then we have the inequality
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 2
9
\Biggl(
2 +
\bigl(
1 -
\bigm| \bigm| H2(1)
\bigm| \bigm| \bigr) 2
1 -
\bigl(
2
\bigm| \bigm| H2(1)
\bigm| \bigm| \bigr) 2 + 4
\bigm| \bigm| c4 - c2(c22 + 2H2(1))
\bigm| \bigm|
\Biggr)
. (2.3)
Proof. Let \Psi (z) be the same as in the proof of Theorem 1 and B0(z) = z2. By the maximum
principle, for each z \in U, we have the inequality | \Psi (z)| \leq | B0(z)| . Therefore,
m(z) =
\Psi (z)
B0(z)
=
2
\Biggl[ \biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\Biggr]
z2
=
= 2
\Bigl[ \bigl(
c3 - c22
\bigr)
+ (2c4 - 4c2c3 + 2c32)z + . . .
\Bigr]
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
ESTIMATES FOR ANALYTIC FUNCTIONS CONCERNED WITH HANKEL DETERMINANT 1211
is analytic function in U and
\bigm| \bigm| m(z)
\bigm| \bigm| \leq 1 for | z| < 1. In particular, we have
| m(0)| = 2| c3 - c22| = 2
\bigm| \bigm| H2(1)
\bigm| \bigm| (2.4)
and
| m\prime (0)| = 2
\bigm| \bigm| \bigm| 2c4 - 4c2c3 + 2c32
\bigm| \bigm| \bigm| = 4
\bigm| \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm| \bigm| .
Furthermore, the geometric meaning of the derivative and the inequality | \Psi (z)| \leq | B0(z)| imply the
inequality
b\Psi \prime (b)
\Psi (b)
= | \Psi \prime (b)| \geq | B\prime
0(b)| =
bB\prime
0(b)
B0(b)
.
The composite function
H(z) =
m(z) - m(0)
1 - m(0)m(z)
is analytic in U, H(0) = 0, | H(z)| < 1 for | z| < 1 and | H(b)| = 1 for b \in \partial U. For p = 1,
from (1.3), we obtain
2
1 +
\bigm| \bigm| H \prime (0)
\bigm| \bigm| \leq | H \prime (b)| = 1 - | m(0)| 2\bigm| \bigm| 1 - m(0)m(b)
\bigm| \bigm| 2 | m\prime (b)| \leq
\leq 1 + | m(0)|
1 - | m(0)|
\bigl\{
| \Psi \prime (b)| - | B\prime
0(b)|
\bigr\}
=
=
1 + 2
\bigm| \bigm| H2(1)
\bigm| \bigm|
1 - 2
\bigm| \bigm| H2(1)
\bigm| \bigm|
\biggl(
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2
\biggr)
.
Since
H \prime (z) =
1 - | m(0)| 2\Bigl(
1 - m(0)m(z)2
\Bigr) m\prime (z)
and \bigm| \bigm| H \prime (0)
\bigm| \bigm| = | m\prime (0)|
1 - | m(0)| 2
=
4
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
1 -
\Bigl(
2
\bigm| \bigm| H2(1)
\bigm| \bigm| \Bigr) 2 ,
we get
2
1 +
4
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
1 -
\bigl(
2
\bigm| \bigm| H2(1)
\bigm| \bigm| \bigr) 2
\leq
1 + 2
\bigm| \bigm| H2(1)
\bigm| \bigm|
1 - 2
\bigm| \bigm| H2(1)
\bigm| \bigm|
\biggl(
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2
\biggr)
and \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 2
9
\Biggl(
2 +
\bigl(
1 - | H2(1)|
\bigr) 2
1 - (2| H2(1)| )2 + 4
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
\Biggr)
.
Theorem 3 is proved.
If f(z) - z a have zeros different from z = 0, taking into account these zeros, inequality (2.3)
can be strengthened in another way. This is given by the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1212 B. N. ÖRNEK
Theorem 4. Let f \in \scrM . Assume that, for some b \in \partial U, f has an angular limit f(b) at b,
f(b) =
2b
3
and f \prime (b) =
2
3
. Let z1, z2, . . . , zn be zeros of the function f(z) - z in U that are different
from zero. Then we have the inequality
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 2
9
\Biggl(
2 +
n\sum
i=1
1 - | zi| 2
| b - zi| 2
+
+
2
\Bigl( \prod n
i=1
| zi| - 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \Bigr) 2\Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2+ 2
\prod n
i=1
| zi|
\bigm| \bigm| \bigm| \bigm| 2 \bigl( c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigr)
+H2(1)
\sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm|
\right) .
(2.5)
Proof. Let \Psi (z) be as in the proof of Theorem 1 and z1, z2, . . . , zn be zeros of the function
f(z) - z in U that are different from zero. Let
B1(z) = z2
n\prod
i=1
z - zi
1 - ziz
.
B(z) is an analytic function in U and | B(z)| < 1 for | z| < 1. By the maximum principle for each
z \in U , we have | \Psi (z)| \leq | B1(z)| . Consider the function
R(z) =
\Psi (z)
B1(z)
= 2
\Biggl[ \biggl(
z
f(z)
\biggr) 2
f \prime (z) - 1
\Biggr]
1
z2
\prod n
i=1
z - ai
1 - aiz
=
= 2
\bigl(
c3 - c22
\bigr)
z2 +
\bigl(
2c4 - 4c2c3 + 2c32
\bigr)
z3 + . . .
z2
\prod n
i=1
z - zi
1 - ziz
=
= 2
\bigl(
c3 - c22
\bigr)
+
\bigl(
2c4 - 4c2c3 + 2c32
\bigr)
z + . . .\prod n
i=1
z - zi
1 - ziz
.
R(z) is analytic in U and | R(z)| < 1 for z \in U. In particular, we have
| R(0)| = 2
| c3 - c22| \prod n
i=1
| zi|
=
2
\bigm| \bigm| H2(1)
\bigm| \bigm| \prod n
i=1
| zi|
and
| R\prime (0)| = 2
\bigm| \bigm| \bigm| \bigm| 2c4 - 4c2c3 + 2c32 +
\bigl(
c3 - c22
\bigr) \sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm| \prod n
i=1
| zi|
.
Moreover, with the simple calculations, we get
b\Psi \prime (b)
\Psi (b)
= | \Psi \prime (b)| \geq | B\prime
1(b)| =
bB\prime
1(b)
B1(b)
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ESTIMATES FOR ANALYTIC FUNCTIONS CONCERNED WITH HANKEL DETERMINANT 1213
and
| B\prime
1(b)| = 2 +
n\sum
i=1
1 - | zi| 2
| b - zi| 2
.
The auxiliary function
\Phi (z) =
R(z) - R(0)
1 - R(0)R(z)
is analytic in the unit disc U, \Phi (0) = 0, | \Phi (z)| < 1 for z \in U and | \Phi (b)| = 1 for b \in \partial U. From
(1.3) for p = 1, we obtain
2
1 + | \Phi \prime (0)|
\leq
\bigm| \bigm| \Phi \prime (b)
\bigm| \bigm| = 1 + | R(0)| 2\bigm| \bigm| \bigm| 1 - R(0)R(b)
\bigm| \bigm| \bigm| 2 | R\prime (b)| \leq
\leq 1 + | R(0)|
1 - | R(0)|
\bigl\{
| \Psi \prime (b)| - | B\prime
1(b)|
\bigr\}
.
Since
| \Phi \prime (0)| = | R\prime (0)|
1 - | R(0)| 2
=
2
\bigm| \bigm| \bigm| \bigm| 2c4 - 4c2c3 + 2c32 +
\bigl(
c3 - c22
\bigr) \sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm| \prod n
i=1
| zi|
1 -
\left( 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \prod n
i=1
| zi|
\right) 2 =
= 2
n\prod
i=1
| zi|
\bigm| \bigm| \bigm| \bigm| 2 \bigl( c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigr)
+H2(1)
\sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm| \Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2 ,
we get
2
1 + 2
\prod n
i=1
| zi|
\bigm| \bigm| \bigm| \bigm| 2 \bigl( c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigr)
+H2(1)
\sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm| \Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2
\leq
\leq
1 +
2
\bigm| \bigm| H2(1)
\bigm| \bigm| \prod n
i=1
| zi|
1 -
2
\bigm| \bigm| H2(1)
\bigm| \bigm| \prod n
i=1
| zi|
\Biggl\{
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2 -
n\sum
i=1
1 - | zi| 2
| b - zi| 2
\Biggr\}
,
2
\biggl( \Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2\biggr)
\Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2 + 2
\prod n
i=1
| zi|
\bigm| \bigm| \bigm| \bigm| 2 \bigl( c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigr)
+H2(1)
\sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1214 B. N. ÖRNEK
\leq
\prod n
i=1
| zi| + 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \prod n
i=1
| zi| - 2
\bigm| \bigm| H2(1)
\bigm| \bigm|
\Biggl\{
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2 -
n\sum
i=1
1 - | zi| 2
| b - zi| 2
\Biggr\}
,
2
\Bigl( \prod n
i=1
| zi| - 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \Bigr) 2\Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2 + 2
\prod n
i=1
| zi|
\bigm| \bigm| \bigm| \bigm| 2 \bigl( c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigr)
+H2(1)
\sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm| \leq
\leq 9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2 -
n\sum
i=1
1 - | zi| 2
| b - zi| 2
and
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 2
9
\left( 2 +
n\sum
i=1
1 - | zi| 2
| b - zi| 2
+
+
2
\Bigl( \prod n
i=1
| zi| - 2
\bigm| \bigm| H2(1)
\bigm| \bigm| \Bigr) 2\Bigl( \prod n
i=1
| zi|
\Bigr) 2
- 4
\bigm| \bigm| H2(1)
\bigm| \bigm| 2 + 2
\prod n
i=1
| zi|
\bigm| \bigm| \bigm| \bigm| 2 \bigl( c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigr)
+H2(1)
\sum n
i=1
1 - | zi| 2
zi
\bigm| \bigm| \bigm| \bigm|
\right) .
Theorem 4 is proved.
If f(z) - z has no zeros different from z = 0 in Theorem 3, the inequality (2.3) can be further
strengthened. This is given by the following theorem.
Theorem 5. Let f \in \scrM and c3 > c22 (c2 > 0, c3 > 0). Also, f(z) - z has no zeros in U
except z = 0. Further assume that, for some b \in \partial U, f has an angular limit f(b) at b, f(b) =
2b
3
and f \prime (b) =
2
3
. Then we have the inequalities
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 4
9
\Biggl(
1 - 1
2
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr)
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr)
-
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
\Biggr)
and \bigm| \bigm| c4 - c2(c
2
2 + 2H2(1))
\bigm| \bigm| \leq \bigm| \bigm| H2(1) \mathrm{l}\mathrm{n}(2H2(1))
\bigm| \bigm| .
Proof. Let c3 > c22 and \Psi (z), m(z) be as in the proof of Theorem 3. Having in mind
inequality (2.4), we denote by \mathrm{l}\mathrm{n}m(z) the analytic branch of the logarithm normed by the condition
\mathrm{l}\mathrm{n}m(0) = \mathrm{l}\mathrm{n}
\bigl(
2(c3 - c22)
\bigr)
= \mathrm{l}\mathrm{n} 2H2(1) < 0.
The function
l(z) =
\mathrm{l}\mathrm{n}m(z) - \mathrm{l}\mathrm{n}m(0)
\mathrm{l}\mathrm{n}m(z) + \mathrm{l}\mathrm{n}m(0)
is analytic in the unit disc U, | l(z)| < 1 for z \in U, l(0) = 0 and | l(b)| = 1 for b \in \partial U. From (1.3)
for p = 1, we obtain
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
ESTIMATES FOR ANALYTIC FUNCTIONS CONCERNED WITH HANKEL DETERMINANT 1215
2
1 + | l\prime (0)|
\leq
\bigm| \bigm| l\prime (b)\bigm| \bigm| = | 2 \mathrm{l}\mathrm{n}m(0)|
| \mathrm{l}\mathrm{n}m(b) + \mathrm{l}\mathrm{n}m(0)| 2
\bigm| \bigm| \bigm| \bigm| m\prime (c)
m(c)
\bigm| \bigm| \bigm| \bigm| =
=
- 2 \mathrm{l}\mathrm{n}m(0)
\mathrm{l}\mathrm{n}2m(0) + \mathrm{a}\mathrm{r}\mathrm{g}2m(b)
\bigl\{
| \Psi \prime (b)| - | B\prime
0(b)|
\bigr\}
.
Since \bigm| \bigm| l\prime (0)\bigm| \bigm| = 1
| 2 \mathrm{l}\mathrm{n}m(0)|
\bigm| \bigm| \bigm| \bigm| m\prime (0)
m(0)
\bigm| \bigm| \bigm| \bigm| = - 1
2 \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) 4 \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
2
\bigm| \bigm| H2(1)
\bigm| \bigm| =
=
- 1
2 \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) 2 \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
H2(1)
=
=
- 1
\mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
H2(1)
,
we have
1
1 -
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \leq - \mathrm{l}\mathrm{n}m(0)
\mathrm{l}\mathrm{n}2m(0) + \mathrm{a}\mathrm{r}\mathrm{g}2m(b)
\biggl(
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2
\biggr)
.
Replacing \mathrm{a}\mathrm{r}\mathrm{g}2m(b) by zero, we take
1
1 -
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \leq - 1
\mathrm{l}\mathrm{n}m(0)
\biggl(
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2
\biggr)
=
- 1
\mathrm{l}\mathrm{n} (2H2(1))
\biggl(
9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2
\biggr)
,
2 -
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr)
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr)
-
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm| \leq 9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm|
and \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 4
9
\Biggl(
1 - 1
2
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr)
H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr)
-
\bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
\Biggr)
.
Similarly, the function l(z) satisfies the assumptions of the Schwarz lemma, we obtain
1 \geq
\bigm| \bigm| l\prime (0)\bigm| \bigm| = | 2 \mathrm{l}\mathrm{n}m(0)|
| \mathrm{l}\mathrm{n}m(0) + \mathrm{l}\mathrm{n}m(0)| 2
\bigm| \bigm| \bigm| \bigm| m\prime (0)
m(0)
\bigm| \bigm| \bigm| \bigm| = - 1
2 \mathrm{l}\mathrm{n}m(0)
\bigm| \bigm| \bigm| \bigm| m\prime (0)
m(0)
\bigm| \bigm| \bigm| \bigm| =
=
- 1
2 \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) 4 \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm|
2
\bigm| \bigm| H2(1)
\bigm| \bigm| =
=
- 1
\mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm| \bigm| \bigm| H2(1)
\bigm| \bigm|
and \bigm| \bigm| c4 - c2
\bigl(
c22 + 2H2(1)
\bigr) \bigm| \bigm| \leq \bigm| \bigm| H2(1) \mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \bigm| \bigm| .
Theorem 5 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1216 B. N. ÖRNEK
Theorem 6. Under hypotheses of Theorem 5, we have
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 4
9
\biggl(
1 - 1
4
\mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \biggr)
.
Proof. From the proof of Theorem 5, using inequality (1.3) for the function l(z), for p = 1, we
obtain
1 \leq
\bigm| \bigm| l\prime (b)\bigm| \bigm| = | 2 \mathrm{l}\mathrm{n}m(0)|
| \mathrm{l}\mathrm{n}m(b) + \mathrm{l}\mathrm{n}m(0)| 2
\bigm| \bigm| \bigm| \bigm| m\prime (b)
m(b)
\bigm| \bigm| \bigm| \bigm| = - 2
\mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \biggl( 9
2
\bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| - 2
\biggr)
and \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \geq 4
9
\biggl(
1 - 1
4
\mathrm{l}\mathrm{n}
\bigl(
2H2(1)
\bigr) \biggr)
.
Theorem 6 is proved.
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Received 29.04.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
|
| id | umjimathkievua-article-907 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:03Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6d/09a439d79724e71fc790f3d38cc59e6d.pdf |
| spelling | umjimathkievua-article-9072025-03-31T08:46:40Z Estimates for analytic functions concerned with Hankel determinant Estimates for analytic functions concerned with Hankel determinant Örnek, B. N. Örnek, B. N. Fekete-Szegˆ functional Hankel determinant Jackís lemma Analytic function Schwarz lemma Fekete-Szegˆ functional Hankel determinant Jackís lemma Analytic function Schwarz lemma UDC 517.5 We give an upper bound of Hankel determinant of the first order $(H_{2}(1))$ for the classes of an analytic function. In addition, an evaluation with the Hankel determinant from below will be given for the second angular derivative of $f(z)$ analytic function. For new inequalities, the results of Jack's lemma and Hankel determinant were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of an angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained. &nbsp; УДК 517.5 Оцiнки для аналiтичних функцiй, пов’язанi з детермiнантом Ганкеля Отримано верхню границю детермінанта Ганкеля першого порядку $(H_{2}(1))$ для класів аналітичної функції. Також встановлено оцінку знизу з детермінантом Ганкеля для другої кутової похідної аналітичної функції $f(z).$ Для отримання нових нерівностей використано лему Джека та детермінант Ганкеля. Крім того, для класу аналітичних функцій на одиничному диску за умови існування кутової границі для межової точки отримано оцінки знизу для модуля кутової похідної. Institute of Mathematics, NAS of Ukraine 2021-09-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/907 10.37863/umzh.v73i9.907 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 9 (2021); 1205 - 1216 Український математичний журнал; Том 73 № 9 (2021); 1205 - 1216 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/907/9105 Copyright (c) 2021 Bülent Nafi ÖRNEK |
| spellingShingle | Örnek, B. N. Örnek, B. N. Estimates for analytic functions concerned with Hankel determinant |
| title | Estimates for analytic functions concerned with Hankel determinant |
| title_alt | Estimates for analytic functions concerned with Hankel determinant |
| title_full | Estimates for analytic functions concerned with Hankel determinant |
| title_fullStr | Estimates for analytic functions concerned with Hankel determinant |
| title_full_unstemmed | Estimates for analytic functions concerned with Hankel determinant |
| title_short | Estimates for analytic functions concerned with Hankel determinant |
| title_sort | estimates for analytic functions concerned with hankel determinant |
| topic_facet | Fekete-Szegˆ functional Hankel determinant Jackís lemma Analytic function Schwarz lemma Fekete-Szegˆ functional Hankel determinant Jackís lemma Analytic function Schwarz lemma |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/907 |
| work_keys_str_mv | AT ornekbn estimatesforanalyticfunctionsconcernedwithhankeldeterminant AT ornekbn estimatesforanalyticfunctionsconcernedwithhankeldeterminant |