Inequalities for complex rational functions
UDC 517.5 For the rational function $r(z)=p(z)/H(z)$ having all its zeros in $|z|\leq 1,$ it is known that\begin{equation*}|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{for}\quad |z|=1,\end{equation*}where $H(z)=\prod_{j=1}^n(z - c_j),$ $|c_j|>1,$ $n$ is a positive int...
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2021
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| author | Bidkham, M. Khojastehnezhad, E. Bidkham, M. Bidkham, M. Khojastehnezhad, E. |
| author_facet | Bidkham, M. Khojastehnezhad, E. Bidkham, M. Bidkham, M. Khojastehnezhad, E. |
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| description | UDC 517.5
For the rational function $r(z)=p(z)/H(z)$ having all its zeros in $|z|\leq 1,$ it is known that\begin{equation*}|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{for}\quad |z|=1,\end{equation*}where $H(z)=\prod_{j=1}^n(z - c_j),$ $|c_j|>1,$ $n$ is a positive integer, $B(z)=H^*(z)/H(z),$ and $H^*(z)=z^n\overline{H(1/\overline{z})}.$In this paper, we improve the above mentioned inequality for the rational function $r(z)$ with all zeros in $|z|\leq 1$ and a zero of order $s$ at the origin. Our main results refine and generalize some known rational inequalities. |
| doi_str_mv | 10.37863/umzh.v73i7.455 |
| first_indexed | 2026-03-24T02:02:44Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i7.455
UDC 517.5
M. Bidkham, E. Khojastehnezhad (Dep. Math., Semnan Univ., Iran)
INEQUALITIES FOR COMPLEX RATIONAL FUNCTIONS
НЕРIВНОСТI ДЛЯ КОМПЛЕКСНИХ РАЦIОНАЛЬНИХ ФУНКЦIЙ
For the rational function r(z) = p(z)/H(z) having all its zeros in | z| \leq 1, it is known that
| r\prime (z)| \geq 1
2
| B\prime (z)| | r(z)| for | z| = 1,
where H(z) =
\prod n
j=1
(z - cj), | cj | > 1, n is a positive integer, B(z) = H\ast (z)/H(z), and H\ast (z) = znH(1/z). In this
paper, we improve the above mentioned inequality for the rational function r(z) with all zeros in | z| \leq 1 and a zero of
order s at the origin. Our main results refine and generalize some known rational inequalities.
Для рацiональної функцiї r(z) = p(z)/H(z), що має всi нулi у | z| \leq 1, виконується нерiвнiсть
| r\prime (z)| \geq 1
2
| B\prime (z)| | r(z)| для | z| = 1,
де H(z) =
\prod n
j=1
(z - cj), | cj | > 1, n — додатне цiле, B(z) = H\ast (z)/H(z) i H\ast (z) = znH(1/z). У цiй роботi
вказану нерiвнiсть удосконалено для рацiональної функцiї r(z) iз нулями у | z| \leq 1 та нулем порядку s у початку
координат. Нашi основнi результати уточнюють та узагальнюють деякi вiдомi рацiональнi нерiвностi.
1. Introduction and statement of results. Everywhere in this paper we assume that m, n \in
\in \{ 1, 2, . . .\} . Let \scrP m be a class of all polynomials of degree at most m. For cj , j = 1, 2, . . . , n,
belong to the complex plane \BbbC , we take H(z) =
\prod n
j=1
(z - cj) and B(z) = H\ast (z)/H(z), where
H\ast (z) = znH(1/z). Here, z = x+ iy = x - iy, x, y are real and i is the imaginary complex unit.
Also, let
\scrR m,n = \scrR m,n(c1, . . . , cn) =
=
\left\{ p(z)/H(z); p \in \scrP m, H(z) =
n\prod
j=1
(z - cj), where | cj | > 1, j = 1, . . . , n
\right\}
denote the class of rational functions with poles at c1, c2, . . . , cn. For m = n, we write \scrR n := \scrR n,n.
Li et al. [4, 5] obtained Bernstein-type inequalities for the rational function r(z). They proved
that if the rational function r(z) \in \scrR n having all its zeros in | z| \leq 1, then, for | z| = 1,
| r\prime (z)| \geq 1
2
| B\prime (z)r(z)| . (1.1)
Let \alpha \in \BbbC is any fixed number. For a polynomial p(z) of degree n, D\alpha p(z), the polar derivative of
p(z) is defined by
D\alpha p(z) = np(z) + (\alpha - z)p\prime (z).
Polynomial D\alpha p(z) is of degree less than or equal n - 1 and
c\bigcirc M. BIDKHAM, E. KHOJASTEHNEZHAD, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7 879
880 M. BIDKHAM, E. KHOJASTEHNEZHAD
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow \infty
\biggl[
D\alpha p(z)
\alpha
\biggr]
= p\prime (z).
In this paper we first give a generalization and refinement of inequality (1.1) by proving the
following theorem.
Theorem 1.1. Let r(z) \in \scrR m,n has all its zeros in | z| \leq k, where k \leq 1, with a zero of order
s at the origin and m \leq n \leq 2(m+ sk)
1 + k
. Then, for any \gamma with | \gamma | \leq 1 and | z| = 1,\bigm| \bigm| \bigm| \bigm| zr\prime (z) + n\gamma
1 + k
r(z)
\bigm| \bigm| \bigm| \bigm| \geq 1
2
\biggl(
| B\prime (z)| + 2(m+ sk) - n(1 + k) + 2n\mathrm{R}\mathrm{e} \gamma
1 + k
\biggr)
| r(z)| .
Remark 1.1. In particular case, if we consider p(z) as a polynomial of degree n, then, for rational
function r(z) =
p(z)
H(z)
=
p(z)
(z - \alpha )n
, we have
r\prime (z) =
\biggl(
p(z)
(z - \alpha )n
\biggr) \prime
= - D\alpha p(z)
(z - \alpha )n+1
,
and, for B(z) =
H\ast (z)
H(z)
,
B\prime (z) = n
| \alpha | 2 - 1
(z - \alpha )2
\biggl(
1 - \alpha z
z - \alpha
\biggr) n - 1
.
Hence, for | z| = 1, we have | B\prime (z)| = n(| \alpha | 2 - 1)
| z - \alpha | 2
.
Now by taking m = n and cj = \alpha , j = 1, 2, . . . , n, in Theorem 1.1, for | z| = 1, we get\bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
1 + k
p(z)
\bigm| \bigm| \bigm| \bigm|
| z - \alpha | n+1
\geq 1
2
\biggl(
n(| \alpha | 2 - 1)
| z - \alpha | 2
+
2sk + n(1 - k) + 2n\mathrm{R}\mathrm{e} \gamma
1 + k
\biggr)
| p(z)|
| z - \alpha | n
, (1.2)
or \bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
1 + k
p(z)
\bigm| \bigm| \bigm| \bigm| \geq 1
2
\biggl(
n(| \alpha | 2 - 1)
| z - \alpha |
+
2sk + n(1 - k) + 2n\mathrm{R}\mathrm{e} \gamma
1 + k
| z - \alpha |
\biggr)
| p(z)| ,
or\bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
1 + k
p(z)
\bigm| \bigm| \bigm| \bigm| \geq 1
2
\biggl(
n(| \alpha | 2 - 1)
1 + | \alpha |
+
2sk + n(1 - k) + 2n\mathrm{R}\mathrm{e} \gamma
1 + k
(| \alpha | - 1)
\biggr)
| p(z)| .
Therefore, we have the following result which is a refinement of the result due to Dewan and Mir [3].
Corollary 1.1. Let p(z) be a polynomial of degree n, having all its zeros in | z| \leq k, where
k \leq 1, with a zero of order s at the origin. Then, for every \alpha with | \alpha | \geq 1, any \gamma with | \gamma | \leq 1 and
| z| = 1, we have\bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
1 + k
p(z)
\bigm| \bigm| \bigm| \bigm| \geq (n+ sk + n\mathrm{R}\mathrm{e} \gamma )(| \alpha | - 1)
1 + k
| p(z)| . (1.3)
By dividing both sides of (1.3) by | \alpha | and letting | \alpha | \rightarrow \infty , we get the following result which is
a generalization of the result due to Aziz and Shah [1].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
INEQUALITIES FOR COMPLEX RATIONAL FUNCTIONS 881
Corollary 1.2. If p(z) is a polynomial of degree n having all its zeros in | z| \leq k, k \leq 1, with a
zero of order s at the origin, then, for any \gamma with | \gamma | \leq 1 and | z| = 1,\bigm| \bigm| \bigm| \bigm| zp\prime (z) + n\gamma
1 + k
p(z)
\bigm| \bigm| \bigm| \bigm| \geq n+ sk + n\mathrm{R}\mathrm{e} \gamma
1 + k
| p(z)| .
If we take m = n and k = 1 in Theorem 1.1, then we have the following refinement of
inequality (1.1).
Corollary 1.3. If r(z) \in \scrR n having all its zeros in | z| \leq 1 with a zero of order s at the origin,
then, for any \gamma with | \gamma | \leq 1 and | z| = 1,\bigm| \bigm| \bigm| zr\prime (z) + n\gamma
2
r(z)
\bigm| \bigm| \bigm| \geq 1
2
(| B\prime (z)| + s+ n\mathrm{R}\mathrm{e} \gamma )| r(z)| . (1.4)
Next, we use \mathrm{m}\mathrm{i}\mathrm{n}| z| =1 | r(z)| to obtain the more precise of inequality (1.4).
Theorem 1.2. If r(z) \in \scrR n having all its zeros in | z| \leq 1, with a zero of order s at the origin,
then, for any \gamma with | \gamma | \leq 1 and | z| = 1,\bigm| \bigm| \bigm| zr\prime (z) + n\gamma
2
r(z)
\bigm| \bigm| \bigm| \geq
\geq 1
2
\biggl(
(| B\prime (z)| + s+ n\mathrm{R}\mathrm{e} \gamma )| r(z)| + (| B\prime (z)| + s+ n\mathrm{R}\mathrm{e} \gamma - | 2s+ n\gamma | ) \mathrm{m}\mathrm{i}\mathrm{n}
| z| =1
| r(z)|
\biggr)
.
Let \mathrm{m}\mathrm{i}\mathrm{n}| z| =1 | r(z)| = r(z0). Similar to (1.2), if we take cj = \alpha , j = 1, 2, . . . , n, in Theorem 1.2,
we have \bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
2
p(z)
\bigm| \bigm| \bigm| \bigm|
| z - \alpha | n+1
\geq 1
2
\biggl( \biggl(
n(| \alpha | 2 - 1)
| z - \alpha | 2
+ s+ n\mathrm{R}\mathrm{e} \gamma
\biggr)
| p(z)|
| z - \alpha | n
+
+
\biggl(
n(| \alpha | 2 - 1)
| z - \alpha | 2
+ s+ n\mathrm{R}\mathrm{e} \gamma - | 2s+ n\gamma |
\biggr)
| p(z0)|
| z0 - \alpha | n
\biggr)
,
or \bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
2
p(z)
\bigm| \bigm| \bigm| \bigm| \geq 1
2
\biggl( \biggl(
n(| \alpha | 2 - 1)
| z - \alpha |
+ (s+ n\mathrm{R}\mathrm{e} \gamma )| z - \alpha |
\biggr)
| p(z)| +
+
\biggl(
n(| \alpha | 2 - 1)
| z - \alpha |
- (| 2s+ n\gamma | - (s+ n\mathrm{R}\mathrm{e} \gamma ))| z - \alpha |
\biggr)
| z - \alpha | n
| z0 - \alpha | n
| p(z0)|
\biggr)
,
or \bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
2
p(z)
\bigm| \bigm| \bigm| \bigm| \geq 1
2
\biggl(
(n+ s+ n\mathrm{R}\mathrm{e} \gamma )(| \alpha | - 1)| p(z)| +
+((n - s+ n| \gamma | - n\mathrm{R}\mathrm{e} \gamma )| \alpha | - (n+ s+ n\mathrm{R}\mathrm{e} \gamma - n| \gamma | ))
\biggl(
| \alpha | - 1
| \alpha | + 1
\biggr) n
| p(z0)|
\biggr)
.
Therefore, the next result is obtained for the polar derivative of a polynomial.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
882 M. BIDKHAM, E. KHOJASTEHNEZHAD
Corollary 1.4. Let p(z) be a polynomial of degree n having all its zeros in | z| \leq 1 with a zero
of order s at the origin. Then, for every \alpha with | \alpha | \geq 1 and any \gamma with | \gamma | \leq 1, | z| = 1,\bigm| \bigm| \bigm| \bigm| zD\alpha p(z) +
n\gamma (\alpha - z)
2
p(z)
\bigm| \bigm| \bigm| \bigm| \geq 1
2
\biggl(
(n+ s+ n\mathrm{R}\mathrm{e} \gamma )(| \alpha | - 1)| p(z)| +
+ [(n - s+ n| \gamma | - n\mathrm{R}\mathrm{e} \gamma )| \alpha | - (n+ s+ n\mathrm{R}\mathrm{e} \gamma - n| \gamma | )]
\biggl(
| \alpha | - 1
| \alpha | + 1
\biggr) n
\mathrm{m}\mathrm{i}\mathrm{n}
| z| =1
| p(z)|
\biggr)
. (1.5)
By dividing both sides of inequality (1.5) by | \alpha | and letting | \alpha | \rightarrow \infty , we have the following
refinement of a result which has been proved by Dewan and Hans [2] (Theorem 1).
Corollary 1.5. If p(z) is a polynomial of degree n, having all its zeros in | z| \leq 1, with a zero of
order s at the origin, then, for any \gamma with | \gamma | \leq 1 and | z| = 1,\bigm| \bigm| \bigm| zp\prime (z) + n\gamma
2
p(z)
\bigm| \bigm| \bigm| \geq n+ s+ n\mathrm{R}\mathrm{e} \gamma
2
| p(z)| + n - s+ n| \gamma | - n\mathrm{R}\mathrm{e} \gamma
2
\mathrm{m}\mathrm{i}\mathrm{n}
| z| =1
| p(z)| .
By referring to the above theorems, the bounds which are obtained for rational function depends
only on the zero of largest modulus and not on the other zeros even if some of them are close to the
origin. Therefore, it would be interesting to obtain a bound which depends on the location of all the
zeros of a rational function. In this connection, we use some known ideas in the literature and obtain
the following interesting result.
Theorem 1.3. If r(z) =
zsp(z)
H(z)
\in \scrR n having all its zeros in | z| \leq 1, where p(z) = b
\prod n - s
i=1
(z -
- bi), | bi| \leq 1, 0 \leq s \leq n, and H(z) =
\prod n
j=1
(z - cj) by | cj | > 1, j = 1, . . . , n, then, for | z| = 1,
| r\prime (z)| \geq 1
2
\left( s+
1 -
\prod n - s
i=1
| bi|
1 +
\prod n - s
i=1
| bi|
+ | B\prime (z)|
\right) | r(z)| . (1.6)
Similar to (1.2), under the formula (1.6) in cj = \alpha , j = 1, 2, . . . , n, we have the following result.
Corollary 1.6. Let p(z) = bzs
\prod n - s
i=1
(z - bi), 0 \leq s \leq n, be a polynomial of degree n having
all its zeros in | z| \leq 1. Then, for every \alpha with | \alpha | \geq 1 and | z| = 1,
| D\alpha p(z)| \geq
| \alpha | - 1
2
\left( n+ s+
1 -
\prod n - s
i=1
| bi|
1 +
\prod n - s
i=1
| bi|
\right) | p(z)| . (1.7)
By dividing both sides of inequality (1.7) by | \alpha | and letting | \alpha | \rightarrow \infty , we have the following
refinement of the result which proved by Zireh [7].
Corollary 1.7. If p(z) = bzs
\prod n - s
i=1
(z - bi), 0 \leq s \leq n, is a polynomial of degree n having all
its zeros in | z| \leq 1, then, for | z| = 1,
| p\prime (z)| \geq 1
2
\left( n+ s+
1 -
\prod n - s
i=1
| bi|
1 +
\prod n - s
i=1
| bi|
\right) | p(z)| .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
INEQUALITIES FOR COMPLEX RATIONAL FUNCTIONS 883
2. Proofs of the theorems. For the proofs of these theorems, we need the following lemma
which is due to Ossermann [6, p. 3514] (Lemma 1).
Lemma 2.1. Let f : D \rightarrow D be an analytic function, where D = \{ z \in \BbbC : | z| < 1\} . Assume
that f(0) = 0 and exists a continuous extendibility of f \prime (z) with | z| < 1 to the point b with | b| = 1.
Then
| f \prime (b)| \geq 2
1 + | f \prime (0)|
.
Proof of Theorem 1.1. By hypothesis we have
r(z) =
zsp(z)
H(z)
=
zs
\prod m - s
i=1
(z - bi)\prod n
j=1
(z - cj)
,
where bi, | bi| \leq k \leq 1, i = 1, . . . ,m - s, are the zeros of r(z). Therefore,
zr\prime (z)
r(z)
+
n\gamma
1 + k
= s+
zp\prime (z)
p(z)
- zH \prime (z)
H(z)
+
n\gamma
1 + k
=
= s+
m - s\sum
i=1
z
z - bi
- zH \prime (z)
H(z)
+
n\gamma
1 + k
. (2.1)
Now we have
\mathrm{R}\mathrm{e}
\biggl(
2zH \prime (z)
H(z)
\biggr)
- n =
n\sum
j=1
2z
z - cj
- 1 =
n\sum
j=1
1 + zcj
1 - zcj
=
=
n\sum
j=1
1 - | cj | 2
| 1 - zcj | 2
= - | B\prime (z)| ,
which implies
\mathrm{R}\mathrm{e}
\biggl(
zH \prime (z)
H(z)
\biggr)
=
n - | B\prime (z)|
2
. (2.2)
Now, for | z| = 1 and | bi| \leq k, where k \leq 1, we have
\mathrm{R}\mathrm{e}
z
z - bi
\geq 1
1 + k
. (2.3)
So, for | z| = 1, by (2.1), (2.2) and (2.3), we obtain
\mathrm{R}\mathrm{e}
\biggl(
zr\prime (z)
r(z)
+
n\gamma
1 + k
\biggr)
= s+\mathrm{R}\mathrm{e}
\Biggl(
m - s\sum
i=1
z
z - bi
\Biggr)
- \mathrm{R}\mathrm{e}
\biggl(
zH \prime (z)
H(z)
\biggr)
+
n\mathrm{R}\mathrm{e}\gamma
1 + k
=
= s+\mathrm{R}\mathrm{e}
\Biggl(
m - s\sum
i=1
z
z - bi
\Biggr)
- n - | B\prime (z)|
2
+
n\mathrm{R}\mathrm{e}\gamma
1 + k
\geq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
884 M. BIDKHAM, E. KHOJASTEHNEZHAD
\geq s+
m - s
1 + k
- n - | B\prime (z)|
2
+
n\mathrm{R}\mathrm{e}\gamma
1 + k
=
=
| B\prime (z)|
2
+
2(m+ sk) - n(1 + k) + 2n\mathrm{R}\mathrm{e}\gamma
2(1 + k)
,
and, hence, \bigm| \bigm| \bigm| \bigm| zr\prime (z)r(z)
+
n\gamma
1 + k
\bigm| \bigm| \bigm| \bigm| \geq \mathrm{R}\mathrm{e}
\biggl(
zr\prime (z)
r(z)
+
n\gamma
1 + k
\biggr)
\geq
\geq | B\prime (z)|
2
+
2(m+ sk) - n(1 + k) + 2\mathrm{R}\mathrm{e}\gamma
2(1 + k)
,
from which we can obtain Theorem 1.1.
Proof of Theorem 1.2. Let m = \mathrm{m}\mathrm{i}\mathrm{n}| z| =1 | r(z)| . If m = 0, then we have the result from
Corollary 1.3. Suppose that m > 0, then m \leq | r(z)| for | z| = 1. If | \lambda | < 1, then it follows by
Rouche’s theorem that the rational function R(z) = r(z) + \lambda mzs has all its zeros in | z| \leq 1 with a
zero of order s at the origin. By applying inequality (1.4), for rational function R(z) for | z| = 1, we
obtain \bigm| \bigm| \bigm| zR\prime (z) +
n\gamma
2
R(z)
\bigm| \bigm| \bigm| \geq 1
2
(| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma )| R(z)|
or \bigm| \bigm| \bigm| zr\prime (z) + \lambda smzs +
n\gamma
2
(r(z) + \lambda mzs)
\bigm| \bigm| \bigm| \geq 1
2
(| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma )| r(z) + \lambda mzs| .
By using a suitable argument of \lambda , we have
| r(z) + \lambda mzs| = | r(z)| + | \lambda mzs| .
Using this equality for the right-hand side of above inequality, we get\bigm| \bigm| \bigm| zr\prime (z) + n\gamma
2
r(z)
\bigm| \bigm| \bigm| + | \lambda |
\bigm| \bigm| \bigm| s+ n\gamma
2
| m| zs
\bigm| \bigm| \bigm| \geq 1
2
(| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma )(| r(z)| + | \lambda | m).
Since | z| = 1, then\bigm| \bigm| \bigm| zr\prime (z) + n\gamma
2
r(z)
\bigm| \bigm| \bigm| \geq 1
2
((| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma )| r(z)| + (| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma - | 2s+ n\gamma | )| \lambda | m).
Now making | \lambda | \rightarrow 1, we get\bigm| \bigm| \bigm| zr\prime (z) + n\gamma
2
r(z)
\bigm| \bigm| \bigm| \geq 1
2
((| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma )| r(z)| + (| B\prime (z)| + s+ n\mathrm{R}\mathrm{e}\gamma - | 2s+ n\gamma | )m),
from which we can obtain Theorem 1.2.
Proof of Theorem 1.3. By similar argument in Theorem 1.1, we can write
r(z) =
zsp(z)
H(z)
=
bzs
\prod n - s
i=1
(z - bi)\prod n
j=1
(z - cj)
,
where bi, | bi| \leq 1, i = 1, . . . , n - s, are the zeros of r(z). Therefore,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
INEQUALITIES FOR COMPLEX RATIONAL FUNCTIONS 885
\mathrm{R}\mathrm{e}
\biggl(
zr\prime (z)
r(z)
\biggr)
= s+\mathrm{R}\mathrm{e}
\biggl(
zp\prime (z)
p(z)
\biggr)
- \mathrm{R}\mathrm{e}
\biggl(
zH \prime (z)
H(z)
\biggr)
. (2.4)
Now we calculate \mathrm{R}\mathrm{e}
\biggl(
zp\prime (z)
p(z)
\biggr)
. Since p(z) is a polynomial of degree (n - s), which has all its
zeros in | z| \leq 1 and implies q(z) = zn - sp(1/z) \not = 0 in | z| < 1, then S(z) =
zp(z)
q(z)
is analytic
function in | z| \leq 1, where S(0) = 0 with | S(z)| = 1 for | z| = 1. Applying Lemma 2.1 to S(z), we
conclude that
| S\prime (z)| \geq 2
1 + | S\prime (0)|
. (2.5)
Now, for S(z) =
zp(z)
q(z)
,
zS\prime (z)
S(z)
= 1 +
zp\prime (z)
p(z)
- zq\prime (z)
q(z)
. (2.6)
Since q(z) = zn - sp(1/z), then
q\prime (z) = (n - s)z(n - s - 1)p(1/z) - z(n - s - 2)p\prime (1/z).
Also, for | z| = 1,
zq\prime (z)
q(z)
= (n - s) -
\biggl(
zp\prime (z)
p(z)
\biggr)
. (2.7)
From (2.6) and (2.7), we get
zS\prime (z)
S(z)
= - (n - s - 1) + 2\mathrm{R}\mathrm{e}
zp\prime (z)
p(z)
for | z| = 1. (2.8)
Also
S(z) =
zp(z)
q(z)
= z
b
b
n - s\prod
i=1
\biggl(
z - bi
1 - biz
\biggr)
,
hence, for | z| = 1,
zS\prime (z)
S(z)
= 1 +
n - s\sum
i=1
z
z - bi
+
n - s\sum
i=1
biz
1 - biz
= 1 +
n - s\sum
i=1
1 - | bi| 2
| z - bi| 2
.
It means that
zS\prime (z)
S(z)
is positive and
zS\prime (z)
S(z)
=
\bigm| \bigm| \bigm| \bigm| zS\prime (z)
S(z)
\bigm| \bigm| \bigm| \bigm| for | z| = 1. Since | S(z)| = 1 for | z| = 1,
hence,
zS\prime (z)
S(z)
= | S\prime (z)| for | z| = 1. (2.9)
By using (2.8) and (2.9), we have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
886 M. BIDKHAM, E. KHOJASTEHNEZHAD
| S\prime (z)| = - (n - s - 1) + 2\mathrm{R}\mathrm{e}
zp\prime (z)
p(z)
for | z| = 1. (2.10)
As
S(z) = z
b
b
n - s\prod
i=1
\biggl(
z - bi
1 - biz
\biggr)
,
hence
| S\prime (0)| =
n - s\prod
i=1
| bi| . (2.11)
From (2.5), (2.10) and (2.11), we get
\mathrm{R}\mathrm{e}
zp\prime (z)
p(z)
\geq n - s - 1
2
+
1
1 +
\prod n - s
i=1
| bi|
for | z| = 1. (2.12)
By combining the relations (2.2), (2.4) and (2.12), the required result is obtained.
Theorem 1.3 is proved.
References
1. A. Aziz, W. M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl., 7, 379 – 391 (2004).
2. K. K. Dewan, S. Hans, Generalization of certain well-known polynomial inequalities, J. Math. Anal. and Appl., 363,
38 – 41 (2010).
3. K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interdisciplinary Math., 10, 525 – 531
(2007).
4. X. Li, A comparison inequality for rational functions, Proc. Amer. Math. Soc., 139, 1659 – 1665 (2011).
5. X. Li, R. N. Mohapatra, R. S. Rodriguez, Bernstein type inequalities for rational functions with prescribed poles,
J. London Math. Soc., 51, 523 – 531 (1995).
6. R. Osserman, A sharp Schwarz inequality on the boundary for functions regular in the disk, Proc. Amer. Math. Soc.,
12, 3513 – 3517 (2000).
7. A. Zireh, Generalization of certain well-known inequalities for the derivative of polynomials, Anal. Math., 41,
117 – 132 (2015).
Received 25.10.18,
after revision — 13.01.20
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
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| id | umjimathkievua-article-455 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:44Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fb/76493444467bc17456c8c88c1b3706fb.pdf |
| spelling | umjimathkievua-article-4552025-03-31T08:47:53Z Inequalities for complex rational functions Inequalities for complex rational functions Inequalities for complex rational functions Bidkham, M. Khojastehnezhad, E. Bidkham, M. Bidkham, M. Khojastehnezhad, E. Rational functions Polynomial Polar derivative Inequality Restricted Zeros Rational functions Polynomial Polar derivative Inequality Restricted Zeros UDC 517.5 For the rational function $r(z)=p(z)/H(z)$ having all its zeros in $|z|\leq 1,$ it is known that\begin{equation*}|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{for}\quad |z|=1,\end{equation*}where $H(z)=\prod_{j=1}^n(z - c_j),$ $|c_j|&gt;1,$ $n$ is a positive integer, $B(z)=H^*(z)/H(z),$ and $H^*(z)=z^n\overline{H(1/\overline{z})}.$In this paper, we improve the above mentioned inequality for the rational function $r(z)$ with all zeros in $|z|\leq 1$ and a zero of order $s$ at the origin. Our main results refine and generalize some known rational inequalities. УДК 517.5 Нерiвностi для комплексних рацiональних функцiй Для раціональної функції $r(z)=p(z)/H(z),$ що має всі нулі у $|z|\leq 1,$ виконується нерівність\begin{equation*}|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{для}\quad |z|=1,\end{equation*}де $H(z)=\prod_{j=1}^n(z-c_j),$ $|c_j|&gt;1,$ $n$ – додатне ціле, $B(z)=H^*(z)/H(z)$ і $H^*(z)=z^n\overline{H(1/\overline{z})}.$У цій роботі вказану нерівність удосконалено для раціональної функції $r(z)$ із нулями у $|z|\leq 1$ та нулем порядку $s$ у початку координат. Наші основні результати уточнюють та узагальнюють деякі відомі раціональні нерівності. Institute of Mathematics, NAS of Ukraine 2021-07-20 Article Article application/pdf application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/455 10.37863/umzh.v73i7.455 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 7 (2021); 879 - 886 Український математичний журнал; Том 73 № 7 (2021); 879 - 886 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/455/9062 https://umj.imath.kiev.ua/index.php/umj/article/view/455/9063 Copyright (c) 2021 M. Bidkham, E. Khojastehnezhad |
| spellingShingle | Bidkham, M. Khojastehnezhad, E. Bidkham, M. Bidkham, M. Khojastehnezhad, E. Inequalities for complex rational functions |
| title | Inequalities for complex rational functions |
| title_alt | Inequalities for complex rational functions Inequalities for complex rational functions |
| title_full | Inequalities for complex rational functions |
| title_fullStr | Inequalities for complex rational functions |
| title_full_unstemmed | Inequalities for complex rational functions |
| title_short | Inequalities for complex rational functions |
| title_sort | inequalities for complex rational functions |
| topic_facet | Rational functions Polynomial Polar derivative Inequality Restricted Zeros Rational functions Polynomial Polar derivative Inequality Restricted Zeros |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/455 |
| work_keys_str_mv | AT bidkhamm inequalitiesforcomplexrationalfunctions AT khojastehnezhade inequalitiesforcomplexrationalfunctions AT bidkhamm inequalitiesforcomplexrationalfunctions AT bidkhamm inequalitiesforcomplexrationalfunctions AT khojastehnezhade inequalitiesforcomplexrationalfunctions |