On an operator preserving inequalities between polynomials
Let $P(z)$ be a polynomial of degree n. We consider an operator $D\alpha$ that maps $P(z)$ into $D\alpha P(z) := nP(z)+(\alpha z)P\prime (z)$ and establish some results concerning the estimates of $| D\alpha P(z)| $ on the disk $| z| = R \geq 1$, and thereby obtain extensions and generalizations...
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Institute of Mathematics, NAS of Ukraine
2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507613477732352 |
|---|---|
| author | Mir, A. Мир, А. |
| author_facet | Mir, A. Мир, А. |
| author_sort | Mir, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:25:58Z |
| description | Let $P(z)$ be a polynomial of degree n. We consider an operator $D\alpha$ that maps $P(z)$ into $D\alpha P(z) := nP(z)+(\alpha z)P\prime (z)$
and establish some results concerning the estimates of $| D\alpha P(z)| $ on the disk $| z| = R \geq 1$, and thereby obtain extensions
and generalizations of a number of well-known polynomial inequalities. |
| first_indexed | 2026-03-24T02:12:06Z |
| format | Article |
| fulltext |
UDC 517.5
A. Mir (Univ. Kashmir, Srinagar, India)
ON AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS
ПРО ОПЕРАТОР, ЩО ЗБЕРIГАЄ НЕРIВНОСТI МIЖ ПОЛIНОМАМИ
Let P (z) be a polynomial of degree n. We consider an operator D\alpha that maps P (z) into D\alpha P (z) := nP (z)+(\alpha - z)P \prime (z)
and establish some results concerning the estimates of | D\alpha P (z)| on the disk | z| = R \geq 1, and thereby obtain extensions
and generalizations of a number of well-known polynomial inequalities.
Нехай P (z) — многочлен степеня n. У роботi розглянуто оператор D\alpha , що вiдображає P (z) в D\alpha P (z) :=
:= nP (z) + (\alpha - z)P \prime (z), та встановлено деякi результати щодо оцiнок | D\alpha P (z)| на крузi | z| = R \geq 1 i, таким
чином, отримано розширення та узагальнення багатьох вiдомих нерiвностей для полiномiв.
1. Introduction. Let \BbbP n denote the class of all complex polynomials of degree at most n. Let Dk -
denote the region inside the disk \BbbT k = \{ z \in \BbbC /| z| = k > 0\} and Dk+ the region outside \BbbT k. For
P \in \BbbP n, set
M(P, k) = \mathrm{m}\mathrm{a}\mathrm{x}
z\in \BbbT k
| P (z)| and m(P, k) = \mathrm{m}\mathrm{i}\mathrm{n}
z\in \BbbT k
| P (z)| .
If P \in \BbbP n, then concerning the estimate of M(P \prime , 1) on \BbbT 1, we have
M(P \prime , 1) \leq nM(P, 1). (1.1)
The above inequality is an immediate consequence of Bernstein’s inequality [3] on the derivative of a
trigonometric polynomial, and is best possible with equality holding for the polynomial P (z) = \lambda zn,
\lambda being a complex number.
If we restrict ourselves to the class of polynomials having no zeros in in the open unit disk, then
the above inequality can be sharpened. In fact, Erdös conjectured, and later Lax [8] proved, that if
P \in \BbbP n and P (z) has all its zeros in \BbbT 1 \cup D1+, then
M(P \prime , 1) \leq n
2
M(P, 1). (1.2)
The above inequality is best possible, and holds with equality for all polynomials having their zeros
on \BbbT 1.
As a refinement of (1.2), Aziz and Dawood [1] proved that if P \in \BbbP n and P (z) has all its zeros
in \BbbT 1 \cup D1+, then
M(P \prime , 1) \leq n
2
\bigl\{
M(P, 1) - m(P, 1)
\bigr\}
. (1.3)
Further, as an extension of (1.3), Jain [7] (see also Dewan and Hans [5]) proved that if P \in \BbbP n
and P (z) has all its zeros in \BbbT 1 \cup D1+, then, for any \beta with | \beta | \leq 1 and z \in \BbbT 1,\bigm| \bigm| \bigm| \bigm| zP \prime (z) +
n\beta
2
P (z)
\bigm| \bigm| \bigm| \bigm| \leq n
2
\biggl\{ \biggl( \bigm| \bigm| \bigm| \bigm| 1 + \beta
2
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| \beta 2
\bigm| \bigm| \bigm| \bigm| \biggr) M(P, 1) -
\biggl( \bigm| \bigm| \bigm| \bigm| 1 + \beta
2
\bigm| \bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \bigm| \beta 2
\bigm| \bigm| \bigm| \bigm| \biggr) m(P, 1)
\biggr\}
. (1.4)
For P \in \BbbP n, the polar derivative D\alpha P (z) of P (z) with respect to the point \alpha is defined as
D\alpha P (z) = nP (z) + (\alpha - z)P \prime (z).
c\bigcirc A. MIR, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8 1061
1062 A. MIR
It is easy to see that D\alpha P (z) is a polynomial of degree at most n - 1, and D\alpha P (z) generalizes
the ordinary derivative in the sense that
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow \infty
\Biggl\{
D\alpha P (z)
\alpha
\Biggr\}
= P \prime (z).
Corresponding to a given nth degree polynomial P (z), we construct a sequence of polar deriva-
tives as follows:
D\alpha 1P (z) = nP (z) + (\alpha 1 - z)P \prime (z),
and
D\alpha k
D\alpha k - 1
. . . D\alpha 1P (z) = (n - k + 1)D\alpha k - 1
D\alpha k - 2
. . . D\alpha 1P (z)+
+(\alpha k - z)
\bigl(
D\alpha k - 1
D\alpha k - 2
. . . D\alpha 1P (z)
\bigr) \prime
, k = 2, 3, . . . , n.
The points \alpha 1, \alpha 2, . . . \alpha k \in \BbbC , k = 1, 2, 3, . . . , n, may or may not be distinct. Like the kth
ordinary derivative P (k)(z) of P (z), the k th polar derivative D\alpha k
D\alpha k - 1
. . . D\alpha 1P (z) of P (z) is a
polynomial of degree at most n - k.
As an extension of inequality (1.3) to the polar derivative of a polynomial, Aziz and Shah [2]
(see also Mir and Baba [11]) showed that if P \in \BbbP n and P (z) has all its zeros in \BbbT 1 \cup D1+, then,
for every \alpha with | \alpha | \geq 1,
M(D\alpha P, 1) \leq
n
2
\Bigl\{ \bigl(
| \alpha | + 1
\bigr)
M(P, 1) -
\bigl(
| \alpha | - 1
\bigr)
m(P, 1)
\Bigr\}
. (1.5)
In the literature, there exist various refinements and generalization of (1.2) – (1.5) and here, we
mention a few of them.
Theorem 1.1 ([9], Theorem 3). If P \in \BbbP n and P (z) has all its zeros in \BbbT 1 \cup D1+, then, for \alpha ,
\beta \in \BbbC with | \alpha | \geq 1, | \beta | \leq 1 and z \in \BbbT 1,\bigm| \bigm| \bigm| \bigm| zD\alpha P (z) + n\beta
\biggl(
| \alpha | - 1
2
\biggr)
P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n
2
\Biggl\{ \Biggl( \bigm| \bigm| \bigm| \bigm| \alpha + \beta
\biggl(
| \alpha | - 1
2
\biggr) \bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| z + \beta
\biggl(
| \alpha | - 1
2
\biggr) \bigm| \bigm| \bigm| \bigm|
\Biggr)
M(P, 1) -
-
\Biggl( \bigm| \bigm| \bigm| \bigm| \alpha + \beta
\biggl(
| \alpha | - 1
2
\biggr) \bigm| \bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \bigm| z + \beta
\biggl(
| \alpha | - 1
2
\biggr) \bigm| \bigm| \bigm| \bigm|
\Biggr)
m(P, 1)
\Biggr\}
. (1.6)
Theorem 1.2 ([6], Theorem 2). If P \in \BbbP n and P (z) has all its zeros in \BbbT 1 \cup D1+, then, for
every complex number \beta with | \beta | \leq 1, 1 \leq s \leq n and z \in \BbbT 1,\bigm| \bigm| \bigm| \bigm| zsP (s)(z) +
\beta n(n - 1) . . . (n - s+ 1)
2s
P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - s+ 1)
2
\biggl\{ \biggl( \bigm| \bigm| \bigm| \bigm| 1 + \beta
2s
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| \beta 2s
\bigm| \bigm| \bigm| \bigm| \biggr) M(P, 1) -
-
\biggl( \bigm| \bigm| \bigm| \bigm| 1 + \beta
2s
\bigm| \bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \bigm| \beta 2s
\bigm| \bigm| \bigm| \bigm| \biggr) m(P, 1)
\biggr\}
. (1.7)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
ON AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS 1063
Theorem 1.3 ([13], Theorem 1.5). If P \in \BbbP n and P (z) has all its zeros in \BbbT 1 \cup Dk+, k \leq 1,
then, for every complex number \beta with | \beta | \leq 1, 1 \leq s \leq n and z \in \BbbT 1, we have
\mathrm{m}\mathrm{a}\mathrm{x}
| z| =1
\bigm| \bigm| \bigm| \bigm| zsP (s)(z) +
\beta n(n - 1) . . . (n - s+ 1)
(1 + k)s
P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - s+ 1)
2
\biggl\{ \biggl(
1
kn
\bigm| \bigm| \bigm| \bigm| 1 + \beta
(1 + k)s
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| \beta
(1 + k)s
\bigm| \bigm| \bigm| \bigm| \biggr) M(P, k) -
-
\biggl(
1
kn
\bigm| \bigm| \bigm| \bigm| 1 + \beta
(1 + k)s
\bigm| \bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \bigm| \beta
(1 + k)s
\bigm| \bigm| \bigm| \bigm| \biggr) m(P, k)
\biggr\}
. (1.8)
2. Statements of results. In this section we state our main results. Their proofs are given in the
next section. From now on, we shall always assume that every P \in \BbbP n is a polynomial of degree
n \geq 2. Our main aim is to extend (1.8) to the polar derivative of a polynomial and thereby obtain a
compact generalization of (1.7) as well. We start by proving the following result.
Theorem 2.1. Let P \in \BbbP n and P (z) has all its zeros in \BbbT k\cup Dk - , k \leq 1. Let t \in \BbbN , t \leq n - 1,
and (\alpha i)
t
i=1 be complex numbers satisfying | \alpha i| \geq k for 1 \leq i \leq t. Then, for every \beta \in \BbbC with
| \beta | \leq 1 and for every z \in \BbbT 1 \cup D1+,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
kn
| z| n
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| m(P, k). (2.1)
Remark 2.1. If we take \alpha 1 = \alpha 2 = . . . = \alpha t = \alpha , divide both sides of (2.1) by | \alpha | t and let
| \alpha | \rightarrow \infty , we obtain the following result.
Corollary 2.1. Let P \in \BbbP n and P (z) has all its zeros in \BbbT k \cup Dk - , k \leq 1. Then, for every \beta
with | \beta | \leq 1, 1 \leq t \leq n - 1 and for every z \in \BbbT 1 \cup D1+,\bigm| \bigm| \bigm| \bigm| ztP (t)(z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
P (z)
\bigm| \bigm| \bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
kn
| z| n
\bigm| \bigm| \bigm| \bigm| 1 + \beta
(1 + k)t
\bigm| \bigm| \bigm| \bigm| m(P, k). (2.2)
Remark 2.2. For k = 1, Corollary 2.1 in particular reduces to a result of Hans and Lal ([6],
Lemma 7) and for | z| = 1, Corollary 2.1 is exactly Theorem 2.1 recently proved by Zireh [13].
Further, for k = 1, Theorem 2.1 reduces to a result of Bidkham and Mezerji ([4], Corollary 3).
Next, we present the following extension of (1.8) to the polar derivative.
Theorem 2.2. Let P \in \BbbP n and P (z) has all its zeros in \BbbT k\cup Dk+, k \leq 1. Let t \in \BbbN , t \leq n - 1,
and (\alpha i)
t
i=1 be complex numbers satisfying | \alpha i| \geq k for 1 \leq i \leq t. Then, for every \beta \in \BbbC with
| \beta | \leq 1 and for every z \in \BbbT 1 \cup D1+,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - t+ 1)
2
\biggl\{ \biggl(
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| +
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1064 A. MIR
+
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| \biggr) M(P, k) -
-
\biggl(
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| -
-
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| \biggr) m(P, k)
\biggr\}
. (2.3)
Remark 2.3. If we take \alpha 1 = \alpha 2 = . . . = \alpha t = \alpha , then divide both sides of (2.3) by | \alpha | t and
let | \alpha | \rightarrow \infty , we recover (1.8). For t = 1, Theorem 2.2 gives the following result.
Corollary 2.2. Let P \in \BbbP n and P (z) has all its zeros in \BbbT k \cup Dk+, k \leq 1, then, for \alpha , \beta \in \BbbC
with | \alpha | \geq k, | \beta | \leq 1 and z \in \BbbT 1 \cup D1+,\bigm| \bigm| \bigm| \bigm| zD\alpha P (z) +
\beta n(| \alpha | - k)
1 + k
P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n
2
\biggl\{ \biggl(
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha +
\beta (| \alpha | - k)
1 + k
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| z + \beta (| \alpha | - k)
1 + k
\bigm| \bigm| \bigm| \bigm| \biggr) M(P, k) -
-
\biggl(
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha +
\beta (| \alpha | - k)
1 + k
\bigm| \bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \bigm| z + \beta (| \alpha | - k)
1 + k
\bigm| \bigm| \bigm| \bigm| \biggr) m(P, k)
\biggr\}
. (2.4)
Remark 2.4. For k = 1, the above Corollary 2.2 simplifies to inequality (1.6). For \beta = 0,
Theorem 2.2 reduces to the following result which gives a generalization of inequality (1.5).
Corollary 2.3. Let P \in \BbbP n and P (z) has all its zeros in \BbbT k\cup Dk+, k \leq 1. Let t \in \BbbN , t \leq n - 1,
and (\alpha i)
t
i=1 be complex numbers satisfying | \alpha i| \geq k for 1 \leq i \leq t. Then, for z \in \BbbT 1, we have\bigm| \bigm| \bigm| \bigm| D\alpha tD\alpha t - 1 . . . D\alpha 1P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - t+ 1)
2
\biggl\{ \biggl(
1
kn
\bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t
\bigm| \bigm| + 1
\biggr)
M(P, k) -
-
\biggl(
1
kn
\bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t
\bigm| \bigm| - 1
\biggr)
m(P, k)
\biggr\}
. (2.5)
Remark 2.5. For k = t = 1, (2.5) reduces to (1.5) and for k = 1, Corollary 2.3 reduces to a
result of Bidkham and Mezerji ([4], Corollary 7). Dividing the two sides of (2.4) by | \alpha | and let
| \alpha | \rightarrow \infty , we have the following generalization of the inequality (1.4).
Corollary 2.4. If P \in \BbbP n and P (z) has all its zeros in \BbbT k \cup Dk+, k \leq 1, then, for \alpha , \beta \in \BbbC
with | \alpha | \geq k, | \beta | \leq 1 and z \in \BbbT 1 \cup D1+,\bigm| \bigm| \bigm| \bigm| zP \prime (z) +
n\beta
1 + k
P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n
2
\biggl\{ \biggl(
| z| n
kn
\bigm| \bigm| \bigm| \bigm| 1 + \beta
1 + k
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| \beta
1 + k
\bigm| \bigm| \bigm| \bigm| \biggr) M(P, k) -
-
\biggl(
| z| n
kn
\bigm| \bigm| \bigm| \bigm| 1 + \beta
1 + k
\bigm| \bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \bigm| \beta
1 + k
\bigm| \bigm| \bigm| \bigm| \biggr) m(P, k)
\biggr\}
. (2.6)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
ON AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS 1065
3. Proofs. We need the following lemmas for the proof of theorems.
Lemma 3.1 ([12], Lemma 2.3). Let P \in \BbbP n and P (z) has all its zeros in \BbbT k\cup Dk - , k \leq 1. Let
(\alpha i)
t
i=1, t \leq n - 1, are complex numbers satisfying | \alpha i| \geq k, 1 \leq i \leq t. Then, for z \in \BbbT 1, we have\bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z)
\bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)| P (z)| . (3.1)
Lemma 3.2. Let P, F \in \BbbP n and F (z) has all its zeros in \BbbT k \cup Dk - , k \leq 1, such that | P (z)| \leq
\leq | F (z)| for z \in \BbbT k. Let t \in \BbbN , t \leq n - 1, and (\alpha i)
t
i=1, are complex numbers satisfying | \alpha i| \geq k,
for 1 \leq i \leq t. Then, for any \beta \in \BbbC with | \beta | \leq 1 and z \in \BbbT 1 \cup D1+,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1F (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)F (z)
\bigm| \bigm| \bigm| \bigm| . (3.2)
Proof of Lemma 3.2. By hypothesis | P (z)| \leq | F (z)| on | z| = k. Hence, for any \alpha \in \BbbC with
| \alpha | < 1, we have | \alpha P (z)| < | F (z)| on the circle | z| = k. Further, all the zeros of F (z) lie in
| z| \leq k, it follows by Rouche’s theorem that all the zeros of G(z) = F (z) + \alpha P (z) with | \alpha | < 1,
also lie in | z| \leq k, k \leq 1. By applying Lemma 3.1 to G(z), we get, for | \alpha i| \geq k, 1 \leq i \leq t, | \alpha | < 1
and | z| = 1, \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1G(z)
\bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)| G(z)| .
Equivalently, for | z| = 1,\bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1F (z) + \alpha ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z)
\bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)| F (z) + \alpha P (z)| . (3.3)
Therefore, for any \beta with | \beta | < 1, we have by Rouche’s theorem, the polynomial
T (z) =
\bigl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1F (z) + \alpha ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z)
\bigr)
+
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)(F (z) + \alpha P (z)) =
=
\biggl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1F (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)F (z)
\biggr)
+
+\alpha
\biggl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\biggr)
\not =
\not = 0 for | z| \geq k. (3.4)
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1066 A. MIR
Since k \leq 1, we have T (z) \not = 0 for | z| \geq 1 as well. Now choosing the argument of \alpha in (3.4)
suitably and letting | \alpha | \rightarrow 1, we get, for | z| \geq 1 and | \beta | < 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1F (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)F (z)
\bigm| \bigm| \bigm| \bigm| .
For \beta with | \beta | = 1, the above inequality holds by continuity.
Lemma 3.2 is proved.
Lemma 3.3. Let P \in \BbbP n, t \in \BbbN , t \leq n - 1, and (\alpha i)
t
i=1 are complex numbers satisfying
| \alpha i| \geq k, for 1 \leq i \leq t. Then, for any complex \beta with | \beta | \leq 1 and | z| \geq 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| +
+kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - t+ 1)
\biggl\{
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| +
+
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| \biggr\} M(P, k), (3.5)
where Q(z) = znP
\biggl(
1
z
\biggr)
.
Proof. Since M(P, k) = \mathrm{m}\mathrm{a}\mathrm{x}| z| =k | P (z)| . It follows by Rouche’s theorem, that for any \gamma with
| \gamma | > 1, the polynomial T (z) = P (z) +
\gamma M(P, k)zn
kn
has all zeros in | z| < k. If we set
S(z) = znT
\biggl(
1
z
\biggr)
= Q(z) +
\gamma M(P, k)
kn
,
then
\bigm| \bigm| \bigm| knS\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| = | T (z)| for | z| = k. Hence, for every complex \eta with | \eta | > 1, the polynomial
W (z) = knS
\Bigl( z
k2
\Bigr)
+\eta T (z) has all its zeros in | z| < k. Therefore, by applying Lemma 3.1 to W (z),
we obtain for | \alpha i| \geq k, 1 \leq i \leq t, t \leq n - 1,\bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1W (z)
\bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)| W (z)| for | z| = 1.
This implies, for any \beta with | \beta | < 1 and | z| = 1,\bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1W (z)
\bigm| \bigm| >
> | \beta | n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)| W (z)| for | z| = 1. (3.6)
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ON AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS 1067
Since by Laguerre’s theorem [10, p. 52], the polynomial D\alpha tD\alpha t - 1 . . . D\alpha 1W (z) has all its zeros in
| z| < k, k \leq 1, for every \alpha i with | \alpha i| \geq k, 1 \leq i \leq t, t \leq n - 1. Rouche’s theorem together with
(3.6) implies that the polynomial
G(z) = ztD\alpha tD\alpha t - 1 . . . D\alpha 1W (z)+
+\beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)W (z) =
= kn
\biggl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1S
\Bigl( z
k2
\Bigr)
+ \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)S
\Bigl( z
k2
\Bigr) \biggr)
+
+\eta
\biggl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1T (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)T (z)
\biggr)
\not =
\not = 0 for | z| \geq k. (3.7)
As k \leq 1, we have G(z) \not = 0 for | z| \geq 1 as well. Hence on choosing the arguement of \eta suitably
in (3.7) and letting | \eta | \rightarrow 1, we get, for | z| \geq 1 and | \beta | < 1,
kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1S
\Bigl( z
k2
\Bigr)
+ \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)S
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1T (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)T (z)
\bigm| \bigm| \bigm| \bigm| . (3.8)
Replacing T (z) by P (z)+
\gamma M(P, k)zn
kn
and S(z) by Q(z)+
\gamma M(P, k)
kn
in (3.8), we get, for | z| \geq 1,
kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+ \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr)
+
+
\gamma
kn
n(n - 1) . . . (n - t+ 1)
\biggl\{
zt +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr\}
M(P, k)
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)+
+
\gamma
kn
n(n - 1) . . . (n - t+ 1)
\biggl\{
\alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr\}
M(P, k)zn
\bigm| \bigm| \bigm| \bigm| . (3.9)
Applying Lemma 3.2 to the right-hand side of (3.9) and choosing the argument of \gamma so that\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)+
+
\gamma
kn
n(n - 1) . . . (n - t+ 1)
\biggl\{
\alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr\}
M(P, k)zn
\bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \gamma kn n(n - 1) . . . (n - t+ 1)
\biggl\{
\alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr\}
M(P, k)zn
\bigm| \bigm| \bigm| \bigm| -
-
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| ,
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1068 A. MIR
we get\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| +
+kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+ \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - t+ 1)| \gamma |
\biggl\{
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| +
+
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| \biggr\} M(P, k). (3.10)
Making | \gamma | \rightarrow 1 and using the continuity for | \beta | = 1 in (3.10), we get the desired result.
Proof of Theorem 2.1. If P (z) has a zeros on | z| = k, then the theorem is trivial. Therefore,
assume that P (z) has all its zeros in | z| < k, k \leq 1, so that m(P, k) > 0 and hence for every \gamma
with | \gamma | < 1, we have \bigm| \bigm| \bigm| \bigm| \gamma m(P, k)zn
kn
\bigm| \bigm| \bigm| \bigm| < | P (z)| for | z| = k.
It follows by Rouche’s theorem, that the polynomial G(z) = P (z) - \gamma m(P, k)zn
kn
of degree n has
all its zeros in | z| < k, k \leq 1. On applying Lemma 3.1 to G(z), we have, for | \alpha i| \geq k, 1 \leq i \leq t,
and | z| = 1,
\bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1G(z)
\bigm| \bigm| \geq n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)| G(z)| ,
i.e., \bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) - \gamma m(P, k)
kn
n(n - 1) . . . (n - t+ 1)\alpha 1\alpha 2 . . . \alpha tz
n
\bigm| \bigm| \bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)
\bigm| \bigm| \bigm| \bigm| P (z) - \gamma m(P, k)zn
kn
\bigm| \bigm| \bigm| \bigm| for | z| = 1. (3.11)
Applying Laguerre’s theorem [10, p. 52] repeatedly, we deduce that for | \alpha i| \geq k, 1 \leq i \leq t, and
| \gamma | < 1, the polynomial D\alpha tD\alpha t - 1 . . . D\alpha 1G(z) has all its zeros in | z| < k, k \leq 1, and therefore for
every complex \beta with | \beta | < 1, the polynomial
T (z) =
\biggl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) - \gamma m(P, k)
kn
n(n - 1) . . . (n - t+ 1)\alpha 1\alpha 2 . . . \alpha tz
n
\biggr)
+
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)
\biggl(
P (z) - \gamma m(P, k)zn
kn
\biggr)
=
=
\biggl(
ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\biggr)
-
- \gamma m(P, k)zn
kn
\biggl\{
n(n - 1) . . . (n - t+ 1)\alpha 1\alpha 2 . . . \alpha t+
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ON AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS 1069
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)
\biggr\}
\not = 0 for | z| \geq k. (3.12)
Since k \leq 1, we have T (z) \not = 0 for | z| \geq 1 as well. Now choosing the argument of \gamma in (3.12)
suitably and letting | \gamma | \rightarrow 1, we get, for | z| \geq 1 and | \beta | < 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \geq
\geq
\bigm| \bigm| \bigm| \bigm| m(P, k)zn
kn
\biggl\{
n(n - 1) . . . (n - t+ 1)\alpha 1\alpha 2 . . . \alpha t+
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)
\biggr\} \bigm| \bigm| \bigm| \bigm|
or \bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) +
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \geq
\geq | z| n
kn
\bigm| \bigm| \bigm| \bigm| n(n - 1) . . . (n - t+ 1)\alpha 1\alpha 2 . . . \alpha t+
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)
\bigm| \bigm| \bigm| \bigm| m(P, k).
For \beta with | \beta | = 1, the above inequality holds by continuity.
Proof of Theorem 2.2. Since m(P, k) = \mathrm{m}\mathrm{i}\mathrm{n}z\in \BbbT k
| P (z)| . Also P (z) has all its zeros in | z| \geq k,
k \leq 1, therefore
m(P, k) \leq | P (z)| for | z| = k.
Hence, it follows by Rouche’s theorem that for m(P, k) > 0 and for any complex \lambda with | \lambda | < 1,
the polynimial h(z) = P (z) - \lambda m(P, k) does not vanish in | z| < k, k \leq 1. Let
g(z) = znh
\biggl(
1
z
\biggr)
= znP
\biggl(
1
z
\biggr)
- \lambda m(P, k)zn = Q(z) - \lambda m(P, k)zn,
then the polynomial g
\Bigl( z
k2
\Bigr)
has all its zeros in | z| \leq k. Also
\bigm| \bigm| \bigm| kng\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| = | h(z)| for | z| = k. By
applying Lemma 3.2 to kng
\Bigl( z
k2
\Bigr)
, we get, for | \alpha i| \geq k, 1 \leq i \leq t, t \leq n - 1, | \beta | \leq 1 and | z| \geq 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1h(z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)h(z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1g
\Bigl( z
k2
\Bigr)
+ \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)g
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| .
Equivalently for | z| \geq 1, we obtain\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z) -
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1070 A. MIR
- \lambda n(n - 1) . . . (n - t+ 1)
\biggl(
zt +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr)
m(P, k)
\bigm| \bigm| \bigm| \bigm| \leq
\leq kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+ \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr)
-
- \lambda
n(n - 1) . . . (n - t+ 1)
k2n
\biggl(
\alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr)
m(P, k)zn
\bigm| \bigm| \bigm| \bigm| . (3.13)
Since P (z) \not = 0 in | z| < k, k \leq 1, we have Q
\Bigl( z
k2
\Bigr)
has all its zeros in | z| \leq k and
kn \mathrm{m}\mathrm{i}\mathrm{n}
| z| =k
\bigm| \bigm| \bigm| Q\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| = \mathrm{m}\mathrm{i}\mathrm{n}
| z| =k
| P (z)| = m(P, k).
Hence by inequality (2.1) of Theorem 2.1 applied to Q
\Bigl( z
k2
\Bigr)
, we get, for | z| \geq 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+
\beta n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| \geq
\geq n(n - 1) . . . (n - t+ 1)
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| \mathrm{m}\mathrm{i}\mathrm{n}
| z| =k
\bigm| \bigm| \bigm| Q\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| =
=
n(n - 1) . . . (n - t+ 1)
k2n
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| m(P, k). (3.14)
Now choosing the arguement of \lambda on the right-hand side of (3.13), such that
kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+
+\beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr)
-
- \lambda
n(n - 1) . . . (n - t+ 1)
kn
\biggl(
\alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr)
m(P, k)zn
\bigm| \bigm| \bigm| \bigm| =
= kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+
+\beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| -
- | \lambda | n(n - 1) . . . (n - t+ 1)
kn
\bigm| \bigm| \bigm| \bigm| \biggl( \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\biggr)
m(P, k)zn
\bigm| \bigm| \bigm| \bigm| ,
which is possible by (3.14), we have from (3.13), for | z| \geq 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| -
- | \lambda | n(n - 1) . . . (n - t+ 1)
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| m(P, k) \leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
ON AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS 1071
\leq kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+
+\beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| -
- | \lambda | | z| nn(n - 1) . . . (n - t+ 1)
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t+
+
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| m(P, k). (3.15)
Letting | \lambda | \rightarrow 1, we obtain from (3.15), for | z| \geq 1,\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z) + \beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| -
- kn
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1Q
\Bigl( z
k2
\Bigr)
+
+\beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)Q
\Bigl( z
k2
\Bigr) \bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - t+ 1)
\biggl\{ \bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| -
- | z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| \biggr\} m(P, k). (3.16)
Combining (3.16) with Lemma 3.3, we get, for | z| \geq 1,
2
\bigm| \bigm| \bigm| \bigm| ztD\alpha tD\alpha t - 1 . . . D\alpha 1P (z)+
+\beta
n(n - 1) . . . (n - t+ 1)
(1 + k)t
(| \alpha 1| - k) . . . (| \alpha t| - k)P (z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq n(n - 1) . . . (n - t+ 1)
\Biggl[ \Biggl\{
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| +
+
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm|
\Biggr\}
M(P, k) -
-
\Biggl\{
| z| n
kn
\bigm| \bigm| \bigm| \bigm| \alpha 1\alpha 2 . . . \alpha t +
\beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm| -
-
\bigm| \bigm| \bigm| \bigm| zt + \beta (| \alpha 1| - k) . . . (| \alpha t| - k)
(1 + k)t
\bigm| \bigm| \bigm| \bigm|
\Biggr\}
m(P, k)
\Biggr]
,
which is equivalent to (2.3).
Theorem 2.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1072 A. MIR
References
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Received 09.12.15,
after revision — 26.09.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
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| id | umjimathkievua-article-1758 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:06Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cc/a41edeb34f159ddd44801b66e63dfacc.pdf |
| spelling | umjimathkievua-article-17582019-12-05T09:25:58Z On an operator preserving inequalities between polynomials Про оператор, що зберiгає нерiвностi мiж полiномами Mir, A. Мир, А. Let $P(z)$ be a polynomial of degree n. We consider an operator $D\alpha$ that maps $P(z)$ into $D\alpha P(z) := nP(z)+(\alpha z)P\prime (z)$ and establish some results concerning the estimates of $| D\alpha P(z)| $ on the disk $| z| = R \geq 1$, and thereby obtain extensions and generalizations of a number of well-known polynomial inequalities. Нехай $P(z)$ — многочлен степеня $n$. У роботi розглянуто оператор $D\alpha$ , що вiдображає $ P(z)$ в $D\alpha P(z) := := nP(z) + (\alpha z)P\prime (z)$, та встановлено деякi результати щодо оцiнок $| D\alpha P(z)| $ на крузi $| z| = R \geq 1$ i, таким чином, отримано розширення та узагальнення багатьох вiдомих нерiвностей для полiномiв. Institute of Mathematics, NAS of Ukraine 2017-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1758 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 8 (2017); 1061-1072 Український математичний журнал; Том 69 № 8 (2017); 1061-1072 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1758/740 Copyright (c) 2017 Mir A. |
| spellingShingle | Mir, A. Мир, А. On an operator preserving inequalities between polynomials |
| title | On an operator preserving inequalities between polynomials |
| title_alt | Про оператор, що зберiгає нерiвностi мiж полiномами |
| title_full | On an operator preserving inequalities between polynomials |
| title_fullStr | On an operator preserving inequalities between polynomials |
| title_full_unstemmed | On an operator preserving inequalities between polynomials |
| title_short | On an operator preserving inequalities between polynomials |
| title_sort | on an operator preserving inequalities between polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1758 |
| work_keys_str_mv | AT mira onanoperatorpreservinginequalitiesbetweenpolynomials AT mira onanoperatorpreservinginequalitiesbetweenpolynomials AT mira prooperatorŝozberigaênerivnostimižpolinomami AT mira prooperatorŝozberigaênerivnostimižpolinomami |