Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$
We show that the derivative of an arbitrary rational function $R$ of degree $n$ that increases on the segment $[−1, 1]$ satisfies the following equality for all $0 < ε < 1$ and $p, q > 1$: $$∥R′∥_{L_p[−1+ε,1−ε]} ≤ C⋅9^{n(1−1/p)}ε^{1/p−1/q−1}∥R∥_{L_q[−1,1]},$$ where the constant...
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| Date: | 2009 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2009
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3132 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509170765135872 |
|---|---|
| author | Vyazovs’ka, M. S. Вязовська, M. С. |
| author_facet | Vyazovs’ka, M. S. Вязовська, M. С. |
| author_sort | Vyazovs’ka, M. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:45:55Z |
| description | We show that the derivative of an arbitrary rational function $R$ of degree $n$ that increases on the segment $[−1, 1]$ satisfies the following equality for all $0 < ε < 1$ and $p, q > 1$:
$$∥R′∥_{L_p[−1+ε,1−ε]} ≤ C⋅9^{n(1−1/p)}ε^{1/p−1/q−1}∥R∥_{L_q[−1,1]},$$
where the constant $C$ depends only on $p$ and $p$. The degree of a rational function $R(x) = P(x)/Q(x)$ is understood as the largest degree among the degrees of the polynomials $P$ and $Q$. |
| first_indexed | 2026-03-24T02:36:51Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDK 517.5
M. S. Vqzovs\ka (Ky]v. nac. un-t im. T. Íevçenka)
OCINKA NORMY POXIDNO}
MONOTONNO} RACIONAL|NO} FUNKCI}
U PROSTORAX Lp
For a derivative of a rational function R of the power n increasing on the interval −[ ]1 1, and for all
0 < ε < 1 and p, q > 1, we show that the inequality
′ ≤ ⋅
− + −
− − −
−[ ]R C R
L
n p p q
Lp q1 1
1 1 1 1 1
1 1
9
ε ε
ε
,
( / ) / /
,[[ ] ,
is satisfied, where the constant C depends only on p and q. The power largest among powers of the
polynomials P and Q is called a power of the rational function R x( ) = P x Q x( )/ ( ) .
Pokazano, çto dlq proyzvodnoj vozrastagwej na otrezke −[ ]1 1, racyonal\noj funkcyy R
stepeny n dlq vsex 0 < ε < 1 y p, q > 1 v¥polnqetsq neravenstvo
′ ≤ ⋅
− + −
− − −
−[ ]R C R
L
n p p q
Lp q1 1
1 1 1 1 1
1 1
9
ε ε
ε
,
( / ) / /
,[[ ] ,
hde postoqnnaq C zavysyt tol\ko ot p y q. Stepen\g racyonal\noj funkcyy R x( ) =
= P x Q x( )/ ( ) naz¥vaem naybol\ßug yz stepenej mnohoçlenov P y Q.
1. Vstup. Naslidkom klasyçno] nerivnosti Bernßtejna [1] [ ocinka poxidno]
mnohoçlena
′ ≤ −[ ]P n P C( ) ,0 1 1 ,
de n poznaça[ stepin\ mnohoçlena P. Vyqvlq[t\sq, wo nemoΩlyvo oderΩaty
prqmyj analoh ci[] teoremy u racional\nij aproksymaci]. Rozhlqnemo funkcig
stepenq 2
g x
x
x
( ) =
+
δ
δ2 2 ,
modul\ qko] ne perevywu[
1
2
pry dovil\nomu δ ta znaçennq poxidno]
′ =g ( )0
1
δ
prqmu[ do neskinçennosti pry δ → 0. Prote dlq racional\nyx
funkcij vykonu[t\sq nerivnist\ Pekars\koho [2], v qkij normy (kvazinormy)
funkci] ta ]] poxidno] rozhlqdagt\sq u prostorax Lp . Poznaçymo
© M. S. VQZOVS|KA, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12 1713
1714 M. S. VQZOVS|KA
f f x dxL a b
p
a
b p
p ,
/
: ( )[ ] =
∫
1
.
Zhidno z nerivnistg Pekars\koho dlq racional\no] funkci] R Lq∈ −[ ]1 1, ste-
penq n ta koΩnoho s ∈N vykonu[t\sq
R cn Rs
L
s
L
p q
( )
, ,−[ ] −[ ]≤
1 1 1 1 ,
de 1 1/ /p q s− = ta 1/q ∉N .
Sytuaciq sutt[vo vidriznq[t\sq dlq vypadku monotonnyx racional\nyx
funkcij. NezvaΩagçy na te, wo porqdky nablyΩennq klasiv monotonnyx
funkcij z Wp
1
racional\nymy funkciqmy bez obmeΩen\ ta monotonnymy racio-
nal\nymy funkciqmy zbihagt\sq [3], isnugt\ obmeΩennq na znaçennq poxidno]
racional\no] monotonno] funkci]. U poperednij statti [4] my oderΩaly nastup-
nyj analoh nerivnosti Bernßtejna dlq monotonnyx racional\nyx funkcij.
Teorema 1. Qkwo R — zrostagça na vidrizku −[ ]1 1, racional\na funkciq
stepenq n, to
′ <
− −[ ]R x
x
R
n
C( ) ,
9
1 1 1 , x ∈ −( , )1 1 .
ZauvaΩennq. U statti [4] teoremu navedeno v dewo inßomu formulgvanni,
odnak z dovedennq lehko vyplyva[ nerivnist\, vkazana vywe.
My uzahal\ng[mo cej rezul\tat dlq prostoriv Lp .
Teorema 2. Qkwo R — zrostagça na vidrizku −[ ]1 1, racional\na funkciq
stepenq n, to
′ ≤− + −[ ]
− − −R C p q RL
n p p q
Lp q1 1
1 1 1 1 19ε ε ε,
( / ) / /( , ) −−[ ]1 1,
dlq vsix 1 < p ≤ ∞ ta 1 < q ≤ ∞.
Nastupna teorema pokazu[, naskil\ky toçnog [ oderΩana ocinka.
Teorema 3. Dlq koΩnoho ε ∈ −( , / )0 1 1 9n
ta p > 1, q > 2 isnu[ zrostagça
na −[ ]1 1, racional\na funkciq
�R stepenq 2n taka, wo
� ′ ≥ ⋅
− + −[ ]
− +( ) −R
L
n p p p
p 1 1
1 4 1 1 11
32
9
ε ε
ε
,
ln ( )/ / /qq
L
R
q
−
−[ ]
1
1 1
�
,
.
2. Dovedennq teoremy 2. Nexaj R — zrostagça na vidrizku −[ ]1 1, racio-
nal\na funkciq stepenq n. Normu v Lp poxidno] R x( ) moΩna ocinyty qk
′ = ′
− + −[ ]
− +
−
∫R R x dxL
p
p
p 1 1
1
1 1
ε ε
ε
ε
,
/
( ) ≤
≤ ′
′
− +
−
− + −[ ]∫ R x dx R x
p
( ) max (
/
,
1
1 1
1 1ε
ε
ε ε
)) /1 1− p =
= R R R xp p( ) ( ) max ( )/
,
/1 1 1
1 1
1 1− − − +( ) ′
− + −[ ]
−ε ε
ε ε
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
OCINKA NORMY POXIDNO} MONOTONNO} RACIONAL|NO} FUNKCI} … 1715
Zvidsy
′ ≤ ′− + −[ ] − + −[ ] − +R R RL
p
C
p
Cp 1 1
1
1 1
1
1 12ε ε ε ε ε,
/
,
/
, −−[ ]
−
ε
1 1/ p
. (1)
Zastosovugçy teoremu 1 do funkci] R x1( ) = R x( / )1 2−( )ε , oderΩu[mo ne-
rivnist\
′ ≤
− − − + −[ ]R x
x
R
n
C( )
/ / , /
9
1 2 1 2 1 2ε ε ε , x ∈ − + −( / , / )1 2 1 2ε ε ,
i, qk naslidok,
max ( )
, / , /x
n
CR x R
∈ − + −[ ] − + −[ ]′ ≤
1 1 1 2 1 2
2
9
ε ε ε εε
. (2)
Oskil\ky funkciq R zrosta[ na vidrizku −[ ]1 1, , to vykonu[t\sq rivnist\
R RC
q
Lq− + −[ ]
−
−[ ]≤1 1
1
1 1ε ε ε,
/
, , (3)
ta analohiçno
R RC
q
Lq− + −[ ]
−
−[ ]≤
1 2 1 2
1
1 12ε ε
ε
/ , /
/
, . (4)
Z nerivnostej (1) – (4) vyplyva[ tverdΩennq teoremy 2, do toho Ω C p q( , ) =
= 21 1 1+ −/ /q pq
.
3. Dovedennq teoremy 3. Wob dovesty teoremu 3, nam potribna dopomiΩna
lema.
Lema 1. Poklademo f x
x
x
( ) =
+1 2 . Nexaj δ ∈ ( , )0 3 , d = 9 – δ i a =
= 24/ δ . Todi funkciq
F x
x
d
d
a
f a x
k
k
k
n
δ ( ) ( )=
−
+
=
∑
1 0
zrosta[ na vidrizku −[ ]1 1, ta zadovol\nq[ nerivnosti
Fδ ( )1 2< , (5)
′ ≥F x dn
δ ( )
1
4
pry x
an
∈
0
1
2
, . (6)
Dovedennq. Dlq poxidno]
′ =
−
+
f x
x
x
( )
( )
1
1
2
2 2
vykonugt\sq taki nerivnosti:
′ ≥ −f x( )
1
8
, (7)
′ ≥ −f x x( ) 1 3 2
, (8)
′ ≥ −f x
x
( )
1
2 . (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
1716 M. S. VQZOVS|KA
Rozhlqnemo funkcig
G x
d
a
f a x
k
k
k
δ ( ) ( )=
= −∞
∞
∑ .
Oçevydno, wo G xδ ( ) [ dyferencijovnog na ( , )0 ∞ . Z nerivnostej ′ ≤f x( ) 1 ,
′ <f x( ) 1 pry x > 1 vyplyva[, wo dlq koΩnoho x a n∈
− , 1
′ ≥ ′F x G xδ δ( ) ( ) .
Krim toho, oçevydno, wo ′F xδ ( ) ≥ 0, x a n∈
−0, . Oskil\ky funkciq G xδ ( ) [
„samopodibnog”, tobto G axδ ( ) = ( / ) ( )a d G xδ , dosyt\ dovesty, wo ′ >G xδ ( ) 0
na vidrizku 1, a[ ] . Rozhlqnemo okremo vypadky x ∈[ ]1 5 4, / , x d∈ 5 4/ , ta
x d a∈ , . Ma[mo
′ = ′
= −∞
∞
∑G x d f a xk k
k
δ ( ) ( ) .
Z nerivnostej (8) i (9) vidpovidno vyplyva[, wo
d f a x
d
x
da
k k
k
′ ≥
−
−
−= −∞
−
∑ ( )
1 2
2
1
1
3
1
,
d f a x
x
d
a d
k k
k
′ ≥ −
−=
∞
∑ ( )
1
2 2
1
.
Na vidrizku 1 5 4, /[ ] vykonu[t\sq nerivnist\ ′ < −f x x( ) / /1 2 2 , qka dozvolq[
ocinyty ′ >G x H xδ ( ) ( )1 pry x ∈[ ]1 5 4, / , de
H x
d
x
da
x
x
d
a d
1
2
2 2 2
1
1
3
1
1
2 2
1
( ) =
−
−
−
+ − −
−
.
Z nerivnosti (7) vyplyva[, wo ′ >G x H xδ ( ) ( )2 pry x d∈ 5 4/ , ,
H x
d
x
da x
d
a d
2
2
2 2 2
1
1
3
1
1
8
1
( ) =
−
−
−
− −
−
.
Na vidrizku x d a∈ , vykonu[t\sq ′ ≥ −f x a x a( / ) /1 , a tomu moΩna ociny-
ty ′G xδ ( ) znyzu funkci[g
H x
d
x
da da
x
ad
x
x
d
x
3
2
2 2
2
2 2
1
1
3
1
1
1
( )
( ) ( )
=
−
−
−
− +
−
+
− 22 2( )a d−
.
Funkci] H1 , H2 , H 3 opukli dohory na vidrizkax 1 5 4, /[ ] , 5 4/ , d ta
d a, vidpovidno. Tomu dosyt\ dovesty, wo vony nabuvagt\ dodatnyx zna-
çen\ na kincqx vidpovidnyx vidrizkiv. Za umov teoremy 6 < d < 9 ta a >
> 4 3/ ⋅d , tomu
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
OCINKA NORMY POXIDNO} MONOTONNO} RACIONAL|NO} FUNKCI} … 1717
H
d da
d
a d
1 2 21
1
1
3
1
0( ) =
−
−
−
−
−
> ,
H H1 25 4 5 4
1
8
1
8 64
0( / ) ( / )= > − +
−
− >
δ
δ
,
H d2
1
8
1
8 72
0( ) > − +
−
− >
δ
δ
,
H d
d
d d a d d a da
3 2 2 2 2
4
1 1
1 6 2
0( ) >
+ −
− − − >
( ) ( )
,
H a
a d d a d a
3 2 2 2 2
1 1
1
1 2 4
0( ) > − +
−
− − − > ,
a otΩe, ′ >G xδ ( ) 0 na vidrizku 1, a[ ] .
Zalyßylosq dovesty nerivnosti (5) ta (6). Oskil\ky f x x( ) /< 1 pry x > 0,
to
F
d
d
a
f a
d
d
a
k
k
k
n
δ ( ) ( )1
1
1
1
10
2=
−
+
<
−
+
=
∑ <
=
∑
k
k
n
2
0
.
Z nerivnosti (8) vyplyva[, wo pry x
an
∈
0
1
2
,
′ =
−
+ ′ > −
=
−∑F x
d
d f a x d ak k
k
n
k k n
δ ( ) ( )
1
1
1
3
40
2 2
>
=
∑ dn
k
n
40
.
Lemu 1 dovedeno.
Dovedennq teoremy 3. Zafiksu[mo p > 1 ta 0 < ε < 1 –
1
9n
. Poklademo
δ =
9
1 2+ p
ta pokaΩemo, wo racional\na funkciq stepenq 2n
�R x F
x
x
F( )
( )
( )=
− +
− −
+δ δ
ε
ε
1
1 1
1
zadovol\nq[ umovy teoremy 3. Z lemy 1 vyplyva[, wo funkciq
�R zrosta[ na
vidrizku −[ ]1 1, . Wob ocinyty normu
� ′
− + −[ ]R
Lp 1 1ε ε,
, vykona[mo zaminu zmin-
nyx y = ( )x − +1 ε / 1 1− −( )( )ε x v intehrali
� ′ =
+ −( )
− −( )
−
−
− +
R x dx
yp
p
p
( )
( )
( )(
1 1
1 1
2 2
2 1
2 2
ε
εε))/( )
( )
2
0
1
1
2ε εε
ε
δ
−− +
−
∫∫ ′F y dyp
.
Z nerivnosti (6) vyplyva[, wo
1 1
1 1
2 2
2 1
1 2
0 + −( )
− −( )
′
−
−
−
∫
( )
( )
( )
/
ε
ε
δ
y
F y d
p
p
a
p
n
yy a dp p n np≥ − − −2 3 1ε .
Oskil\ky
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
1718 M. S. VQZOVS|KA
da p p p− −= −1 1 2 19 24/ / /( )δ δ =
= 9 8 1
1
2 1
2 1 91 1 2 1 4 1⋅ −
+
+ ≥− − − +/ / ln ( )( )p p p
p
p // p ,
to
� ′ ≥ ⋅
− + −[ ]
− +( ) −R
L
n p p p
p 1 1
1 4 1 1 11
8
9
ε ε
ε
,
ln ( ) / /
. (10)
Wob ocinyty znaçennq
�R
Lq −[ ]1 1,
, rozhlqnemo funkcig
H x
x x
x
( )
, ,
, ,
=
+ ∈ −[ ]
∈[ ]
1 1 0
1 0 1
pry
pry
dlq qko] vykonu[t\sq
�R x F H
x
x
( ) ( )
( )
≤
− +
− −
2 1
1
1 1
δ
ε
ε
.
Ocinymo
H
x
x
dx
q
− +
− −
−
∫
1
1 11
1 ε
ε( )
=
=
ε
ε
ε
ε
( )
( )
( )
( )
x
x
dx
x
x
q+
− −
+
+
− −
−
∫
1
1 1
1
1 11
0
+
−
∫
0
1 ε
ε
q
dx .
Oskil\ky
ε
ε
( )
( )
x
x
+
− −
1
1 1
< ε pry x ∈ −[ ]1 0, , perßyj dodanok ne perevywu[ εq
.
Wob ocinyty druhyj dodanok, vykona[mo zaminu x = 1 – ε y. Todi pry y ∈[ ]1 1, /ε
ε
ε
ε
ε
( )
( )
x
x
y
y y y
+
− −
=
−
+ −
<
+
1
1 1
2
1
2
1
.
Tomu
ε
ε
εε ε
( )
( ) ( )
/
x
x
dx
y
q q
q
+
− −
≤
+
−
∫ ∫
1
1 1
2
10
1
1
1
ddy =
=
2
1
1
2
1
1 1
21 1
q
q qq
ε
ε
ε
−
−
+
<− −( / )
.
Zvidsy
H
x
x
dx
q
q− +
− −
≤ + + ≤
−
∫
1
1 1
2 4
1
1 ε
ε
ε ε ε ε
( )
.
OtΩe,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
OCINKA NORMY POXIDNO} MONOTONNO} RACIONAL|NO} FUNKCI} … 1719
�R
L
q q
q −[ ]
+≤
1 1
2 2 12
,
/ /ε . (11)
Z nerivnostej (10) ta (11) vyplyva[ tverdΩennq teoremy 3.
1. Bernßtejn N. S. Ob ocenkax proyzvodn¥x mnohoçlenov: Soçynenyq. — M.: Yzd-vo AN
SSSR, 1952. – T. 1. – S. 370 – 425.
2. Pekarskyj A. A. Ocenky proyzvodn¥x racyonal\n¥x funkcyj v L p −[ ]1 1, // Mat. zametky.
– 1986. – 39, # 3. – S. 388 – 394.
3. Bondarenko A. V. On monotone rational approximation // J. Approxim. Theory. – 2005. – 135. –
P. 54 – 69.
4. Bondarenko A. V., Viazovska M. S. Bernstein type inequality in monotone rational approximation //
East J. Approxim. – 2005. – 11. – P. 103 – 108.
OderΩano 18.06.09,
pislq doopracgvannq — 03.09.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
|
| id | umjimathkievua-article-3132 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:36:51Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/77/3c37850f644ee5bc4e451c72eb7a3577.pdf |
| spelling | umjimathkievua-article-31322020-03-18T19:45:55Z Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ Оцінка норми похідної монотонної раціональної функції у просторах $L_p$ Vyazovs’ka, M. S. Вязовська, M. С. We show that the derivative of an arbitrary rational function $R$ of degree $n$ that increases on the segment $[−1, 1]$ satisfies the following equality for all $0 < ε < 1$ and $p, q > 1$: $$∥R′∥_{L_p[−1+ε,1−ε]} ≤ C⋅9^{n(1−1/p)}ε^{1/p−1/q−1}∥R∥_{L_q[−1,1]},$$ where the constant $C$ depends only on $p$ and $p$. The degree of a rational function $R(x) = P(x)/Q(x)$ is understood as the largest degree among the degrees of the polynomials $P$ and $Q$. Показано, что для производной возрастающей на отрезке $[−1, 1]$ рациональной функции $R$ степени $n$ для всех $0 < ε < 1$ и $p, q > 1$ выполняется неравенство $$∥R′∥_{L_p[−1+ε,1−ε]} ≤ C⋅9^{n(1−1/p)}ε^{1/p−1/q−1}∥R∥_{L_q[−1,1]},$$ где постоянная $C$ зависит только от $p$ и $p$. Степенью рациональной функции $R(x) = P(x)/Q(x)$ называем наибольшую из степеней многочленов $P$ и $Q$. Institute of Mathematics, NAS of Ukraine 2009-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3132 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 12 (2009); 1713–1719 Український математичний журнал; Том 61 № 12 (2009); 1713–1719 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3132/3012 https://umj.imath.kiev.ua/index.php/umj/article/view/3132/3013 Copyright (c) 2009 Vyazovs’ka M. S. |
| spellingShingle | Vyazovs’ka, M. S. Вязовська, M. С. Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ |
| title | Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ |
| title_alt | Оцінка норми похідної монотонної раціональної функції у просторах $L_p$ |
| title_full | Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ |
| title_fullStr | Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ |
| title_full_unstemmed | Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ |
| title_short | Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$ |
| title_sort | estimation of the norm of the derivative of a monotone rational function in the spaces $l_p$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3132 |
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