Improvement of one inequality for algebraic polynomials
We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $...
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| Date: | 2009 |
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| 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/3015 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509030305234944 |
|---|---|
| author | Nesterenko, A. N. Tymoshkevych, T. D. Chaikovs'kyi, A. V. Нестеренко, О. Н. Тимошкевич, Т. Д. Чайковський, А. В. |
| author_facet | Nesterenko, A. N. Tymoshkevych, T. D. Chaikovs'kyi, A. V. Нестеренко, О. Н. Тимошкевич, Т. Д. Чайковський, А. В. |
| author_sort | Nesterenko, A. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:43:20Z |
| description | We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $k = 1$. We also propose some other new pairs $(n, k)$ for which this inequality holds. Some conditions on the polynomial $P_{n+k}$ under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed. |
| first_indexed | 2026-03-24T02:34:37Z |
| format | Article |
| fulltext |
UDK 517.518
O. N. Nesterenko, T. D. Tymoßkevyç, A. V. Çajkovs\kyj
(Ky]v. nac. un-t im. T. Íevçenka)
POSYLENNQ ODNI{} NERIVNOSTI
DLQ ALHEBRA}ÇNYX MNOHOÇLENIV
We prove that the inequality g n L( / ) – ,⋅ [ ]1 1 1 Pn k L+ [ ]1 1 1– ,
≤ 2
1 1 1
gPn k L+ [ ]– ,
, where g : –1[ ,
1] → R is a monotone odd function and Pn k+ is an algebraic polynomial of degree not higher than
n + k, is true for all natural n if k = 0 and for all natural n ≥ 2 if k = 1. Other new pairs (n, k) are
found for which this inequality is also true. Some conditions on polynomials Pn k+ are established
under which the inequality becomes an equality. Some generalizations of the considered inequality are
obtained.
Dokazano, çto neravenstvo g n L( / ) – ,⋅ [ ]1 1 1 Pn k L+ [ ]1 1 1– ,
≤ 2
1 1 1
gPn k L+ [ ]– ,
, hde g : –1[ ,
1] → R � monotonnaq neçetnaq funkcyq, a Pn k+ � alhebrayçeskyj mnohoçlen stepeny ne
v¥ße n + k, v¥polnqetsq dlq vsex natural\n¥x n pry k = 0 y dlq vsex natural\n¥x n ≥ 2
pry k = 1, najden¥ druhye nov¥e par¥ (n, k), dlq kotor¥x ono ymeet mesto. Ustanovlen¥ ne-
kotor¥e uslovyq na mnohoçlen Pn k+ , pry kotor¥x πto neravenstvo prevrawaetsq v ravenstvo.
Poluçen¥ nekotor¥e obobwenyq πtoho neravenstva.
Vstup. Formulgvannq osnovnyx rezul\tativ. Nexaj g : R → R � monoton-
na neparna funkciq ta
g xn( ) : = g nx( ), n ∈N , x ∈R .
U roboti [1] (lema 5.1) dovedeno, wo dlq koΩnoho alhebra]çnoho mnohoçlena
P = Pn – 2 stepenq ne vywe n – 2, n ≥ 2, vykonu[t\sq nerivnist\
g PL L1 11 1 1 1– , – ,[ ] [ ] ≤ 2
1 1 1g Pn L – ,[ ] . (1)
U statti [2] otrymano posylennq ta uzahal\nennq ci[] nerivnosti. Zokrema, vsta-
novleno, wo pry n ≥ 7 nerivnist\ (1) spravdΩu[t\sq dlq alhebra]çnyx mnoho-
çleniv P stepenq ne vywe n. Metog dano] roboty [ dovedennq c\oho tverdΩen-
nq dlq vsix natural\nyx n, znaxodΩennq novyx par (n, k), dlq qkyx nerivnist\
(1) vykonu[t\sq u vypadku mnohoçlena P stepenq ne vywe n + k, uzahal\nennq
]] na vypadok funkcij bahat\ox zminnyx, a takoΩ vstanovlennq umov na mno-
hoçleny, dlq qkyx nerivnist\ peretvorg[t\sq na rivnist\.
Teorema 1. Nexaj n ∈N . Dlq koΩnoho alhebra]çnoho mnohoçlena Pn ste-
penq ne vywe n vykonu[t\sq nerivnist\
g PL n L1 11 1 1 1– , – ,[ ] [ ] ≤ 2
1 1 1g Pn n L – ,[ ]. (2)
Naslidok. Qkwo b > 0, to dlq koΩnoho alhebra]çnoho mnohoçlena Pn
stepenq ne vywe n vykonu[t\sq nerivnist\
g PL b b n L b b1 1– , – ,[ ] [ ] ≤ 2
1
b g Pn n L b b– ,[ ] .
Dlq formulgvannq inßyx rezul\tativ vvedemo take oznaçennq.
Oznaçennq. Vymirnu za Lebehom funkcig g : – ,1 1[ ] → R nazvemo dopusty-
mog zi stalog 1 / n, de n ∈N , qkwo:
1) g � parna funkciq na – ,1 1
n n
;
© O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2 231
232 O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ
2) g � nespadna na 0 1,
n
;
3) ess inf ( )
/1 1n x
g x
≤ ≤
≥ g
n
1
.
ZauvaΩymo, wo monotonna neparna funkciq [ dopustymog zi stalog 1 / n,
n ∈N .
Budemo doslidΩuvaty nerivnist\
g
n
P
L
L
⋅
[ ] [ ]
1
11 1
1 1
– ,
– , ≤ 2
1 1 1gP L – ,[ ], (3)
qka dlq neparnyx monotonnyx funkcij g : R → R ta mnohoçlena P [ rivno-
syl\nog nerivnosti (1) abo (2) v zaleΩnosti vid stepenq mnohoçleniv.
Teorema 2. 1. Dlq koΩnoho k ≥ – 1 isnu[ N k( ) ∈N take, wo dlq vsix n ≥
≥ N k( ) dovil\nyj alhebra]çnyj mnohoçlen P = Pn k+ stepenq ne vywe n + k
zadovol\nq[ nerivnist\ (3) dlq bud\-qko] funkci] g, dopustymo] zi stalog
1 / n.
2. Dlq koΩnoho n ∈N isnu[ K n( ) ∈N take, wo pry vsix n ≥ K n( ) dlq
deqkoho alhebra]çnoho mnohoçlena P = Pn k+ stepenq ne vywe n + k i deqko]
funkci] g, dopustymo] zi stalog 1 / n, nerivnist\ (3) [ xybnog.
ZauvaΩennq. 1. Qkwo dlq pary (n0 , k0) spravdΩu[t\sq tverdΩennq 1
teoremy 2, to vono spravdΩu[t\sq i dlq inßyx par (n0 , k) , de k ≤ k0.
2. Z lem 2 i 3, navedenyx nyΩçe, ta zauvaΩen\ do nyx vyplyva[, wo moΩna
poklasty N (– 1) = N (0) = 1, N (1) = 2, N (2) = 8, N (3) = 9, N (4) = 11, K (1) = 1,
K (2) = 2, K (3) = 3. Z lemy 2 takoΩ moΩna otrymaty qvnu ocinku dlq N k( ) pry
koΩnomu k ∈N .
3. Z dovedennq teoremy vyplyva[ navit\ bil\ß syl\ne tverdΩennq, niΩ tver-
dΩennq p. 1: dlq koΩnoho n ≥ – 1 isnu[ ˜ ( )N k ∈N take, wo dlq vsix n ≥ ˜ ( )N k
dovil\nyj alhebra]çnyj mnohoçlen Pn k+ stepenq ne vywe n + k zadovol\nq[
xoça b odnu z nerivnostej (4) � (6) dlq bud\-qko] funkci] g, dopustymo] zi sta-
log 1 / n.
n
k
n
k
– 1 0 1 2 3 4 – 1 0 1 2 3 4
1 + + � � � � 1 +
2 + + + � � � 2 +
3 + + + + � � 3 +
4 + + + + + 4 +
5 + + + + 5 +
6 + + + 6 +
7 + + + 7 + +
8 + + + + 8 + +
9 + + + + + 9 + +
10 + + + + + 10 + + +
11 + + + + + + 11 + + +
Dlq malyx znaçen\ par (n, k) u tablyci zliva znakom � + � poznaçeno pary,
dlq qkyx (3) vykonu[t\sq, znakom � – � � pary, dlq qkyx nerivnist\ (3) [ xybnog
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
POSYLENNQ ODNI{} NERIVNOSTI DLQ ALHEBRA}ÇNYX MNOHOÇLENIV 233
(poroΩni miscq � nevidomi vypadky). Dlq porivnqnnq u tablyci sprava navede-
no analohiçni rezul\taty z roboty [2].
Dovedennq teoremy 2 i zauvaΩennq 1 � 3 spyragt\sq na teoremy 3, 4, qki, od-
nak, magt\ i samostijnyj interes.
Teorema 3. Nexaj n ∈N , P ∈ L1 1 1– ,[ ], L x( ) : = P x( ) + P x(– ) ,
x ∈ – ,1 1[ ].
1. Nexaj funkciq g : – ,1 1[ ] → R dopustyma zi stalog 1 / n. Nerivnist\
(3) spravdΩu[t\sq todi j lyße todi, koly vykonu[t\sq nerivnist\
n g x dx L x dx
n
0
1
0
1/
( ) ( )∫ ∫ ≤
0
1
∫ L x g x dx( ) ( ) . (4)
2. Nerivnist\ (3) spravdΩu[t\sq dlq vsix funkcij g : – ,1 1[ ] → R , dopus-
tymyx zi stalog 1 / n, todi j lyße todi, koly vykonu[t\sq nerivnist\
P L u u1 – ,[ ] ≤ nu P L1 1 1– ,[ ], u
n
∈
0 1, . (5)
3. Dlq toho wob nerivnist\ (3) spravdΩuvalasq dlq vsix funkcij g : –1[ ,
1] → R, dopustymyx zi stalog 1 / n, dostatn\o, wob vykonuvalas\ nerivnist\
L x( ) ≤ n L t dt
0
1
∫ ( ) , x
n
∈
0 1, . (6)
Teorema 4. Nexaj N ∈N .
1. Dlq vymirno] za Lebehom funkci] g : – ,1 1[ ] → R tako], wo
–
( )
1
1
∫ g x dx ≠
≠ 0, velyçynu
P
gP
L
L
1
1
1 1
1 1
– ,
– ,
[ ]
[ ]
vyznaçeno dlq vsix nenul\ovyx mnohoçleniv P stepenq ne vywe N. Sered mno-
hoçleniv P, dlq qkyx cq velyçyna nabuva[ svoho najbil\ßoho znaçennq, moΩna
vybraty mnohoçlen P0, qkyj ma[ (z uraxuvannqm kratnosti) N koreniv na
– ,1 1[ ].
2. Pry fiksovanomu u ∈ 0 1,( ] velyçynu
P
P
L u u
L
1
1 1 1
– ,
– ,
[ ]
[ ]
vyznaçeno dlq vsix nenul\ovyx mnohoçleniv P stepenq ne vywe N. Sered mno-
hoçleniv P, dlq qkyx cq velyçyna nabuva[ svoho najbil\ßoho znaçennq, moΩna
vybraty mnohoçlen P0, qkyj ma[ (z uraxuvannqm kratnosti) N koreniv na
– ,1 1[ ].
3. Nexaj n ∈N . Velyçynu
sup ( )
( )
, /x n
L x
L t dt
∈[ ]
∫
0 1
0
1 ,
de L x( ) : = P x( ) + P x(– ) , x ∈ – ,1 1[ ], vyznaçeno dlq vsix nenul\ovyx mnoho-
çleniv P stepenq ne vywe N. Sered mnohoçleniv P, dlq qkyx cq velyçyna na-
buva[ svoho najbil\ßoho znaçennq, moΩna vybraty mnohoçlen P0, wo ma[ (z
uraxuvannqm kratnosti) N koreniv na – ,1 1[ ].
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
234 O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ
TverdΩennq p. 3 zalyßa[t\sq pravyl\nym, qkwo vkazanu velyçynu rozhlqda-
ty lyße na mnoΩyni nenul\ovyx parnyx mnohoçleniv stepenq ne vywe N abo na
mnoΩyni nenul\ovyx neparnyx mnohoçleniv stepenq ne vywe N.
ZauvaΩennq 4. Qkwo neobxidno dovesty, wo odyn iz drobiv, vkazanyx u
teoremi, ne perevywu[ zadanu stalu dlq vsix vidpovidnyx mnohoçleniv, dosyt\
dovodyty ce tverdΩennq lyße dlq mnohoçleniv, wo magt\ N riznyx koreniv na
– ,1 1[ ]. Dijsno, mnoΩyna takyx mnohoçleniv u rivnomirnij metryci [ wil\nog u
mnoΩyni vidpovidnyx mnohoçleniv, qki magt\ N koreniv na – ,1 1[ ].
Teorema 5. Nexaj m ∈N , kj ∈Z , kj ≥ – 1, N kj j( ) : = ˜ ( )N kj , ostanng ve-
lyçynu vyznaçeno v zauvaΩenni 3, nj ≥ N kj j( ) , j ∈ 1{ , 2, … , m} , a vymirna za
Lebehom funkciq g : – ,1 1[ ]m → R dlq koΩnoho j ∈ 1{ , 2, … , m} qk funkciq
odni[] j-] zminno] pry vsix fiksovanyx inßyx zminnyx [ dopustymog zi stalog
1/n . Todi dlq dovil\noho alhebra]çnoho mnohoçlena P, stepin\ qkoho po j-j
zminnij ne perevywu[ nj + kj pry koΩnomu j ∈ 1{ , 2, … , m} , vykonu[t\sq ne-
rivnist\
g
n n n
P
m L
L
m
m
⋅ ⋅ … ⋅
[ ]( )
[ ]( )
1 2 1 1
1 1
1
1
, , ,
– ,
– , ≤ 2
1 1 1
m
LgP m– ,[ ]( ). (7)
Teoremu 4 vstanovyv A. V. Çajkovs\kyj, lemu 3 � T. D. Tymoßkevyç, teore-
mu 5 � O. N. Nesterenko. Inßi rezul\taty otrymano spil\no.
Dovedennq osnovnyx rezul\tativ. Dovedennq teoremy 3. Qkwo Pg ∉
∉ L1 1 1– ,[ ], to prava çastyna nerivnosti (3) dorivng[ + ∞ i nerivnist\ vyko-
nu[t\sq, tomu nadali prypuska[mo, wo Pg ∈ L1 1 1– ,[ ]. Bez obmeΩennq zahal\-
nosti vvaΩatymemo, wo g x( ) = g
n
1
, x ∈
1 1
n
,
.
Ma[mo
Pg L1 1 1– ,[ ] =
0
1
∫ P x g x dx( ) ( ) +
0
1
∫ P x g x dx(– ) (– ) =
=
0
1
∫ +( )P x P x g x dx( ) (– ) ( ) =
0
1
∫ L x g x dx( ) ( ) .
Z inßoho boku,
g
n
P
L
L
⋅
[ ] [ ]
1
11 1
1 1
– ,
– , =
– –
( )
1
1
1
1
∫ ∫
g x
n
dx P x dx = 2
0
1
0
1
n g x dx L x dx
n/
( ) ( )∫ ∫ .
Zvidsy vyplyva[, wo nerivnosti (3) i (4) [ rivnosyl\nymy.
Nexaj vykonu[t\sq spivvidnoßennq (6). Nerivnist\ (4) rivnosyl\na nerivnosti
0
1
0
1/
( ) – ( ) ( )
n
L x n L t dt g x dx∫ ∫
+
1
1
/
( ) ( )
n
L x g x dx∫ ≥ 0. (8)
Vraxovugçy spivvidnoßennq (6) i dopustymist\ funkci] g, pomiça[mo, wo li-
va çastyna nerivnosti (8) bude najmenßog, qkwo g x( ) = g
n
1
= const, x ∈
∈ – ,1 1[ ]. Ale dlq takyx funkcij liva çastyna (8) dorivng[ nulg.
Nexaj vykonu[t\sq spivvidnoßennq (5). Bez obmeΩennq zahal\nosti vvaΩa-
tymemo, wo g x( ) ≥ 0, x ∈ – ,1 1[ ], g x( ) = 1, x ∈
1 1
n
,
, g( )0 = 0, a takoΩ wo g
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
POSYLENNQ ODNI{} NERIVNOSTI DLQ ALHEBRA}ÇNYX MNOHOÇLENIV 235
[ absolgtno neperervnog. Nexaj takoΩ P L1 1 1– ,[ ] = 1.
Z odnoho boku, ma[mo
g
n L
⋅
[ ]1 1 1– ,
= n g x dx
n
n
– /
/
( )
1
1
∫ = 2 1 1
0
1
n
n
g
n
xg x dx
n
′
∫– ( )
/
=
= 2 – 2
0
1
n xg x dx
n/
( )∫ ′ .
Z inßoho boku,
gP L1 1 1– ,[ ] = 1 –
–
– ( ) ( )
1
1
1∫ ( )g x P x dx =
= 1 –
– /
/
– ( ) ( )
1
1
1
n
n
g x P x dx∫ ( ) =
= 1 +
– / – /
( ) ( )
1
0
1n n
x
g u du P x dx∫ ∫ ′
–
0
1 1/ /
( ) ( )
n
x
n
g u du P x dx∫ ∫ ′
=
= 1 +
– /
( ) ( )
1
0 0
n u
P x dx g u du∫ ∫
′ –
0
1/
( ) ( )
n
u
u
P x dx g u du∫ ∫
′ =
= 1 –
0
1 0/
–
( ) ( )
n
u
P x dx g u du∫ ∫
′ –
0
1
0
/
( ) ( )
n u
P x dx g u du∫ ∫
′ =
= 1 –
0
1/
–
( ) ( )
n
u
u
P x dx g u du∫ ∫
′ .
Zvidsy z uraxuvannqm spivvidnoßennq (5) otrymu[mo nerivnist\ (3). Qkwo neriv-
nist\ (5) porußu[t\sq pry deqkomu u, moΩna vybraty funkcig g tak, wob ]]
poxidna na 0 1,[ ] bula vidminnog vid nulq lyße pry arhumentax, blyz\kyx do u.
Todi porußyt\sq nerivnist\ (3).
Teoremu 3 dovedeno.
Dovedennq teoremy 4. 1. MiΩ mnohoçlenamy vyhlqdu P x( ) = a0 + a1x + …
… + a xN
N i toçkamy (a0, a1, … , aN ) ∈ RN +1 isnu[ vza[mno odnoznaçna vidpo-
vidnist\. Mnohoçlen, wo vidpovida[ vektoru
r
a N∈ +R 1, poznaçatymemo çerez
Pa
r , a toçku, wo vidpovida[ mnohoçlenu P, � çerez
r
a P( ) . Rozhlqnemo funkcig
F : R
N + { }1 0\ → (0, + ∞), zadanu formulog
F a( )
r
=
P
gP
a L
a L
r
r
1
1
1 1
1 1
– ,
– ,
[ ]
[ ]
.
Oskil\ky F ne zming[ znaçennq pry domnoΩenni
r
a na nenul\ove çyslo, to vsi
znaçennq vona nabuva[ pry
r
a ∈ Ω : =
S
r
0 1,( ) � RN +1, de
S
r
0 1,( ) � odynyçna
sfera z centrom u poçatku koordynat. Na kompakti Ω funkciq F dosqha[ svo-
ho najbil\ßoho znaçennq. OtΩe, F dosqha[ najbil\ßoho znaçennq i na
R
N + { }1 0\ . Vyberemo odnu z toçok
r
a0, de vono dosqha[t\sq, tak, wob mnoho-
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
236 O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ
çlen Pa
r
0
mav maksymal\no moΩlyvu kil\kist\ koreniv na – ,1 1[ ] (z uraxuvan-
nqm kratnosti).
Nexaj
P xa
r
0
( ) = Q x0( ) R x0( ) , de Q x0( ) � mnohoçlen, vsi koreni qkoho le-
Ωat\ na – ,1 1[ ], R0 � mnohoçlen, vsi koreni qkoho leΩat\ na C \ – ,1 1[ ]. Bez ob-
meΩennq zahal\nosti vvaΩatymemo, wo R x0( ) > 0, x ∈ – ,1 1[ ]. Prypustymo, wo
R0 ne [ stalog. Qk i na poçatku dovedennq, vstanovymo vza[mno odnoznaçnu
vidpovidnist\ miΩ mnohoçlenamy R stepenq ne vywe s i toçkamy
r
b s∈ +R 1. Ne-
xaj R x0( ) =
R x
b
r
0
( ) = b0
0 + b x1
0 + … + b xs
s0 .
Poznaçymo
Ω1 : =
r
rb R x xs
b
∈ > ∈[ ]{ }+R 1 0 1 1( ) , – , .
MnoΩyna Ω1 [ vidkrytog. Dijsno, qkwo prypustyty, wo cq mnoΩyna ne [ vid-
krytog, to isnu[
r
b m{ : m ≥ 1} � Rs +1
\ Ω1 taka, wo
r
b m( ) →
r
′ ∈b Ω1 , m → ∞ .
Ale todi isnu[ mnoΩyna xm{ : m ≥ 1} � – ,1 1[ ] taka, wo
R x
b mm
r
( ) ( ) ≤ 0. Bez ob-
meΩennq zahal\nosti moΩna vvaΩaty, wo xm → x0 ∈ – ,1 1[ ], m → ∞ . Zvidsy
R x
b
r
′
( )0 ≤ 0. Ce supereçyt\ tomu, wo
r
′ ∈b Ω1 .
Rozhlqnemo funkcig
H b( )
r
= –
–
( )
( ) ( )
1
1 0
0 1
1
1 0
0 1
∫
∫
+ + … +( )
+ + … +( )
Q x b b x b x dx
Q x g x b b x b x dx
s
s
s
s
,
r
b s∈ +R 1
\ 0{ }.
U koΩnij toçci
r
b ∈Ω1 znaçennq H zbiha[t\sq z deqkym znaçennqm funkci] F,
tomu
sup ( )r
r
b
H b
∈Ω1
≤
sup ( )
r
r
a
F a
∈Ω
. Z inßoho boku, v toçci
r
b0
1∈Ω H b( )
r
0 = F a( )
r0 =
=
sup ( )
r
r
a
F a
∈Ω
. OtΩe,
H b( )
r
0 =
sup ( )r
r
b
H b
∈Ω1
=
sup ( )
r
r
a
F a
∈Ω
.
Zvidsy vyplyva[, wo v toçci
r
b0 funkciq H ma[ lokal\nyj (nestrohyj) maksy-
mum na Ω1. Ale dlq drobovo-linijno] funkci] ce moΩlyvo lyße qkwo H �
stala. V takomu vypadku znaçennq H b( )
r
0 dosqha[t\sq i na meΩax vidkryto]
mnoΩyny Ω1, tobto isnu[
r
b ∈∂Ω1 takyj, wo H b( )
r
= H b( )
r
0 ; pry c\omu
∃ ∈[ ]x – ,1 1 :
R x
b
r( ) = 0.
OtΩe, maksymum funkci] F dosqha[t\sq dlq mnohoçlena
Q R
b
0 r , qkyj ma[ na
– ,1 1[ ] bil\ße koreniv, niΩ
Pa
r
0
. Supereçnist\.
OtΩe, R0 � stala i
Pa
r
0
ma[ N koreniv na – ,1 1[ ].
2. Dovedennq povnistg povtorg[ dovedennq p. 1 z vidpovidnog zaminog
funkcij F, H.
3. Qk i pry dovedenni p. 1, pokazu[mo, wo cej drib dosqha[ maksymumu. Viz\-
memo mnohoçlen iz najbil\ßog kil\kistg koreniv, dlq qkoho maksymum dosqha-
[t\sq. Dlq n\oho vyberemo toçku x0 ∈ 0 1,
n
, v qkij dosqha[t\sq supremum u
çysel\nyku. Rozhlqdagçy funkcig
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
POSYLENNQ ODNI{} NERIVNOSTI DLQ ALHEBRA}ÇNYX MNOHOÇLENIV 237
H b( )
r
=
Q x R x Q x R x
Q x R x dx
b b
b
0
0 0
0
0 0
1
1 0
( ) ( ) (– ) (– )
( ) ( )
–
r r
r
+
∫
,
povtorg[mo mirkuvannq z dovedennq p. 1.
U vypadku, koly rozhlqdagt\sq lyße parni çy lyße neparni mnohoçleny,
mirkuvannq analohiçni, lyße mnohoçlen R0 obov�qzkovo [ parnym i ma[ vyhlqd
b0
0 + b x1
0 2 + … + b xs
s0 2 .
Teoremu 4 dovedeno.
Dovedennq teoremy 2 spyra[t\sq na rqd lem.
Lema 1. Nexaj n ∈N , n ≥ 2, εn : = arcsin 1
n
, ϕn : =
π ε– 2
4
n , Fn( )α = 1
sinα
+
+ 1
2sin( – )ϕ αn
, α ∈ ( , )0 2ϕ . Todi
∃ ∈( ]! ,α ϕn n0 : Fn n( )α = π
εn
.
Dovedennq. ZauvaΩymo, wo funkciq
Fn( )α = 1
sinα
+ 1
2sin( – )ϕ αn
=
2
2
sin
cos( – ) –
cos
cos( – )
ϕ
ϕ α ϕ
ϕ α
n
n
n
n
, α ϕ∈( ]0, n ,
spada[. Krim toho,
∀ ≥n 2: sinϕ ε
πn
n≥ 2
,
tobto Fn n( )ϕ ≤ π
εn
, otΩe,
∃ ∈( ]! ,α ϕn n0 : Fn n( )α = π
εn
.
Lema 2. Nexaj n ∈N , n ≥ 2, k ∈Z, k ≥ – 1 i v poznaçennqx poperedn\o]
lemy
ε
ε αn
n n
n k+ + +
+
3
2
1
2 cos( )
≤ π
2
. (9)
Todi dovil\nyj alhebra]çnyj mnohoçlen P = Pn k+ stepenq ne vywe n + k zado-
vol\nq[ nerivnist\ (5).
Dovedennq. Nexaj D um( ) = 1
2
+ cos u + cos 2u + … + cos m u =
sin /
sin( / )
m u
u
+( )1 2
2 2
,
u ∈R , m ∈N , � qdro Dirixle. Rozhlqnemo T tn k+ +1( ) : = sin t Pn k+ (cos t), t ∈R,
� neparnyj tryhonometryçnyj polinom porqdku ne vywe n + k + 1. Dlq n\oho
magt\ misce rivnosti
T xn k+ +1( ) = 1
1 1π
π
π
–
( – ) ( )∫ + + + +D x t T t dtn k n k =
= – ( ) ( )
–
1
1 1π
π
π
∫ + + + ++D x t T t dtn k n k , x ∈R ,
zvidky
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
238 O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ
T xn k+ +1( ) = 1
2
1 1
1π
π
π
–
( – ) – ( )
( )∫ + + + +
+ +
+D x t D x t
T t dtn k n k
n k , x ∈R .
ZauvaΩymo, wo z opuklosti donyzu funkci] arcsin na 0
2
, π
vyplyva[ ne-
rivnist\
arcsinu ≤ u
u
u
arcsin 0
0
, u u∈[ ]0 0, , 0 < u0 ≤ 1. (10)
Vykorystovugçy ]] ta poperedn[ zobraΩennq dlq Tn k+ +1, dlq u ∈ 0 1,
n
ma[mo
Tn k L u u+ + +[ ]1 2 21 π π/ – arcsin , / arcsin
=
=
π
π
π
π
π
/ – arcsin
/ arcsin
–
( – ) – ( )
( )
2
2
1 1
1
1
2
u
u
n k n k
n k
D x t D x t
T t dt dx
+
+ + + +
+ +∫ ∫
+
≤
≤ 1
21
0 2
2
1 1
1π π π π π
π
T
D x t D x t
dxn k L
t u
u
n k n k
+ + [ ] ∈[ ]
+
+ + + +∫
+
– ,
, / – arcsin
/ arcsin
sup
( – ) – ( )
≤
≤ 1
1
1π π π
T un k L+ + [ ]– ,
arcsin ×
× sup ( – ) – ( )
, , – / arcsint x u
n k n kD x t D x t
∈[ ] ≤
+ + + + +
0 2
1 1
π π
≤
≤ 1 1
1
1π π π
T nu
n
Kn k L n k+ + [ ] +– ,
arcsin = 1
1
1π
ε
π π
T nu Kn k L n n k+ + [ ] +– ,
,
de
Kn k+ : = sup ( – ) – ( )
, , – /t x
n k n k
n
D x t D x t
∈[ ] ≤
+ + + + +
0 2
1 1
π π ε
.
Vraxovugçy cg ocinku ta rivnosti
Tn k L+ + [ ]1
1 – ,π π
= 2
1 1 1
Pn k L+ [ ]– ,
,
Tn k L u u+ + +[ ]1 2 21 π π/ – arcsin , / arcsin
= Pn k L u u+ [ ]1 – ,
, u ∈ 0 1,
n
,
baçymo, wo dlq vstanovlennq nerivnosti (5) dosyt\ dovesty, wo
Kn k+ ≤ π
ε2 n
.
Rozhlqnemo mnoΩynu par
Ωn = ( , ) – – , –α β π α β ε β α π
2
0≤ ≤ ≤{ }n =
= ( , ) – , max( , – ), min( , – – )α β α π ϕ β α ϕ α α π π ϕ α≤ ∈ +[ ]{ }2
2 2n n n .
Poznaçymo α : =
x t–
2
, β : =
x t+
2
, de t ∈ 0, π[ ], x – π
2
≤ εn. Todi (α, β) ∈
∈ Ωn . Rozhlqnemo taki vypadky:
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
POSYLENNQ ODNI{} NERIVNOSTI DLQ ALHEBRA}ÇNYX MNOHOÇLENIV 239
1) α < 0. Todi za vlastyvostqmy qdra Dirixle
D Dn k n k+ + + +1 12 2( ) – ( )α β = D Dn k n k+ + + +1 12 2 2(– ) – ( – )α π β ,
pryçomu (– α, π – β)∈ Ωn . Tomu cej vypadok zvodyt\sq do vypadku α ≥ 0.
2) α ∈ α ϕn n,[ ]. Todi β ∈ 2ϕn[ – α, π – 2ϕn – α] i
D Dn k n k+ + + +1 12 2( ) – ( )α β ≤ 1
2 sinα
+ 1
2 sinβ
≤ 1
2
Fn( )α ≤
≤ 1
2
Fn n( )α = π
ε2 n
.
3) α ∈ ϕn( , π
2
– ϕn]. Todi β ∈ α[ , π – 2ϕn – α] i
D Dn k n k+ + + +1 12 2( ) – ( )α β ≤ 1
2 sinα
+ 1
2 sinβ
≤ 1
sinα
≤ 1
sinϕn
≤ π
ε2 n
.
4) α ∈ 0, αn[ ) . Todi β ∈ 2ϕn[ – α, π – 2ϕn – α]. Poznaçymo δ : = β – π
2
. To-
di δ ∈ –εn[ – α, εn – α]. Zvidsy
D Dn k n k+ + + +1 12 2( ) – ( )α β ≤ Dn k+ +1 2( )α + Dn k+ +1 2( )β =
=
sin( )
sin
cos ( )
2 2 1
2
2 1
n k
n k
+ + + + +( )α
α
α +
cos ( )
cos
2 1
2
n k+ + +( )δ δ
δ
≤
≤
2 2 1
2
n k+ +
+ 1 + 1
2 cosδ
≤ n + k + 3
2
+ 1
2 cos( )ε αn n+
.
OtΩe, v usix vypadkax Kn k+ ≤ π
ε2 n
.
Lemu 2 dovedeno.
Dovedennq teoremy 2. P. 1 vyplyva[ z lemy 2, oskil\ky liva çastyna
nerivnosti (9) pry fiksovanomu k ≥ – 1 prqmu[ do 1 pry n → ∞.
Druhyj punkt ci[] teoremy vyplyva[ z toho, wo dlq funkcij
g x( ) =
0 0 1
2
1 1
2
1
, , ,
, , ,
x
n
x
n
∈
∈
P x( ) =
1 2 0 1
2
0 1
2
1
– , , ,
, , ,
nx x
n
x
n
∈
∈
nerivnist\ (3) [ xybnog, a funkcig P moΩna qk zavhodno dobre rivnomirno na-
blyzyty mnohoçlenamy.
Teoremu 2 dovedeno.
ZauvaΩennq. 5. Liva çastyna nerivnosti (9) [ zrostagçog funkci[g po k i
spadnog po n. Dijsno, n nε spada[ za nerivnistg (10), drib
ε
ε α
n
n ncos( )+
spa-
da[, tomu wo
∀ ≥n 2: ϕ ϕn n< +1, ∀α ∈ 0, ϕn( ]: Fn( )α > Fn +1( )α ,
otΩe,
∀ ≥n 2: Fn n( )α +1 > Fn n+ +1 1( )α = π
εn +1
> π
εn
= Fn n( )α .
Zvidsy z uraxuvannqm monotonnoho spadannq funkci] Fn ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
240 O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ
∀ ≥n 2: αn +1 < αn.
Tomu qkwo nerivnist\ (9) spravdΩu[t\sq dlq deqko] pary (n0, k0), to vona
spravdΩu[t\sq dlq vsix par (n, k0), n ≥ n0, ta dlq vsix par (n0, k), – 1≤ k ≤ k0.
6. Bezposeredni pidraxunky livo] çastyny nerivnosti (9) pokazugt\, wo moΩ-
na poklasty N (– 1) = 3, N (0) = 4, N (1) = 6, N (2) = 8, N (3) = 9, N (4) = 11.
Lema 3. Nerivnist\ (3) spravdΩu[t\sq dlq dovil\no] dopustymo] zi stalog
1/n funkci] g i vsix mnohoçleniv stepenq ne vywe n + k, qkwo para ( n, k)
nabuva[ odnoho zi znaçen\ (1, 0), (2, 1), (3, 2), (4, 3), (5, 2). Ce tverdΩennq [
xybnym dlq par (1, 1), (2, 2), (3, 3).
Dovedennq. Nexaj P � mnohoçlen stepenq ne vywe n + k , a L x( ) =
= P x( ) + P x(– ) , x ∈ – ,1 1[ ].
1. U vypadku (n, k) = (1, 0) funkciq L [ monotonno nespadnog na 0 1,[ ], qk
i funkciq g . Tomu L zadovol\nq[ spivvidnoßennq (4), wo vyplyva[ z neriv-
nosti Çebyßova [3] (teorema 236), zvidky za teoremog 3 ma[mo nerivnist\ (3).
2. Z uraxuvannqm totoΩnostej a b+ + a b– = a + b + a b– ,
max ,a b{ } =
a b a b+ + –
2
, a, b ∈R , otryma[mo, wo koly P x( ) = P∗ + P∗∗, de
P∗ � neparnyj mnohoçlen, P∗∗ � parnyj mnohoçlen, to L x( ) = 2 max ( )P x∗{ ,
P x∗∗ }( ) , x ∈R .
Zvidsy vyplyva[, wo oskil\ky
max ( ) , ( )
0
1
0
1
∫ ∫
f x dx h x dx ≤
0
1
∫ { }max ( ), ( )f x h x dx , f, h ∈ C 0 1,[ ]( ),
to dlq dovedennq spivvidnoßennq (6) dosyt\ vstanovyty nerivnist\
P x( ) ≤ n P t dt
0
1
∫ ( ) , x
n
∈
0 1, , (11)
dlq vsix alhebra]çnyx mnohoçleniv P stepenq ne vywe n + k, qki [ abo parnymy,
abo neparnymy funkciqmy. Krim toho, perevirqty cg nerivnist\, z ohlqdu na za-
uvaΩennq 4, dostatn\o dlq mnohoçleniv, wo magt\ n + k riznyx koreniv na
– ,1 1[ ].
Oçevydno takoΩ, wo koly nerivnist\ (11) vykonu[t\sq dlq mnohoçlena P
pry deqkomu n = n0, to vona vykonu[t\sq dlq c\oho mnohoçlena i pry n ≥ n0.
3. Oçevydno, wo dlq P x( ) = x nerivnist\ (11) vykonu[t\sq pry n = 2. Dlq
P x( ) = x2 – b2 , b ∈ 0 1,( ], spivvidnoßennq (11) takoΩ spravdΩu[t\sq, oskil\ky
2
0
1 2∫ t – b dt2 = 8
3
3b + 2
3
– 2 2b ≥ max b2{ , 1
4
– b2} ≥ x b2 2– , x ∈ 0 1
2
,
(per-
ßa nerivnist\ dovodyt\sq metodamy dyferencial\noho çyslennq, a druha [ oçe-
vydnog). Takym çynom, vraxovugçy teoremu 3, oderΩu[mo nerivnist\ (3) u vy-
padku (n, k) = (2, 0).
4. Nexaj P x( ) = x3 – b x2 , b ∈ 0 1,( ]. Todi nerivnist\ (11) vykonu[t\sq, tomu
wo 2
0
1 3 2∫ t b t dt– = 2
0
2 3b
b t t dt∫ ( – ) + 2
1 3 2
b
t b t dt∫ ( – ) = 2
4
4b
– b4
4
+ b4
2
+ 1
4
–
– b2
2
= b4 + 1
2
– b2 ≥ x3 – b x2 , x ∈ 0 1
2
,
. Ostannq nerivnist\ [ pravyl\nog,
bo b4 + (x – 1)b2 + 1
2
– x3 ≥ 0 i b4 – (x + 1) b2 + 1
2
+ x3 ≥ 0, x ∈ 0 1
2
,
, wo
vstanovlg[t\sq metodamy analizu.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
POSYLENNQ ODNI{} NERIVNOSTI DLQ ALHEBRA}ÇNYX MNOHOÇLENIV 241
Takym çynom, vraxovugçy teoremu 3, otrymu[mo nerivnist\ (3) u vypadku (n,
k) = (2, 1) i u vypadku (n, k) = (3, 0).
5. Vstanovymo nerivnist\ (11) dlq neparnyx mnohoçleniv P t( ) = P tm2 1+ ( ) =
= t m2 1+ + am –1 t m2 1– + … + a t0 , t ∈R, m ∈N . Poznaçymo G s( ) : = sm +
+ am –1 sm –1 + … + a0, s ∈R . Dali vykonu[mo zaminu s = t2 i vykorystovu[mo
nerivnist\ Bernßtejna dlq poxidno] alhebra]çnoho mnohoçlena [4] (hl. V , § 2,
naslidok do teoremy 1):
n P t dt
0
1
∫ ( ) = n G s ds
2
0
1
∫ ( ) ≥ n G u du
s
s
2 1 1
0
max ( )
– ,∈[ ] ∫ ≥ n G u du
s
s
2 1 1
0
max ( )
– ,∈[ ] ∫ ≥
≥ n x
m
G u du x
s
2
1
1
2 2
0
2– ( )
( ) ( )
+
′
∫ = n x
m
G x
2
1
1
4
2–
( )
+
≥
≥ n
m n
x G x
2
4
2
2 1
1 1
( )
– ( )
+
=
n
m
P x
4 1
2 2
–
( )
+
, x
n
∈
0 1, .
Qkwo n = 4, k = 3, m ≤ 2, to
n
m
4 1
2 2
–
+
≥
80
2 3⋅
≥ 1. Qkwo n ≥ 3, k = 2, 2m + 1 ≤
≤ n + k, to
n
m
4 1
2 2
–
+
≥
n
n
4 1
3
–
+
≥ 1. Tomu nerivnist\ (11) vykonu[t\sq dlq vsix
neparnyx mnohoçleniv stepenq ne vywe n + k u vypadku (n, k) = (4, 3) ta pry
n ≥ 3 i k = 2.
6. Wob dovesty perßu çastynu lemy, zalyßylos\ pereviryty nerivnist\ (11)
dlq parnyx alhebra]çnyx mnohoçleniv stepenq 4 pry n = 3 i stepenq 6 pry n =
= 4.
Rozhlqnemo funkci]
R b c x1( , , ) = 3
0
1
2 2 2 2∫ ( – )( – )t b t c dt –
– ( – )( – )x b x c2 2 2 2 , b, c ∈[ ]0 1, , x ∈
0 1
3
, ,
R a b c x2( , , , ) = 4
0
1
2 2 2 2 2 2∫ ( – )( – )( – )t a t b t c dt –
� ( – )( – )( – )x a x b x c2 2 2 2 2 2 , a, b, c ∈[ ]0 1, , x ∈
0 1
4
, .
Dlq nyx z vykorystannqm komp�gtera dovedemo, wo vony nevid�[mni na svo]x
mnoΩynax vyznaçennq. Dlq c\oho pryrivng[mo do nulq poxidnu po x, z otryma-
noho rivnqnnq vyraΩa[mo x çerez inßi zminni, a zminni a, b, c perebyra[mo z
dostatn\o dribnym krokom. Zvidsy za teoremog 3 vyplyva[ perßa çastyna lemy.
7. Kontrpryklady dlq vidpovidnyx par: qkwo (n, k) = (1, 1), to g =
=
X – , – / / ,1 1 2 1 2 1[ ] [ ]U , P x( ) = 1 – x2; qkwo (n, k) = (2, 2), to g =
X – , – / / ,1 1 5 1 5 1[ ] [ ]U ,
P x( ) = (0,49 – x2) (0,81 – x2); qkwo (n, k) = (3, 3), to g =
X – , – / / ,1 1 30 1 30 1[ ] [ ]U ,
P x( ) = (0,25 – x2) (0,64 – x2) (0,81 – x2).
Lemu 3 dovedeno.
ZauvaΩennq 7. Na pidstavi zauvaΩennq 1 nerivnist\ (3) vykonu[t\sq takoΩ
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
242 O. N. NESTERENKO, T. D. TYMOÍKEVYÇ, A. V. ÇAJKOVS|KYJ
dlq par (1, – 1), (2, – 1), (2, 0), (3, 0), (3, 1), (4, 1), (4, 2), (5, 1) i [ xybnog dlq
(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4). Ce razom iz zauvaΩennqm 6 svidçyt\ pro
pravyl\nist\ zapovnennq tablyci na poçatku statti.
Lema 4. Nexaj vykonu[t\sq umova teoremy 5. Nexaj takoΩ funkciq P ∈
∈ L m
1
11 1– , –[ ]( ) dlq koΩnoho j ∈ 1{ , 2, … , m} qk funkciq odni[] j-] zminno]
pry vsix fiksovanyx inßyx zminnyx zadovol\nq[ odnu z umov (4) � (6) z zaminog n
na nj , pryçomu ci umovy pry riznyx j moΩut\ buty riznymy, ale pry fiksova-
nomu j povynni zalyßatys\ tymy samymy pry riznomu fiksuvanni inßyx zmin-
nyx. Todi spravdΩu[t\sq nerivnist\ (7).
Dovedennq provodyt\sq indukci[g po m doslivnym povtorennqm mirkuvan\
z dovedennq lemy 5 z roboty [2], qkwo vraxuvaty, wo za umov dovodΩuvano] lemy
pry m ≥ 2 funkciq – ,1 1[ ] � y � g
n
⋅
1
,
⋅
n2
, … ,
⋅
nm –1
, y
L m
[ ]( )1 1 1– ,
[ dopusty-
mog zi stalog 1/nm, a funkciq – ,1 1[ ] � y � P ⋅( , … , ⋅ , y L m) [ ]( )1
11 1– , – zado-
vol\nq[ tu samu umovu z (4) � (6), wo j usi funkci] – ,1 1[ ] � y � P x( 1, … , xm –1,
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ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 2
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| id | umjimathkievua-article-3015 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:37Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/db/5d97ca2f60331569a0affa6ac8923ddb.pdf |
| spelling | umjimathkievua-article-30152020-03-18T19:43:20Z Improvement of one inequality for algebraic polynomials Посилення однієї нерівності для алгебраїчних многочленів Nesterenko, A. N. Tymoshkevych, T. D. Chaikovs'kyi, A. V. Нестеренко, О. Н. Тимошкевич, Т. Д. Чайковський, А. В. We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $k = 1$. We also propose some other new pairs $(n, k)$ for which this inequality holds. Some conditions on the polynomial $P_{n+k}$ under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed. Доказано, что неравенство $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, где $g : [-1, 1]→ℝ$ — монотонная нечетная функция, а $P_{n+k}$ — алгебраический многочлен степени не выше $n + k$, выполняется для всех натуральных $n$ при $k = 0$ и для всех натуральных $n ≥ 2$ при $k = 1$, найдены другие новые пары $(n, k)$, для которых оно имеет место. Установлены некоторые условия на многочлен $P_{n+k}$, при которых это неравенство превращается в равенство. Получены некоторые обобщения этого неравенства. Institute of Mathematics, NAS of Ukraine 2009-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3015 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 2 (2009); 231-242 Український математичний журнал; Том 61 № 2 (2009); 231-242 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3015/2779 https://umj.imath.kiev.ua/index.php/umj/article/view/3015/2780 Copyright (c) 2009 Nesterenko A. N.; Tymoshkevych T. D.; Chaikovs'kyi A. V. |
| spellingShingle | Nesterenko, A. N. Tymoshkevych, T. D. Chaikovs'kyi, A. V. Нестеренко, О. Н. Тимошкевич, Т. Д. Чайковський, А. В. Improvement of one inequality for algebraic polynomials |
| title | Improvement of one inequality for algebraic polynomials |
| title_alt | Посилення однієї нерівності для алгебраїчних многочленів |
| title_full | Improvement of one inequality for algebraic polynomials |
| title_fullStr | Improvement of one inequality for algebraic polynomials |
| title_full_unstemmed | Improvement of one inequality for algebraic polynomials |
| title_short | Improvement of one inequality for algebraic polynomials |
| title_sort | improvement of one inequality for algebraic polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3015 |
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