Comparison theorems for some nonsymmetric classes of functions
We prove comparison theorems of the Kolmogorov type for some nonsymmetric classes of functions.
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| Date: | 2008 |
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| Main Authors: | , , , |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3212 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509261666189312 |
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| author | Motornaya, O. V. Motornyi, V. P. Моторная, О. В. Моторный, В. П. Моторная, О. В. Моторный, В. П. |
| author_facet | Motornaya, O. V. Motornyi, V. P. Моторная, О. В. Моторный, В. П. Моторная, О. В. Моторный, В. П. |
| author_sort | Motornaya, O. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:23Z |
| description | We prove comparison theorems of the Kolmogorov type for some nonsymmetric classes of functions. |
| first_indexed | 2026-03-24T02:38:18Z |
| format | Article |
| fulltext |
UDK 517.5
V. P.�Motorn¥j (Dnepropetr. nac. un-t),
O. V. Motornaq (Kyev. nac. un-t ym. T. Íevçenko)
TEOREMÁ SRAVNENYQ DLQ NEKOTORÁX
NESYMMETRYÇNÁX KLASSOV FUNKCYJ
The comparison theorems of the Kolmogorov type for some nonsymmetric classes of functions are
proved.
Dovedeno teoremy porivnqnnq typu Kolmohorova dlq deqkyx nesymetryçnyx klasiv funkcij.
Oboznaçym çerez Wp
r
, r N∈ , p ≥ 1, klass funkcyj f, zadann¥x na otrezke
– ,1 1[ ], (r – 1)-q proyzvodnaq kotor¥x absolgtno neprer¥vna, a f r
p
( ) ≤ 1.
Kak ob¥çno, v sluçae p = ∞ polahaem, çto f xr( ) ( ) ≤ 1 poçty vsgdu na otrezke
– ,1 1[ ]. Çerez Wn
0,±
, n = 0, 1, … , r N∈ , oboznaçym klass funkcyj f, zadann¥x
na otrezke – ,1 1[ ], takyx, çto max ( ),±{ }f t 0 ≤ 1 poçty vsgdu na otrezke – ,1 1[ ],
y ortohonal\n¥x lgbomu mnohoçlenu stepeny ne v¥ße n.
V sylu sootnoßenyj dvojstvennosty [1, s. 46] dlq nayluçßyx odnostoron-
nyx pryblyΩenyj
E fn
∓( )1 = sup ( ) ( )
,
–h Wn
f t h t dt
∈ ± ∫
0
1
1
, (1)
hde E fn
–( )1 (sootvetstvenno E fn
+( )1) — nayluçßee snyzu (sootvetstvenno
sverxu) pryblyΩenye funkcyy f L∈ 1 alhebrayçeskymy mnohoçlenamy v pro-
stranstve L1. Dlq lgboho r N∈ poloΩym
Wn
r,± = h t
r
t u h u du h Wr
r
n( )
( – )!
– ( ) ,– ,
–
= ( ) ∈
+
±∫1
1
1 0
1
1
,
hde
x a r– –( )+
1 =
x a x a
x a
r– , ,
, ,
–( ) ≥
<
1
0
— useçennaq stepen\.
Hranyc¥ znaçenyj funkcyj yz klassov Wn
r,±
v fyksyrovannoj toçke a ∈
∈ (– 1, 1) poluçym yz ravenstv
sup ( )
, –h Wn
r
h a
∈
= 1
1
2
2 1
1
1
1
1
( – )!
( – ) , ,
( – ) , ,
–
– –r
E x a r k
E x a r k
n
r
n
r
+
+
+
( ) =
( ) = +
(2)
inf ( )
, –h Wn
r
h a
∈
=
–
( – )!
( – ) , ,
( – ) , ,
– –
–
1
1
2
2 1
1
1
1
1
r
E x a r k
E x a r k
n
r
n
r
+
+
+
( ) =
( ) = +
(3)
© V. P. MOTORNÁJ, O. V. MOTORNAQ, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7 969
970 V. P. MOTORNÁJ, O. V. MOTORNAQ
sup ( )
,h Wn
r
h a
∈ +
= 1
1
2
2 1
1
1
1
1
( – )!
( – ) , ,
( – ) , ,
– –
–r
E x a r k
E x a r k
n
r
n
r
+
+
+
( ) =
( ) = +
(4)
inf ( )
,h Wn
r
h a
∈ +
=
–
( – )!
( – ) , ,
( – ) , .
–
– –
1
1
2
2 1
1
1
1
1
r
E x a r k
E x a r k
n
r
n
r
+
+
+
( ) =
( ) = +
(5)
V rabote [2] ustanovlen¥ neravenstva
E
x a
rn
r
∓ ( – )
( – )!
–
+
1
1
1
≤
sup ( ) –t r
r
r
D t a
n
∓( ) ( )1 2
+ C
a
n
r
r
r
1 2 2
1
–
( – )( ) +
+ , (6)
hde D tr( ) = 2B tr( ), a B tr( ) — funkcyq Bernully:
B tr( ) =
cos( – )kt r
kr
k
π 2
1=
∞
∑ .
Sçytaq r fyksyrovann¥m, oboznaçaem Cr
∗ = max
1 1≤ ≤ +k r
kC y polahaem
Cr a, = 1 +
2
1 2 2
C
n a
r
r r
∗
( ) +
–
– ( – ) . (7)
Tohda yz ravenstv (2), (3) y neravenstva (6) dlq lgboj funkcyy h ∈ Wn
r, –
sle-
dugt ocenky dlq znaçenyj funkcyy h v toçke a ∈ (– 1, 1):
C
D t a
n
r a
t r
r
r,
min ( ) –1 2( )
≤ h a( ) ≤ C
D t a
n
r a
t r
r
r,
max ( ) –1 2( )
. (8)
Sootvetstvenno, yz ravenstv (4), (5) y neravenstva (6) dlq lgboj funkcyy h ∈
∈ Wn
r,+
sledugt ocenky dlq znaçenyj funkcyy h v toçke a ∈ (– 1, 1):
C
D t a
n
r a
t r
r
r,
min – ( ) –( ) ( )1 2
≤ h a( ) ≤ C
D t a
n
r a
t r
r
r,
max – ( ) –( ) ( )1 2
. (9)
Dlq λ > 0 rassmotrym funkcyy φλ, ( )r t = λ λ– ( )r
rD t y ψλ, ( )r t =
= – ( )–λ λr
rD t . Oçevydno, çto r-q proyzvodnaq (tam, hde ona suwestvuet) funk-
cyy φλ, ( )r t ravna – 1, a funkcyy ψλ, ( )r t ravna 1. Çerez W r
(– , )
, –
∞ ∞ (sootvet-
stvenno Wr
(– , )
,
∞ ∞
+
) oboznaçym klass funkcyj, zadann¥x na vsej dejstvytel\-
nojFosy, ( – )r 1 -q proyzvodnaq kotor¥x lokal\no absolgtno neprer¥vna, a
f tr( )( ) ≥ – 1 (sootvetstvenno f tr( )( ) ≤ 1) poçty vsgdu. Yzvestna sledugwaq
teorema Xermandera [3].
Teorema 1. Pust\ h ∈ W r
(– , )
, –
∞ ∞ , r = 2, 3, … , y dlq nekotoroho λ > 0
min ( ),φλ r t < h t( ) < max ( ),φλ r t . Esly v nekotor¥x toçkax x, y R∈ h x( ) =
= φλ, ( )r y , to ′h x( ) ≤ ′φλ, ( )r y , esly ′φλ, ( )r y > 0, y ′h x( ) ≥ ′φλ, ( )r y , esly
′φλ, ( )r y < 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
TEOREMÁ SRAVNENYQ DLQ NEKOTORÁX NESYMMETRYÇNÁX … 971
Sootvetstvugwee utverΩdenye ymeet mesto dlq funkcyj yz klassa
W r
(– , )
,
∞ ∞
+
. V πtom sluçae vmesto funkcyj φλ, ( )r t sleduet yspol\zovat\ funkcyy
ψλ, ( )r t . Funkcyy φλ, ( )r t y ψλ, ( )r t prynqto naz¥vat\ funkcyqmy sravnenyq.
Esly n ≥ r – 1, to v sylu ortohonal\nosty mnohoçlenov stepeny n funk-
cyy h t( ) yz klassov Wn
r,±
udovletvorqgt uslovyqm h k( )(– )1 = hk ( )1 = 0, k =
= 0, 1, … , r – 1, blahodarq kotor¥m, doopredelqq kaΩdug funkcyg h t( ) yz
klassa Wn
r,±
nulem vne otrezka – ,1 1[ ], moΩno sçytat\, çto Wn
r,± � W r
(– , )
,
∞ ∞
±
dlq r ≥ 1. Yz neravenstva (8) (sootvetstvenno (9)) sleduet, çto dlq lgboj
funkcyy h t( ) ∈ Wn
r, –
(sootvetstvenno h t( ) ∈ Wn
r,+
) ymegt mesto neravenstva
min ( ),φn r t < C h tr,
– ( )0
1 < max ( ),φn r t
(sootvetstvenno
min ( ),ψn r t < C h tr,
– ( )0
1 < max ( ),ψn r t ).
Poπtomu teorema sravnenyq ymeet mesto dlq lgboj par¥ C hr,
–
0
1 ∈ Wn
r, –
y
φn r t, ( ) (sootvetstvenno C hr,
–
0
1 ∈ Wn
r,+
y ψn r t, ( )).
Oçevydno, çto teoremaF1 dast hrubug ocenku, esly m¥ budem ocenyvat\
proyzvodnug funkcyy h t( ) yz klassa Wn
r,±
v toçkax, udalenn¥x ot toçky 0.
V svqzy s neobxodymost\g ymet\ analoh teorem¥ Xermandera, pozvolqgwyj
dostatoçno xoroßo ocenyvat\ proyzvodnug funkcyy h t( ) yz klassa Wn
r,±
dlq
vsex x ∈ (– 1, 1), vvedem sledugwee opredelenye.
Opredelenye 1. Pust\ h ∈ Wn
r, –
, x ∈ (– 1, 0). Dyfferencyruemaq funk-
cyq f naz¥vaetsq funkcyej sravnenyq dlq h v toçke x, esly min ( )t f t ≤
≤ h t( ) ≤ max ( )t f t , t ∈ – ,1 x( ], f x( ) = h x( ) y sign ′f x( ) = sign ′h x( ) .
Vvedem sledugwye oboznaçenyq: λ( )x = n x1 2– , ν( )x = π 1 2– /x n .
Zametym, çto 2ν( )x est\ peryod funkcyj φλ( ), ( )x r t y ψλ( ), ( )x r t . Netrudno do-
kazat\ (sm. [4], lemmuF1), çto dlq lgboho x ∈ –1 +( π n , –π n) suwestvuet
x∗ ∈ x x,( , + )π n takoe, çto
x∗ = x + ν( )x∗
, (10)
y pry πtom bol\ßomu znaçenyg x sootvetstvuet bol\ßee x∗
.
Lemma 1. Pust\ h ∈ Wn
k,–
, k = 1, 2, … , r, x ∈ –1 +( π n , –π n). V slu-
çaqx k = 4m + 1, h x( ) < 0; k = 4m, ′h x( ) > 0; k = 4m – 1, h x( ) > 0; k = 4m – 2,
′h x( ) < 0 funkcyej sravnenyq dlq h budet funkcyq
f tr k, , ( )1 = C t xr x x k, ( ), ( – )φλ ,
a v sluçaqx k = 4m + 1, h x( ) > 0; k = 4m, ′h x( ) < 0; k = 4m – 1, h x( ) < 0; k =
= 4m – 2, ′h x( ) > 0 — funkcyq
f tr k, , ( )2 = C x t xr x x k, ( ), ( ) –φ νλ +( ),
hde v kaΩdom yz sluçaev x v¥brano tak, çto 0 < x – x < λ( )x , a velyçyna
Cr x, opredelena ravenstvom (7).
Dokazatel\stvo. Rassmotrym osnovn¥e sluçay. Pust\ k = 4m + 1, h x( ) <
< 0 y y ∈ x x, ∗[ ]. Tohda funkcyq C tr y y k, ( ), ( )φλ v¥pukla vnyz na otrezke ν( )y[ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
972 V. P. MOTORNÁJ, O. V. MOTORNAQ
2ν( )y ] y v sylu neravenstv (8)
0 > h x( ) > min ( ), ( ),t r y y kC tφλ .
Poπtomu suwestvuet edynstvennoe çyslo α( )y takoe, çto 0 < α( )y < ν( )y y v
toçke 2ν( )y – α( )y funkcyq C tr y y k, ( ), ( )φλ prynymaet znaçenye h x( ) , a znak
proyzvodnoj sovpadaet so znakom proyzvodnoj h x( ) . Oçevydno, çto funkcyq
α( )y neprer¥vna na otrezke x x, ∗[ ]. Na πtom otrezke zadadym ewe funkcyg
g y( ) = x + α( )y – y. Poskol\ku g x( ) = α( )x > 0, a g x( )∗ = x + α( )x∗ – x∗ =
= – ( )ν x∗ + α( )x∗ < 0, suwestvuet toçka x ∈ x x, ∗[ ] takaq, çto g x( ) = 0, sle-
dovatel\no, x = x + α( )x .
PoloΩym
f tr k, , ( )1 = C t xr x x k, ( ), –φλ ( ).
V sylu neravenstv (8)
min ( ), ,t r kf t1 ≤ h t( ) ≤ max ( ), ,t r kf t1 ∀ t ∈ (– , )1 x , f xr k, , ( )1 = h x( ),
y znak proyzvodnoj f tr k, , ( )1 v toçke x sovpadaet s sign ′h x( ) .
Rassmotrym sluçaj k = 4m + 1, h x( ) > 0. Yspol\zuq v¥puklost\ vverx
funkcyy C tr y y k, ( ), ( )φλ na otrezke 0, ( )ν y[ ] y neravenstva (8), dlq kaΩdoho y
yz otrezka x x, ∗[ ] opredelqem edynstvennoe çyslo α( )y takoe, çto 0 < α( )y <
< ν( )y y v toçke ν( )y – α( )y funkcyq C tr y y k, ( ), ( )φλ prynymaet znaçenye
h x( ) , a znak proyzvodnoj sovpadaet so znakom proyzvodnoj ′h x( ). Kak y v pre-
d¥duwem sluçae, opredelqem çyslo x y polahaem
f tr k, , ( )2 = C x t xr x x k, ( ), ( ) –φ νλ +( ). (11)
Oçevydno, çto dlq funkcyy f tr k, , ( )2 v¥polnqgtsq uslovyq 1 – 3, opredelqg-
wye funkcyg sravnenyq dlq funkcyy h t( ) v toçke x.
Teper\ rassmotrym sluçaj k = 4m, ′h x( ) < 0. Pust\ y ∈ x x, ∗[ ]. Poskol\ku
v sylu neravenstv (8) çyslo h x( ) naxodytsq meΩdu naymen\ßym y naybol\ßym
znaçenyqmy funkcyy C tr y y k, ( ), ( )φλ y na otrezke 0, ( )ν y[ ] πta funkcyq ub¥-
vaet, suwestvuet edynstvennoe çyslo α( )y takoe, çto 0 < α( )y < ν( )y y
C yr y y k, ( ), ( )φ νλ ( – α( )y ) = h x( ) . Dalee, kak y v¥ße, opredelqem çyslo x y za-
daem funkcyg sravnenyq f tr k, , ( )2 ravenstvom (11). Ostal\n¥e sluçay ana-
lohyçn¥.
Lemma dokazana.
Zameçanye 1. Funkcyy sravnenyq f tr k, , ( )1 y f tr k, , ( )2 opredelen¥ tak,
çto proyzvodnaq funkcyy sravnenyq dlq h t( ) v toçke x qvlqetsq funkcyej
sravnenyq dlq ′h t( ) v toçke x1 ∈ x( – ν( ),x x), esly tol\ko ′h x( )1 = ′f xr k, , ( )1 1
(sootvetstvenno ′h x( )1 = ′f xr k, , ( )2 1 ) y sign ′′h x( )1 = sign ′′f xr k, , ( )1 1 (sootvetstven-
no sign ′′h x( )1 = sign ′′f xr k, , ( )2 1 ).
Dejstvytel\no, pust\ k = 4m + 1, h x( ) < 0, funkcyq f tr k, , ( )1 qvlqetsq
funkcyej sravnenyq dlq h t( ) v toçke x y ′h x( )1 = ′f xr k, , ( )1 1 , a takΩe
sign ′′h x( )1 = sign ′′f xr k, , ( )1 1 , hde x1 ∈ x( – ν( ),x x). No sign ′′f xr k, , ( )1 1 = 1, tak kak
funkcyq f tr k, , ( )1 v¥pukla vverx na yntervale x( – ν( ),x x). Sledovatel\no,
dlq ′h t( ) neobxodymo brat\ funkcyg sravnenyq f tr m, , ( )4 1 , kotoraq, oçevydno,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
TEOREMÁ SRAVNENYQ DLQ NEKOTORÁX NESYMMETRYÇNÁX … 973
sovpadaet s ′ +f tr m, , ( )4 1 1 . V sluçae k = 4m – 1, h x( ) > 0 rassuΩdenyq analo-
hyçn¥.
Rassmotrym sluçaj k = 4m, ′h x( ) > 0. V sylu opredelenyq funkcyej srav-
nenyq dlq h t( ) v toçke x qvlqetsq f tr k, , ( )1 . PredpoloΩym, çto ′h x( )1 =
= ′f xr k, , ( )1 1 , a sign ′′h x( )1 = sign ′′f xr k, , ( )1 1 , hde x1 ∈ x( – ν( ),x x). Poskol\ku
′f tr k, , ( )1 > 0 na yntervale x( – ν( ),x x), to ′h x( )1 > 0. Sledovatel\no, dlq
′h t( ) v toçke x1 neobxodymo brat\ funkcyg sravnenyq f tr m, – , ( )4 1 1 , a ona sov-
padaet s ′f tr m, , ( )4 1 . Tot Ωe rezul\tat poluçaem v sluçae k = 4m – 2, ′h x( ) < 0, a
takΩe v sluçaqx, kohda funkcyej sravnenyq qvlqetsq funkcyq f tr k, , ( )2 .
Zameçanye 2. Esly x ∈ π n[ , 1 – π n], to postroenye funkcyj sravne-
nyq analohyçno. V πtom sluçae çyslo x∗
opredelqetsq ravenstvom x∗ = x –
– ν( )x∗
, a çyslo x udovletvorqet neravenstvu 0 < x – x < ν( )x . Takym Ωe ob-
razom opredelqgtsq (y dokaz¥vaetsq yx suwestvovanye) funkcyy sravnenyq
dlq funkcyj yz klassa Wn
r,+
.
Opredelenye 2. Pust\ h t( ) y f t( ) — dyfferencyruem¥e funkcyy na
yntervale (– 1, 1), v nekotoroj toçke x ∈ (– 1, 1) h x( ) = f x( ) y proyzvodn¥e
v πtoj toçke ymegt odynakov¥j znak. Budem hovoryt\, çto h t( ) yzmenqetsq
v toçke x b¥stree funkcyy f t( ), esly sign h t( )( – f t( )) = sign ( ′f x( )(t – x))
dlq vsex t yz nekotoroj okrestnosty toçky x.
Oçevydno, çto esly ′h x( ) > ′f x( ) , to h t( ) yzmenqetsq v toçke x b¥st-
ree funkcyy f t( ). No h t( ) moΩet yzmenqt\sq v toçke x b¥stree, esly
′h x( ) = ′f x( ) . Esly Ωe h t( ) ne yzmenqetsq b¥stree v toçke x, to ′h x( ) ≤
≤ ′f x( ) .
Teorema 2. Pust\ h ∈ Wn
k,–
, k = 1, 2, … , r. Esly ω( )t qvlqetsq funk-
cyej sravnenyq dlq h t( ) v toçke x, t.Fe . ω( )t est\ f tr k, , ( )1 y l y f tr k, , ( )2 ,
to h t( ) ne yzmenqetsq b¥stree ω( )t v toçke x y, sledovatel\no, ′h x( ) ≤
≤ ′ω ( )x . Pry πtom v sluçae k = 1 sleduet sravnyvat\ znaçenyq proyzvodn¥x
tol\ko v toçkax, v kotor¥x proyzvodnaq otrycatel\naq.
Dokazatel\stvo v osnovnom sovpadaet s dokazatel\stvom teorem¥F1 yz
[4]. V sluçae k = 1 ′h x( ) ≥ – 1, a vo vsex toçkax, hde h x( ) = C tr x x, ( ), ( )φλ 1 , pro-
yzvodnaq funkcyy C tr x x, ( ), ( )φλ 1 ravna – ,Cr x < – 1. Poπtomu v toçke x h t( )
ub¥vaet ne b¥stree funkcyy sravnenyq. Pust\ k = 2. PredpoloΩym, çto teo-
rema ne ymeet mesta. Tohda najdutsq funkcyq h ∈ Wn
2,–
y toçka x takye, çto
h x( ) = f xr, , ( )2 1 , esly ′h x( ) < 0, yly h x( ) = f xr, , ( )2 2 , esly ′h x( ) > 0, y h t( ) yz-
menqetsq b¥stree funkcyy sravnenyq v toçke x. V pervom sluçae raznost\
h t( )F– f tr, , ( )2 1 v nekotoroj toçke y ∈ x( – ν( ),x x) ymeet maksymum, a vo vtorom
sluçae raznost\ h t( ) – f tr, , ( )2 2 ymeet maksymum v toçke z yntervala x( , x +
+ ν( )x ) . Sledovatel\no, ′h y( ) – ′f yr, , ( )2 1 = 0 y ′′h y( ) – ′′f yr, , ( )2 1 ≤ 0 yly ′h z( ) –
–F ′f zr, , ( )2 2 = 0 y ′′h z( ) – ′′f zr, , ( )2 2 ≤ 0. Poslednye neravenstva nevozmoΩn¥, tak
kak yz opredelenyq klassa Wn
2,–
sleduet, çto dlq vsex t, hde proyzvodnaq su-
westvuet, ′′h t( ) – ′′ω ( )t > 0. Dalee prymenym metod matematyçeskoj yndukcyy.
PredpoloΩym, çto teorema ymeet mesto dlq k – 1, y dokaΩem, çto ona spraved-
lyva dlq k . Pust\ snaçala k — neçetnoe çyslo, dlq opredelennosty k = 4m –
– 1. Sluçaj k = 4m + 1 analohyçen. PredpoloΩym, çto teorema ne ymeet mes-
ta. Tohda najdutsq funkcyq h ∈ Wn
m4 1– ,–
y toçka x takye, çto h x( ) =
= f xr k, , ( )1 , esly h x( ) > 0, yly h x( ) = f xr k, , ( )2 , esly h x( ) < 0, y h t( ) yzme-
nqetsq b¥stree sootvetstvugwej funkcyy sravnenyq v toçke x. Dalee budem
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
974 V. P. MOTORNÁJ, O. V. MOTORNAQ
rassmatryvat\ sluçaj h x( ) > 0. V sluçae h x( ) < 0 rassuΩdenyq analohyçn¥.
Esly ′h x( ) > 0, to v nekotoroj toçke y ∈ ( , )x x raznost\ h t( ) – f tr k, , ( )1 ymeet
maksymum, a esly ′h x( ) < 0, to πta raznost\ ymeet maksymum v nekotoroj toçke
z ∈ x( – ν( ),x x) . Sledovatel\no, ′h y( ) – ′f yr k, , ( )1 = 0 y ′′h y( ) – ′′f yr k, , ( )1 ≤ 0
yly ′h z( ) – ′f zr k, , ( )1 = 0 y ′′h z( ) – ′′f zr k, , ( )1 ≤ 0. Esly v poslednyx neravenstvax
ymeet mesto znak strohoho neravenstva, to srazu poluçaem protyvoreçye s ut-
verΩdenyem teorem¥ dlq k = 4m – 2. Esly v πtyx neravenstvax ymeet mesto
znak ravenstva, to raznost\ ′h t( ) – ′f tr k, , ( )1 v toçke y yly z menqet znak s
plgsa na mynus, a πto oznaçaet, çto ′h t( ) yzmenqetsq b¥stree ′f tr k, , ( )1 v toçke
y yly z, çto protyvoreçyt utverΩdenyg teorem¥ dlq k = 4m – 2.
Rassmotrym teper\ sluçaj k = 4m. Snova predpolahaem, çto suwestvugt
funkcyq h ∈ Wn
m4 ,–
y toçka x, dlq kotor¥x teorema neverna. Pust\ dlq
opredelennosty ′h x( ) > 0 (sluçaj ′h x( ) < 0 analohyçen). PredpoloΩym sna-
çala, çto ′′f xr m, , ( )4 1 ≤ 0. Tohda ′′f tr m, , ( )4 1 ≤ 0 y na yntervale ( , )x x y v nekoto-
roj toçke y ∈ ( , )x x raznost\ h t( ) – f tr m, , ( )4 1 ymeet maksymum. Sledovatel\-
no, ′h y( ) = ′f yr m, , ( )4 1 y raznost\ ′h t( ) – ′f tr m, , ( )4 1 v toçke y menqet znak s plg-
sa na mynus, t.Fe. ′h t( ) yzmenqetsq b¥stree v toçke y, çto protyvoreçyt utver-
Ωdenyg teorem¥ dlq k = 4m – 1. Pust\ teper\ ′′f xr m, , ( )4 1 > 0. Tohda ′′f tr m, , ( )4 1 >
> 0, y na yntervale x( – ν( ),x x) y v nekotoroj toçke z ∈ x( – ν( ),x x) raz-
nost\ h t( ) – f tr m, , ( )4 1 ymeet mynymum. Sledovatel\no, ′h z( ) = ′f zr m, , ( )4 1 y raz-
nost\ ′h t( ) – ′f tr m, , ( )4 1 v toçke z menqet znak s mynusa na plgs, t.Fe. ′h t( ) yz-
menqetsq b¥stree ′f tr m, , ( )4 1 v toçke z, çto protyvoreçyt utverΩdenyg teo-
rem¥ dlq k = 4m – 1.
Teorema dokazana.
Opredelenye 3. Pust\ h ∈ Wn
r, –
y ( , )a b � (–1 + π n, π n). Funkcyej
sravnenyq dlq h na yntervale ( , )a b budem naz¥vat\ lgbug funkcyg f t( ),
zadannug y dyfferencyruemug na dejstvytel\noj osy, takug, çto esly v ne-
kotor¥x toçkax x ∈ ( , )a b y y R∈ 1 h x( ) = f y( ) , to ′h x( ) ≤ ′f y( ) pry uslo-
vyy, çto znaky proyzvodn¥x sovpadagt.
Teorema 3. Pust\ h ∈ Wn
r, –
y ( , )a b � (–1 + π n, π n). Funkcyej srav-
nenyq dlq h na yntervale ( , )a b budet funkcyq
f t( ) = C tr a b r, ( ),
( )φλ ∗ , (12)
hde Cr a, opredelqetsq ravenstvom (7), a b∗
— ravenstvom (10).
Dokazatel\stvo. Pust\ h ∈ Wn
r, –
y x ∈ ( , )a b . Rassmotrym funkcyg
sravnenyq ω( )t dlq h t( ) v toçke x, t. Fe. ω( )t est\ f tr k, , ( )1 yly f tr k, , ( )2 , y
voz\mem lgboe y R∈ 1
, dlq kotoroho h x( ) = ω( )x = f y( ) y znaky proyzvodn¥x
v πtyx toçkax sovpadagt. Poskol\ku Cr x, < Cr a, , prymenqq teoremuF1 dlq
funkcyj C tr a,
– ( )1 ω y φλ( ),
( )
b r
t∗ , poluçaem
C xr a,
– ( )1 ′ω ≤ ′ ∗φλ( ),
( )
b r
y .
Sledovatel\no, v sylu teorem¥F2
′h x( ) ≤ ′ω ( )x ≤ C yr a b k, ( ),
( )′ ∗φλ = ′f y( ) .
TeoremaF3 dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
TEOREMÁ SRAVNENYQ DLQ NEKOTORÁX NESYMMETRYÇNÁX … 975
Zameçanyq. 3. Yz dokazatel\stva teorem¥F3 sleduet, çto funkcyej srav-
nenyq dlq h t( ) na yntervale ( , )a b budet takΩe funkcyq vyda f t t( – )0 dlq
lgboho t R0
1∈ , a ′f t t( – )0 budet funkcyej sravnenyq dlq ′h t( ) na πtom Ωe
yntervale ( , )a b .
4. Esly b ∈ (–π n , π n), to v kaçestve funkcyy sravnenyq dlq h t( ) na
yntervale ( , )a b sleduet vzqt\ funkcyg f t( ) = C tr n r, , ( )0 φ , tak kak teore-
maFXermandera dlq funkcyj C h tr,
– ( )0
1
y φn r t, ( ) v πtom sluçae πkvyvalentna
teoremeF3.
5. Esly f t( ) qvlqetsq funkcyej sravnenyq dlq h t( ) na yntervale ( , )a b ,
to funkcyq z t( ) = f dt( ) , d > 1, budet funkcyej sravnenyq dlq h t( ) na ynter-
vale ( , )a b . Pry πtom esly v nekotor¥x toçkax x ∈ ( , )a b y y R∈ 1
h x( ) = z y( ) ,
to ′h x( ) < ′z y( ) pry uslovyy, çto znaky proyzvodn¥x sovpadagt. Dejstvy-
tel\no, pust\ y1 = dy. Tohda f y( )1 = f dy( ) = z y( ) = h x( ) y v sylu teorem¥F3
′h x( ) ≤ ′f y( )1 = ′f dy( ) < d f dy′( ) = ′z y( ) .
1. Kornejçuk N. P., Lyhun A. A., Doronyn V. H. FApproksymacyq s ohranyçenyqmy. – Kyev:
Nauk. dumka, 1982. – 250 s.
2. Motorn¥j V. P., Motornaq O. V.F Ob odnostoronnem pryblyΩenyy useçenn¥x stepenej
alhebrayçeskymy mnohoçlenamy v srednem // Tr. Mat. yn-ta RAN. – 2005. – 248. – S. 185 –
193.
3. Hermander L. A new proof and a generalization of inequality of Bohr // Math. scand. – 1954. –
# 2. – P. 33 – 45.
4. Motorn¥j V. P., Motornaq O. V.6 Nayluçßee pryblyΩenye klassov dyfferencyruem¥x
funkcyj alhebrayçeskymy mnohoçlenamy v srednem // Tr. Mat. yn-ta RAN. – 1995. – 210. –
S. 171 – 188.
Poluçeno 03.11.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
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| id | umjimathkievua-article-3212 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:18Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cf/e2261e8dc7a8fe80ed8600fdee84efcf.pdf |
| spelling | umjimathkievua-article-32122020-03-18T19:48:23Z Comparison theorems for some nonsymmetric classes of functions Теоремы сравнения для некоторых несимметричных классов функций Motornaya, O. V. Motornyi, V. P. Моторная, О. В. Моторный, В. П. Моторная, О. В. Моторный, В. П. We prove comparison theorems of the Kolmogorov type for some nonsymmetric classes of functions. Доведено теореми порівняння типу Колмогорова для деяких несиметричних класів функцій. Institute of Mathematics, NAS of Ukraine 2008-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3212 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 7 (2008); 969–975 Український математичний журнал; Том 60 № 7 (2008); 969–975 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3212/3170 https://umj.imath.kiev.ua/index.php/umj/article/view/3212/3171 Copyright (c) 2008 Motornaya O. V.; Motornyi V. P. |
| spellingShingle | Motornaya, O. V. Motornyi, V. P. Моторная, О. В. Моторный, В. П. Моторная, О. В. Моторный, В. П. Comparison theorems for some nonsymmetric classes of functions |
| title | Comparison theorems for some nonsymmetric classes of functions |
| title_alt | Теоремы сравнения для некоторых несимметричных классов функций |
| title_full | Comparison theorems for some nonsymmetric classes of functions |
| title_fullStr | Comparison theorems for some nonsymmetric classes of functions |
| title_full_unstemmed | Comparison theorems for some nonsymmetric classes of functions |
| title_short | Comparison theorems for some nonsymmetric classes of functions |
| title_sort | comparison theorems for some nonsymmetric classes of functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3212 |
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