Inequalities for derivatives of functions in the spaces Lp
The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved: $$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\f...
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| Date: | 2008 |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509301817212928 |
|---|---|
| author | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_facet | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. |
| author_sort | Kofanov, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:15Z |
| description | The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved:
$$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt
\left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$
where $\varphi_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q \geq 1$ or for any $q > 0$ in the cases of $r = 2$ and $r = 3.$
As a corollary, well-known A. A. Ligun's inequality for functions $x \in L^{r}_{\infty}$ of the form
$$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r}} ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q < \infty,$$
is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$ |
| first_indexed | 2026-03-24T02:38:56Z |
| format | Article |
| fulltext |
UDK 517.5
V. A. Kofanov (Dnepropetr. nac. un-t)
NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ
V PROSTRANSTVAX Lp
The following sharp inequality for local norms of functions x Lr∈ ∞ ∞, ( )R is proved:
1
b a
x t dtq
a
b
−
′∫ ( ) ≤
1
1
0
1
π
ϕ
ϕ
π
r
q L
r
r
r
q
r
q
rt dt
x
x−
∞
−
∞∫ ∞
( )
( ) ( )R
, r ∈N ,
where ϕr is the perfect Euler spline, takes place on intervals [ , ]a b of monotonicity of the function x
for q ≥ 1 or for any q > 0 in the cases of r = 2 and r = 3.
As a corollary, well-known A. A. Ligun’s inequality for functions x Lr∈ ∞ of the form
x x xk
q
r k q
r
k r
k r r k r( )
/
/ ( ) /
≤
−
∞
− ∞
−
∞
ϕ
ϕ 1
1 , k r, ∈N , k r< , 1 ≤ < ∞q ,
is proved for q ∈[ , )0 1 in the cases of r = 2 and r = 3.
Otrymano novu toçnu nerivnist\ dlq lokal\nyx norm funkcij x Lr∈ ∞ ∞, ( )R :
1
b a
x t dtq
a
b
−
′∫ ( ) ≤
1
1
0
1
π
ϕ
ϕ
π
r
q L
r
r
r
q
r
q
rt dt
x
x−
∞
−
∞∫ ∞
( )
( ) ( )R
, r ∈N ,
de ϕr — ideal\nyj splajn Ejlera, na promiΩkax [ , ]a b monotonnosti x dlq vypadku q ≥ 1, a
takoΩ dlq dovil\nyx q > 0 u vypadkax r = 2 ta r = 3 .
Qk naslidok, vidomu nerivnist\ A. A. Lyhuna dlq periodyçnyx funkcij x Lr∈ ∞
x x xk
q
r k q
r
k r
k r r k r( )
/
/ ( ) /
≤
−
∞
− ∞
−
∞
ϕ
ϕ 1
1 , k r, ∈N , k r< , 1 ≤ < ∞q ,
dovedeno dlq q ∈[ , )0 1 u vypadkax r = 2 ta r = 3 .
1. Vvedenye. Symvolom G budem oboznaçat\ otrezok [ a, b ] , dejstvytel\nug
os\ R yly okruΩnost\ T, realyzovannug v vyde otrezka [ 0, 2 π ] s otoΩdestv-
lenn¥my koncamy. Budem rassmatryvat\ prostranstva L Gp( ) yzmerym¥x
funkcyj x : G → R takyx, çto x L Gp( ) < ∞ , hde
x L Gp( ) : =
x t dt p
x t p
p
G
p
t G
( ) , ,
sup ( ) , ,
/
∫( ) < < ∞
= ∞
∈
1
0esly
eslyvrai
y esly lebehova mera µ G mnoΩestva G koneçna, to
x L G0( ) : = exp ( ) ln ( )µG x t dt
G
− ∫
1 .
Çerez L Gr
∞( ) oboznaçym prostranstvo funkcyj, ymegwyx lokal\no abso-
lgtno neprer¥vn¥e proyzvodn¥e do ( )r − 1 -ho porqdka, pryçem x L Gr( ) ( )∈ ∞ .
PoloΩym
© V. A. KOFANOV, 2008
1338 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ V PROSTRANSTVAX Lp 1339
L Gr
∞ ∞, ( ) : = L G L Gr
∞ ∞( ) ( )∩ y W Gr
∞ ∞, ( ) : = x L G xr r∈ ≤{ }∞ ∞ ∞,
( )( ): 1 .
Esly µ G < ∞ , to, oçevydno, L Gr
∞ ∞, ( ) = L Gr
∞( ) y W Gr
∞ ∞, ( ) = W Gr
∞( ) . V slu-
çae G = T vmesto Lr
∞( )T , W r
∞( )T y x Lp( )T budem pysat\ Lr
∞ , W r
∞ y x p
sootvetstvenno.
Symvolom ϕr t( ) oboznaçym r-j 2 π -peryodyçeskyj yntehral s nulev¥m
srednym znaçenyem na peryode ot funkcyy ϕ0( )t = sgn sin t .
A.DA.DLyhun [1] dlq funkcyj x Lr∈ ∞, k r, ∈N , k < r , q ∈ [ 1, + ∞ ) , dokazal
sledugwee neravenstvo:
x k
q
( ) ≤
ϕ
ϕ
r k q
r
k r
k r r k r
x x
−
∞
− ∞
−
∞1
1
/
/ ( ) /
. (1)
V dannoj rabote dokazano, çto neravenstvo (1) soxranqet sylu dlq proyz-
vol\n¥x q ≥ 0 v sluçae funkcyj x Lr∈ ∞ maloj hladkosty ( r = 2 yly r = 3 ).
Pry πtom predloΩen metod dokazatel\stva neravenstva (1) dlq vsex q ≥ 0, os-
novann¥j na neravenstvax dlq lokal\n¥x norm proyzvodn¥x. ∏tot metod pry-
menqlsq avtorom ranee v druhyx sluçaqx. V çastnosty, s eho pomow\g poluçe-
n¥ naybolee obwye toçn¥e neravenstva typa Bernßtejna dlq polynomov y
splajnov [2, 3]. Nam potrebuetsq sledugwyj çastn¥j sluçaj neravenstva dlq
lokal\n¥x norm, poluçennoho v πtyx rabotax. Pust\ x Lr∈ ∞ ∞, ( )R , r ≥ 2, a çys-
la a, b ∈ R takov¥, çto
′x a( ) = ′x b( ) = 0, ′x t( ) > 0, t ∈ ( a, b ) . (2)
Tohda dlq lgboho q ≥ 1
′∫ x t dtq
a
b
( ) ≤ ϕ
ϕ
π
r
q L
r
r q
r r
q
rt dt
x
x−
∞
− +
∞
−
∫ ∞
1
0
1 1
1
( ) ( )
( )
( )R . (3)
Neravenstvo (3) ne v¥polnqetsq dlq q < 1. V dannoj rabote dokazano (teo-
remaD1), çto dlq funkcyj x Lr∈ ∞ ∞, ( )R pry uslovyqx (2) ymeet mesto sledug-
waq modyfykacyq neravenstva (3):
1
b a
x t dtq
a
b
−
′∫ ( ) ≤ 1
1
0
1
π
ϕ
ϕ
π
r
q L
r
r
r
q
r
q
rt dt
x
x−
∞
−
∞∫ ∞
( ) ( ) ( )R . (4)
Odnako dannoe neravenstvo, v otlyçye ot neravenstva (3), soxranqet sylu pry
vsex q ≥ 0 po krajnej mere dlq funkcyj x Lr∈ ∞ maloj hladkosty. Yz nera-
venstva (4) neposredstvenno sleduet neravenstvo (1) dlq vsex q ≥ 0 (teore-
maD2).
2. Osnovnaq lemma. Budem hovoryt\, çto funkcyq ϕ α β∈ ∞L1 [ , ] qvlqetsq
funkcyej sravnenyq dlq funkcyy f L a b∈ ∞
1 [ , ], esly f L a b∞[ , ] ≤ ϕ α βL∞[ , ] y
yz uslovyq f ( )ξ = ϕ η( ) , ξ ∈[ , ]a b , η α β∈[ , ], sleduet neravenstvo ′f ( )ξ ≤
≤ ′ϕ η( ) (esly ukazann¥e proyzvodn¥e suwestvugt).
Pust\ γ ∈ ( 0, 1 ) , a funkcyq ϕ ∈ ∞L1 0 1[ , ] udovletvorqet uslovyqm ϕ ( )0 = 0,
′ϕ ( )t > 0 poçty vsgdu na ( 0, 1 ) y
γ ϕ ( 1 ) ≤
0
1
∫ ϕ ( )t dt . (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1340 V. A. KOFANOV
Çerez Kγ ϕ( ) oboznaçym klass funkcyj f L∈ ∞
1 0 1[ , ], dlq kotor¥x funkcyq ϕ
qvlqetsq funkcyej sravnenyq, y f ( )0 = 0, ′f t( ) ≥ 0 poçty vsgdu na ( 0, 1 ) .
Lemma&1. Pust\ q, γ ∈ ( 0, 1 ) , a funkcyq ϕ udovletvorqet opysann¥m
v¥ße uslovyqm. Tohda dlq lgboj funkcyy f K∈ γ ϕ( ) v¥polnqetsq neraven-
stvo
f t dt
f t dt
q
q
( )
( )
0
1
0
1
∫
∫[ ]γ ≤
ϕ
ϕ
γ
q
q
t dt
t dt
( )
( )
0
1
0
1
∫
∫[ ]
. (6)
Dokazatel\stvo. Dlq proyzvol\noj funkcyy f K∈ γ ϕ( ), f ≠ 0, poloΩym
F ( f ) : =
f t dt
f t dt
q
q
( )
( )
0
1
0
1
∫
∫[ ]γ .
Oçevydno, çto
f t f t( ) ( )1 2− ≤ ϕ L t t
∞
−[ , ]0 1 1 2 , t1, t2 ∈ [ 0, 1 ] , dlq f K∈ γ ϕ( ).
Krome toho, f L∞[ , ]0 1 ≤ ϕ ( )1 . Poπtomu klass Kγ ϕ( ) kompakten y, sledova-
tel\no, funkcyonal F ( f ) dostyhaet na nem svoej verxnej hrany. PokaΩem, çto
tol\ko funkcyq ϕ qvlqetsq πkstremal\noj v zadaçe
F ( f ) → sup, f K∈ γ ϕ( ). (7)
Zafyksyruem f K∈ γ ϕ( ), f ≠ ϕ, y dokaΩem, çto f ne moΩet b¥t\ πkstre-
mal\noj v zadaçe (7). Poskol\ku ϕ qvlqetsq funkcyej sravnenyq dlq f, to
f t( ) ≤ ϕ( )t , t ∈ [ 0, 1 ] . PoloΩym
a = a ( f ) : = sup [ , ] : ( ) ( )t f t t∈ ={ }0 1 ϕ .
Tak kak f ≠ ϕ, to a ∈ [ 0, 1 ) . Qsno, çto f ( a ) = ϕ ( a ) . Pust\, dalee,
b = b ( f ) : = sup [ , ] : ( ) ( )t a f t a∈ ={ }1 ϕ .
VozmoΩn¥ dva sluçaq: 1) b ∈ [ a, 1 ) y 2) b = 1.
Pust\ snaçala b ∈ [ a, 1 ) . Tohda dlq lgboho ε ∈ [ 0, 1 – b ) suwestvuet edyn-
stvennoe (v sylu strohoj monotonnosty ϕ ) çyslo δ ( ε ) takoe, çto
ϕ δ ε( ( ))a + = f b( )+ ε . (8)
Tem sam¥m na [ 0, 1 – b ) opredelena funkcyq δ ε( ) , neprer¥vnaq y neub¥vag-
waq, tak kak f y ϕ neprer¥vn¥ y ne ub¥vagt na [ 0, 1 ] . Pry πtom v sylu opre-
delenyq b suwestvuet ε0 takoe, çto
µ ε ε ε∈ ′ + >{ }( , ] : ( )0 01 f b > 0 ∀ ε1 ≤ ε0 .
A tak kak ′ +f b( )ε = ′ ′ +δ ε ϕ δ ε( ) ( ( ))a y ϕ stroho vozrastaet, to y
µ ε ε δ ε∈ ′ >{ }( , ] : ( )0 01 > 0 ∀ ε1 ≤ ε0 . (9)
Otmetym takΩe, çto δ ( ε ) ≤ ε na ( 0, ε0 ] . ∏to sleduet yz toho, çto ϕ qvlqetsq
funkcyej sravnenyq dlq f. Pry πtom v sylu opredelenyq a ( f ) , esly b ( f ) =
= a ( f ) , to δ ( ε ) < ε v ( 0, ε0 ] . Takym obrazom, esly ∆ : = b – a, to
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ V PROSTRANSTVAX Lp 1341
∆ + ε – δ ( ε ) > 0, ε ∈ ( 0, ε0 ] . (10)
Dlq lgboho ε ∈ ( 0, ε0 ] opredelym funkcyg
f tε( ) : =
ϕ δ ε
ϕ δ ε δ ε ε
ε
( ) [ , ( )],
( ( )) [ ( ), ],
( ) [ , ].
t t a
a t a b
f t t b
pry
pry
pry
∈ +
+ ∈ + +
∈ +
0
1
Qsno, çto f Kε γ ϕ∈ ( ). PokaΩem, çto
F f( )ε0
> F f( ) . (11)
Otsgda budet sledovat\, çto f ne qvlqetsq πkstremal\noj funkcyej v zadaçe
(7). PoloΩym F ( ε ) : = F f( )ε , ε ∈ ( 0, ε0 ] . Qsno, çto F ( f ) = lim ( )ε ε→0 F . Dlq
dokazatel\stva (11) dostatoçno ubedyt\sq v tom, çto
µ ε ε ε∈ ′ >{ }( , ] : ( )0 00 F > 0. (12)
V sylu opredelenyq F ( ε ) y fε ymeem
F ( ε ) =
ϕ ϕ δ ε ε δ ε
ϕ ϕ δ ε ε δ ε
δ ε
ε
δ ε
ε
γ
qa q q
b
a
b
q
t dt a f t dt
t dt a f t dt
( ) ( ( )) ( ) ( )
( ) ( ( )) ( ) ( )
( )
( )
0
1
0
1
+
+
+
+
∫ ∫
∫ ∫
+ + + −[ ] +
+ + + −[ ] +
∆
∆
.
PoloΩym dlq kratkosty
Iq( )ε : =
0
1a
q q
b
qt dt a f t dt
+
+
∫ ∫+ + + −[ ] +
δ ε
ε
ϕ ϕ δ ε ε δ ε
( )
( ) ( ( )) ( ) ( )∆ .
Tohda ln ( )F ε = ln ( ) ln ( )I q Iq ε γ ε− 1 . Zametym, çto vsledstvye (8)
′Iq( )ε = ϕ δ ε δ ε ϕ δ ε ϕ δ ε δ ε ε δ εq qa q a a( ( )) ( ) ( ( )) ( ( )) ( ) ( )+ ′ + + ′ + ′ + −[ ]−1 ∆ +
+ ϕ δ ε δ ε ϕ δ εq qa a( ( )) ( ) ( ( ))+ − ′[ ] − +1 =
= q a aqϕ δ ε ϕ δ ε δ ε ε δ ε− + ′ + ′ + −[ ]1( ( )) ( ( )) ( ) ( )∆ .
Poπtomu
′F
F
( )
( )
ε
ε
=
′
−
′I
I
q
I
I
q
q
( )
( )
( )
( )
ε
ε
γ ε
ε
1
1
= q a R′ + ′ + −[ ]ϕ δ ε δ ε ε δ ε ε( ( )) ( ) ( ) ( )∆ , (13)
hde
R( )ε =
ϕ δ ε
ε
γ
ε
q
q
a
I I
− + −
1
1
( ( ))
( ) ( )
.
Sledovatel\no,
R a( ) ( ( ))ε ϕ δ ε+ =
I
a
I
a
q
q
( )
( ( ))
( )
( ( ))
ε
ϕ δ ε
γ ε
ϕ δ ε+
−
+
− −1
1
1
=
=
0
1 1a q
b
q
t
a
dt
f t
a
dt
+
+
−
∫ ∫+
+ + −[ ] +
+
δ ε
ε
ϕ
ϕ δ ε
ε δ ε
ϕ δ ε
( )
( )
( ( ))
( ) ( )
( ( ))
∆ –
– γ ϕ
ϕ δ ε
ε δ ε
ϕ δ ε
δ ε
ε0
1 1a
b
t
a
dt
f t
a
dt
+
+
−
∫ ∫+
+ + −[ ] +
+
( )
( )
( ( ))
( ) ( )
( ( ))
∆ .
PoloΩym
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1342 V. A. KOFANOV
C : =
I
a
q
q
( )
( ( ))
ε
ϕ δ ε+
I
a
1( )
( ( ))
ε
ϕ δ ε+
.
(Zametym, çto C > 0. ) Tohda
CR a( ) ( ( ))ε ϕ δ ε+ = ( ) ( )1 − + −[ ]γ ε δ ε∆ +
+
0
a q
t
a
t
a
dt
+
∫ +
−
+
δ ε
ϕ
ϕ δ ε
γ ϕ
ϕ δ ε
( )
( )
( ( ))
( )
( ( ))
+
+
b
q
f t
a
f t
a
dt
+
∫ +
−
+
ε
ϕ δ ε
γ
ϕ δ ε
1
( )
( ( ))
( )
( ( ))
. (14)
Poskol\ku γ < 1, vsledstvye (10) ( ) ( )1 − + −[ ]γ ε δ ε∆ > 0, ε ∈ ( 0, ε0 ) .
Pust\ snaçala a = 0. Zametym, çto
ϕ
ϕ δ ε
( )
( ( ))
t
< 1 dlq t ∈ ( 0, δ ( ε )) . Qsno
takΩe, çto δ ( ε ) → 0 pry ε → 0. Poπtomu
0
δ ε
ϕ
ϕ δ ε
γ ϕ
ϕ δ ε
( )
( )
( ( ))
( )
( ( ))∫ −
t t
dt
q
→ 0, ε → 0.
S druhoj storon¥,
f t
f
( )
( ( ))δ ε
≥ 1 dlq t ∈ ( b + ε, 1 ) , a ϕ ( δ ( ε )) → ϕ ( 0 ) = 0
pry ε → 0. Poπtomu
b
q
f t f t
dt
+
∫ −
ε
ϕ δ ε
γ
ϕ δ ε
1
( )
( ( ))
( )
( ( ))
≥ ( ) ( )
( ( ))
1
1
−
+
∫γ
ϕ δ ε
εb
q
f t
dt → ∞ , ε → 0.
Na osnovanyy yzloΩennoho v¥ße pry uçete (14) moΩno zaklgçyt\, çto
R ( ε ) > 0 v nekotoroj okrestnosty ( , )0 ′ε (umen\ßaq (esly nuΩno) ε0 , moΩno
sçytat\, çto ε ε0 < ′ ). Yz πtoho fakta, sootnoßenyj (13), (9), (10) y strohoj
monotonnosty ϕ sleduet (12) v sluçae a = 0.
Pust\ teper\ a > 0. PokaΩem, çto R ( 0 ) > 0. Qsno, çto δ ( 0 ) = 0. Poπtomu
v sylu (14)
C R ( 0 ) ϕ ( a ) = ( )( ) ( ) ( )1 1− − + −γ γb a z z q ,
hde
z q( ) : =
0
1a q
b
q
t
a
dt
f t
a
dt∫ ∫
+
ϕ
ϕ ϕ
( )
( )
( )
( )
.
Zametym, çto ′′z q( ) > 0, t. e. funkcyq z q( ) v¥puklaq na [ 0, 1 ] . Poπtomu dlq
dokazatel\stva neravenstva R ( 0 ) > 0 dostatoçno ustanovyt\, çto
γ max ( ), ( )z z0 1{ } < z b a( ) ( )( )1 1+ − −γ .
Dlq πtoho, v svog oçered\, dostatoçno dokazat\ neravenstvo
γ z ( 0 ) < z b a( ) ( )( )1 1+ − −γ ,
kotoroe πkvyvalentno sledugwemu:
γ ( )1 − +b a < ( )( )
( )
( )
( )
( )
1
0
1
− − + +∫ ∫γ ϕ
ϕ ϕ
b a
t
a
dt
f t
a
dt
a
b
,
yly
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ V PROSTRANSTVAX Lp 1343
0
1a
b
t dt f t dt b a a∫ ∫+ + − −ϕ γ ϕ( ) ( ) ( ) ( ) > 0.
Poskol\ku
b
f t dt
1
∫ ( ) ≥ f b b( )( )1 − = ϕ ( )( )a b1 − ,
dlq dokazatel\stva neravenstva R ( 0 ) > 0 dostatoçno pokazat\, çto
ξb a( ) : =
0
1
a
t dt a a∫ + − −ϕ γ ϕ( ) ( ) ( ) > 0.
Zametym, çto
′ξb a( ) = ϕ ϕ γ ϕ( ) ( ) ( ) ( )a a a a− + − − ′1 = ( ) ( )1 − − ′a aγ ϕ .
Poπtomu ξb a( ) ≥ min ( ), ( )ξ ξb b b0{ } , a ∈ [ 0, b ] . No ξb( )0 = 0, ξb b( ) =
= ϕ γ ϕ( ) ( ) ( )t dt b b
b
0
1∫ + − − : = η( )b . PokaΩem, çto η( )b > 0, b ∈ ( 0, 1 ) . Qsno,
çto ′η ( )b = ( ) ( )1 − − ′b bγ ϕ . Poπtomu η( )b > min ( ), ( )η η0 1{ } . No η( )0 = 0,
η( )1 = ϕ γϕ( ) ( )t dt
0
1
1∫ − ≥ 0 vsledstvye (5). Sledovatel\no, ξb b( ) = η( )b > 0,
b ∈ ( 0, 1 ) . Tohda y ξb a( ) > 0 dlq a ∈ ( 0, b ] . Ytak, R ( 0 ) > 0. V sylu nepre-
r¥vnosty R ( ε ) otsgda sleduet, çto pry lgbom fyksyrovannom a ∈ ( 0, 1 )
R ( ε ) > 0 v nekotoroj okrestnosty ( 0, ε ( a )) (umen\ßaq (esly nuΩno) ε0 ,
moΩno sçytat\, çto ε0 < ε ( a ) ) . Yz πtoho fakta s uçetom (13), (9), (10) y
strohoj monotonnosty ϕ sleduet v¥polnenye (12) v sluçae b ∈ [ a, 1 ) .
Ostalos\ rassmotret\ sluçaj b = 1. B πtom sluçae f ( t ) = ϕ ( t ) dlq t ∈ [ 0,
a ] y f ( t ) = ϕ ( a ) dlq t ∈ [ a, 1 ] . PoloΩym ∆ : = 1 – a y dlq ε ∈ ( 0, ∆ ) oprede-
lym funkcyg
f tε( ) : =
ϕ ε
ϕ ε ε
( ) [ , ],
( ) [ , ].
t t a
a t a
pry
pry
∈ +
+ ∈ +
0
1
Oçevydno, çto f Kε γ ϕ∈ ( ). Pust\, kak y ran\ße, F ( ε ) : = F f( )ε . PokaΩem, çto
y v rassmatryvaemom sluçae v¥polneno sootnoßenye (12) s nekotor¥m ε0 . Kak
y v sluçae b < 1, dokaz¥vaetsq analoh ravenstva (13):
′F
F
( )
( )
ε
ε
= q a R′ + −[ ]ϕ ε ε ε( ) ( )∆ ,
hde
R ( ε ) =
ϕ ε
ε
γ
ε
q
q
a
I I
− + −
1
1
( )
( ) ( )
, Iq( )ε : =
0
a
q qt dt a
+
∫ + + −[ ]
ε
ϕ ϕ ε ε( ) ( ) ∆ .
Otsgda sleduet analoh ravenstva (14):
C R ( ε ) ϕ ( a + ε ) = ( )
( )
( )
( )
( )
1
0
− −[ ] +
+
−
+
+
∫γ ε ϕ
ϕ ε
γ ϕ
ϕ ε
ε
∆
a q
t
a
t
a
dt , C > 0.
Yz πtoho ravenstva, kak y v sluçae b < 1, sleduet, çto R ( ε ) > 0 v nekotoroj
okrestnosty ( 0, ε ′ ) . Poπtomu, kak y v sluçae b < 1, v¥polneno (12). Sledova-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1344 V. A. KOFANOV
tel\no, funkcyq f, dlq kotoroj a = a ( f ) < 1, ne moΩet b¥t\ πkstremal\noj v
zadaçe (7). Takym obrazom, dlq πkstremal\noj funkcyy f neobxodymo a ( f ) =
= 1. ∏to oznaçaet, v sylu opredelenyq a ( f ) , çto f = ϕ.
Lemma dokazana.
3. Neravenstva dlq proyzvodn¥x suΩenyq funkcyj x ∈∈∈∈ Lr
∞∞ ∞∞, (R) na ot-
rezky yx monotonnosty.
Nam potrebugtsq nekotor¥e vspomohatel\n¥e utverΩdenyq. Dlq λ > 0
poloΩym ϕλ, ( )r t : = λ ϕ λ−r
r t( ) .
Lemma&2. Pust\ r ∈ N , q, λ > 0, α — nul\ splajna ϕλ,r . Tohda funkcyq
f ( y ) : = 1
y
t dt
y
r
q
α
α
λϕ
+
∫ , ( )
stroho vozrastaet na [ / ],0 2π λ .
Dokazatel\stvo. Ymeem
ln f ( y ) = ln ( ) ln,
α
α
λϕ
+
∫ −
y
r
q
t dt y .
Sledovatel\no,
′f y
f y
( )
( )
=
ϕ α
ϕ
λ
λα
α
,
,
( )
( )
r
q
r
qy
y
t dt y
+
−+
∫
1 .
Poskol\ku ϕλ, ( )r t stroho vozrastaet na [ / ],α α π λ+ 2 , to
α
α
λϕ
+
∫
y
r
q
t dt, ( ) < y yr
q
ϕ αλ, ( )+ .
Poπtomu ′f y( ) > 0 dlq y ∈( / ),0 2π λ , çto y zaverßaet dokazatel\stvo lemm¥.
Lemma&3. Pust\ r ∈ N , r ≥ 2, x W a br∈ ∞[ , ], ′ +x a( ) = ′ −x b( ) = 0, ′x t( ) >
> 0 dlq t ∈ ( a, b ) . Tohda pry lgbom q > 0 otnoßenye
( ) ( )
( )
b a x t dt
x t dt
q
a
b
a
b
r
r
q
− ′
′
−
−
∫
∫
1
1
ne zavysyt ot dlyn¥ otrezka [ a, b ] .
Dokazatel\stvo. Dlq x W a br∈ ∞[ , ] rassmotrym funkcyg y ( t ) : = γ −r ×
× x t a( )γ + , hde γ = b – a . Qsno, çto y r
L
( )
[ , ]∞ 0 1
= x r
L a b
( )
[ , ]∞
. Poπtomu y
prynadleΩyt Wr
∞[ , ]0 1 . Pry πtom ′ +y ( )0 = ′ −y ( )1 = 0, ′y t( ) > 0 dlq t ∈ ( 0,
1 ) . Krome toho,
′∫ y t dtq( )
0
1
= γ γ− − ′ +∫ ( ) ( )r q
x t a dt1
0
1
= γ
γ
− − ′∫( ) ( )r q q
a
b
x s ds1 =
= γ − − −
−
′∫( ) ( )r q q
a
b
b a
x s ds1 1 1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ V PROSTRANSTVAX Lp 1345
V çastnosty,
′∫ y t dt( )
0
1
= γ − ′∫r
a
b
x s ds( ) .
Poπtomu
( ) ( )
( )
b a x t dt
x t dt
q
a
b
a
b
r
r
q
− ′
′
−
−
∫
∫
1
1 =
′
′
∫
∫
−
y t dt
y t dt
q
r
r
q
( )
( )
0
1
0
1
1 .
Lemma dokazana.
Dlq f L a b∈ 1[ , ] symvolom r f t( , ), t ∈ [ 0, b – a ] , oboznaçym ub¥vagwug pe-
restanovku funkcyy f (sm., naprymer, [4]).
Çerez ϕω,r−1 oboznaçym suΩenye splajna ϕω,r−1 na [ α , α + π / ω ] , hde α
— nul\ ϕω,r−1.
Lemma&4. Pust\ r = 2 yly r = 3, ω = π . Tohda funkcyq ϕ ( t ) : =
: = r tr( , ),ϕω − −1 1 , opredelennaq na [ 0, 1 ] , udovletvorqet uslovyqm lemm¥D1,
t. e. ϕ ( )0 0= , ′ϕ ( )t > 0 poçty vsgdu na ( 0, 1 ) , y v¥polneno neravenstvo
(5) s γ = −r
r
1
.
Dokazatel\stvo. V¥polnenye uslovyj ϕ ( )0 0= , ′ >ϕ ( )t 0 na ( 0, 1 ) oçe-
vydno. Dlq dokazatel\stva (5) zametym, çto
ϕ ( )1 = ϕω,r− ∞1 = ω ϕ− −
− ∞
( )r
r
1
1 ,
a
ϕ ( )t dt
0
1
∫ = r t dtr( , ),ϕω −∫ 1
0
1
= ϕω
α
α π ω
,
/
( )r t dt−
+
∫ 1 =
= 2 1
1 1
− −
−ω ϕr
r =
2 1
0
2
− −ω ϕ
π
r
r� = 2ω ϕ−
∞
r
r = 2ω−r
rK ,
hde Kr r:= ∞ϕ — konstanta Favara. Teper\ neravenstvo (5) prynymaet vyd
r
r
Kr
−
−
1
1 ≤ 2
π
Kr , y eho spravedlyvost\ lehko sleduet yz yzvestn¥x ravenstv
K1 2= π / , K2
2 8= π / , K3
3 24= π / .
Lemma&5. Pust\ r ∈ N , r ≥ 2, x W r∈ ∞ ∞, ( )R . Pust\, dalee, çyslo λ > 0
udovletvorqet uslovyg
x L∞( )R = ϕλ,r ∞
, (15)
a çysla a, b ∈ R takov¥, çto x ′ ( a ) = x ′ ( b ) = 0, ′x t( ) > 0 dlq t ∈ ( a, b ) .
Tohda dlq lgboho q ≥ 1, a v sluçae, r = 2 yly r = 3 y dlq lgboho q > 0
1
b a
x t dtq
a
b
−
′∫ ( ) ≤ λ
π
ϕλ
π λ
,
/
( )r
q
t dt−∫ 1
0
. (16)
Dokazatel\stvo. Zafyksyruem x W r∈ ∞ ∞, ( )R y promeΩutok [ a, b ] , na ko-
torom x udovletvorqet uslovyqm lemm¥. Rassmotrym try sluçaq: 1) b – a ≤
≤ π / λ , q > 0; 2) b – a > π / λ , q ≥ 1; 3) b – a > π / λ , q ∈ ( 0, 1 ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1346 V. A. KOFANOV
Pust\ snaçala b – a ≤ π / λ , q > 0. Zametym, çto yz uslovyq (15) y teorem¥
sravnenyq Kolmohorova [5] sleduet neravenstvo
′
∞
x L ( )R ≤ ϕλ,r− ∞1 . (17)
Pust\ α — nul\ splajna ϕλ,r−1. Perexodq, esly nuΩno, k sdvyhu funkcyy x ,
moΩem sçytat\, çto [ a, b ] ⊂ [ α , α + π / λ ] . Tohda yz neravenstva (17) y uslo-
vyj ′x a( ) = ′x b( ) = 0 v sylu teorem¥ sravnenyq Kolmohorova sleduet, çto
′x t( ) ≤ ′ −ϕλ, ( )r t1 , t ∈ [ a, b ] . Poπtomu dlq lgboho q > 0
′∫ x t dtq
a
b
( ) ≤ ϕλ
α
α π λ
,
/
( )r
q
t dt−
+
∫ 1 . (18)
V¥berem c ∈ ( a, b ) tak, çto
′∫ x t dtq
a
c
( ) = ′∫ x t dtq
c
b
( ) . (19)
Vsledstvye (18) y (19) suwestvuet y ∈( / ],0 2π λ takoe, çto
′∫ x t dtq
a
c
( ) = ϕλ
α
α
, ( )r
q
y
t dt−
+
∫ 1 , ′∫ x t dtq
c
b
( ) = ϕλ
α π λ
α π λ
,
/
/
( )r
q
y
t dt−
+ −
+
∫ 1 . (20)
Otsgda s pomow\g teorem¥ sravnenyq Kolmohorova v¥vodym neravenstva
c – a ≥ y y b – c ≥ y . Qsno, çto yntehral¥ v prav¥x çastqx (20) ravn¥. Po-
πtomu
′∫ x t dtq
a
c
( ) ≤ c a
y
t dtr
q
y
−
−
+
∫ ϕλ
α
α
, ( )1 , ′∫ x t dtq
c
b
( ) ≤ b c
y
t dtr
q
y
−
−
+
∫ ϕλ
α
α
, ( )1 .
Prymenqq takΩe lemmuD2, poluçaem
′∫ x t dtq
a
b
( ) = ′∫ x t dtq
a
c
( ) + ′∫ x t dtq
c
b
( ) ≤ c a
y
t dtr
q
y
−
−
+
∫ ϕλ
α
α
, ( )1 +
+ b c
y
t dtr
q
y
−
−
+
∫ ϕλ
α
α
, ( )1 ≤ ( ) ( ),
/
b a t dtr
q
− −
+
∫2
1
2
λ
π
ϕλ
α
α π λ
,
çto ravnosyl\no (16).
Pust\ teper\ b – a > π / λ , q ≥ 1. Poskol\ku x W r∈ ∞ ∞, ( )R , prymenqq nera-
venstvo (3) y uçyt¥vaq uslovye (15), poluçaem
a
b
qx t dt∫ ′( ) ≤
0
1
1 1
π
ϕ
ϕ∫ −
∞
− +
∞
r
q L
r
r q
r
t dt
x
( ) ( )
( )
R =
=
0
1
1 1π
ϕ λ∫ −
−
− +
r
q r
r q
rt dt( ) ( )
( )
=
0
1
π λ
λϕ
/
, ( )∫ −r
q
t dt (21)
(spravedlyvost\ posledneho ravenstva lehko proverqetsq zamenoj peremenn¥x v
poslednem yntehrale). Yz (21) neposredstvenno sleduet (16) v sylu uslovyq b –
– a > π / λ .
Rassmotrym, nakonec, sluçaj b – a > π / λ , q ∈ ( 0, 1 ) . Zametym, çto utverΩ-
denye lemm¥ ravnosyl\no neravenstvu
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NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ V PROSTRANSTVAX Lp 1347
( ) ( )
( )
b a x t dt
x
q
a
b
L
r
r
q
− ′−
−
∫
∞
1
1
R
≤
( / ) ( ),
/
,
ω π ϕ
ϕ
ω
π ω
ω
r
q
r
r
r
q
t dt−
∞
−
∫ 10
1 ,
pryçem pravaq çast\ ne zavysyt ot ω . DokaΩem v rassmatryvaemom sluçae bo-
lee syl\noe neravenstvo
( ) ( )
( )
b a x t dt
x t dt
q
a
b
a
b
r
r
q
− ′
′
−
−
∫
∫
1
1 ≤
( / ) ( )
( )
,
/
,
/
ω π ϕ
ϕ
ω
π ω
ωβ
β π ω
r
q
r
r
r
q
t dt
t dt
−
−
+
−
∫
∫
10
1
1 , (22)
hde β — nul\ splajna ϕω,r−1. Levaq çast\ πtoho neravenstva ne zavysyt ot
dlyn¥ otrezka [ a, b ] v sylu lemm¥D3. Poπtomu moΩno sçytat\, çto [ a, b ] =
= [ 0, 1 ] (esly πto ne tak, dostatoçno perejty k funkcyy γ γ− +r x t a( ), γ = b –
– a ). Krome toho, esly x — suΩenye x na [ a, b ] , to r x L b aq
( , ) [ , ]′ ⋅ −0 =
= ′x L a bq[ , ], q > 0, y (22) moΩno predstavyt\ v vyde
r x t dt
r x t dt
q
r
r
q
( , )
( , )
′
′
∫
∫
−
0
1
0
1
1 ≤
( / ) ( , )
( , )
,
/
,
/
ω π ϕ
ϕ
ω
π ω
ω
π ω
r t dt
r t dt
q
r
r
r
r
q
−
−
−
∫
∫
10
10
1 ,
hde ϕω,r−1 — suΩenye splajna ϕω,r−1 na [ / ],β β π ω+ .
V sylu neravenstva (17) y teorem¥ sravnenyq Kolmohorova splajn ϕλ,r−1
qvlqetsq funkcyej sravnenyq dlq ′x . Tohda sohlasno teoreme o proyzvodnoj
perestanovky [4] (predloΩenyeD1.3.2) funkcyq r tr( , ),ϕλ −1 , opredelennaq na
[ / ],0 π λ , qvlqetsq funkcyej sravnenyq dlq perestanovky r x t( , )′ , opredelen-
noj na [ 0, 1 ] . Vsledstvye predpoloΩenyq b – a = 1 > π λ/ . PoloΩym ω = π .
Tohda ω < λ , y, sledovatel\no, funkcyq r tr( , ),ϕω −1 tem bolee budet funkcy-
ej sravnenyq dlq r x t( , )′ . Perexodq k vozrastagwym perestanovkam ϕ ( t ) : =
: = r tr( , ),ϕω − −1 1 y f ( t ) : = r x t( , )′ −1 , poluçaem funkcyy ϕ , f L∈ 1 0 1[ , ], udov-
letvorqgwye uslovyqm lemm¥D1 s γ = r
r
− 1
(esly prynqt\ vo vnymanye lem-
muD4). V sylu lemm¥D1 ymeet mesto neravenstvo (6), kotoroe v dannom sluçae
ravnosyl\no dokaz¥vaemomu neravenstvu (22).
Lemma dokazana.
Teorema&1. Pust\ r ∈ N , r ≥ 2, x Lr∈ ∞ ∞, ( )R , a çysla a, b ∈ R udovlet-
vorqgt uslovyqm x ′ ( a ) = x ′ ( b ) = 0, ′x t( ) > 0 dlq t ∈ ( a, b ) . Tohda dlq
lgboho q ≥ 1, a v sluçae r = 2 yly r = 3 y dlq lgboho q > 0 ymeet mes-
to neravenstvo
1
b a
x t dt
a
b
q
−
′∫ ( ) ≤ 1
1
0
1
π
ϕ
ϕ
π
r
q L
r
r
r
q
r
L
q
rt dt
x
x−
∞
−
∫ ∞
∞
( ) ( ) ( )
( )
R
R
. (23)
Dokazatel\stvo. Zafyksyruem x Lr∈ ∞ ∞, ( )R y çysla a, b ∈ R , udovlet-
vorqgwye uslovyqm teorem¥. Poskol\ku neravenstvo (23) odnorodno, moΩno
sçytat\, çto
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1348 V. A. KOFANOV
x r
L
( )
( )∞ R
= 1. (24)
Tohda x Wr∈ ∞ ∞, ( )R . V¥berem çyslo λ > 0 tak, çto
x L∞ ( )R = ϕλ,r ∞
. (25)
Tohda v sylu lemm¥D5
1
b a
x t dt
a
b
q
−
′∫ ( ) ≤
λ
π
ϕ
π λ
λ
0
1
/
, ( )∫ −r
q
t dt . (26)
Teper\ yz (25) y (26) v¥vodym
( ) ( )
( )
b a x t dt
x
q
a
b
L
r
r
q
− ′−
−
∫
∞
1
1
R
≤
( / ) ( ),
/
,
λ π ϕ
ϕ
λ
π λ
λ
r
q
r
r
r
q
t dt−
∞
−
∫ 10
1 =
( / ) ( )1 10
1
π ϕ
ϕ
π
r
q
r
r
r
q
t dt−
∞
−
∫
.
Yz posledneho neravenstva s uçetom (24) sleduet (23).
Teorema dokazana.
4. Neravenstva dlq peryodyçeskyx funkcyj.
Teorema&2. Pust\ r = 2, k = 1 yly r = 3, k = 1, 2. Tohda dlq lgboj
funkcyy x Lr∈ ∞ y lgboho q ∈ [ 0, 1 )
x k
q
( ) ≤
ϕ
ϕ
r k q
r
k r
k r r k r
x x
−
∞
− ∞
−
∞1
1
/
/ ( ) /
. (27)
Neravenstvo (27) toçnoe y obrawaetsq v ravenstvo dlq funkcyj vyda
x ( t ) = a nt brϕ ( )+ , a, b ∈ R , n ∈ N .
Dokazatel\stvo. Poskol\ku dlq lgboj funkcyy f ∈ L1 ymeet mesto ra-
venstvo f 0 = lim ( ) /
q
q
qf→
−
0
12π (sm., naprymer, [6, s.D188]), dostatoçno do-
kazat\ (27) dlq q > 0.
Zafuksyruem x Lr∈ ∞ y pust\ c — proyzvol\n¥j nul\ proyzvodnoj x k( )
.
Rassmotrym sovokupnost\ vsex otrezkov [ , ] [ , ]a b c cj j ⊂ + 2π takyx, çto
x ak
j
( )( ) = x bk
j
( )( ) = 0, x tk( )( ) > 0, t a bj j∈( , ).
Qsno, çto
( )b aj j
j
−∑ ≤ 2 π , x k
q
q( ) =
j a
b
k q
j
j
x t dt∑ ∫ ( )( ) . (28)
Ocenym yntehral¥ x t dtk q
a
b
j
j ( )( )∫ v (28) s pomow\g neravenstva (23), prymenen-
noho k funkcyy x Lk r k( )−
∞
− +∈1 1. Pry πtom dlq kratkosty poloΩym
S : =
1
0
1
1
1
1
π
ϕ
ϕ
π
∫ −
−
∞
− + ∞
−
− +
∞
− +
r k
q
k
r k
r k
r k
q
r
q
r kt dt
x
x( )
( )
( )
y zametym, çto
2
0
π
ϕ∫ −r k
qt dt( ) = ϕr k q
q
− .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
NERAVENSTVA DLQ PROYZVODNÁX FUNKCYJ V PROSTRANSTVAX Lp 1349
Tohda yz (28) v¥vodym ocenku
x k
q
q( ) ≤ ( )b a Sj j
j
−∑ ≤ 2 π S = ϕ
ϕr k q
q
k
r k
r k
r k
q
r
q
r k
x
x−
−
∞
− + ∞
−
− +
∞
− +
( )
( )
1
1
1
1, (29)
kotoraq v sluçae k = 1 πkvyvalentna dokaz¥vaemomu neravenstvu (27). Esly
Ωe k = 2, to, ocenyvaq normu x k( )−
∞
1
v (29) s pomow\g neravenstva Kolmo-
horova [5], poluçaem
x k
q
q( ) ≤ ϕ
ϕr k q
q
r
r k
r r
k
r
r k
r k
q
r
q
r k
x
x x−
∞
∞
− +
∞
−
−
− +
∞
− +
1
1 1
1( ) ( ) =
= ϕ
ϕr k q
q
r
r k
r
q
r
k
r
qx
x−
∞
∞
−
∞
( ) .
Neravenstvo (27) dokazano. Eho toçnost\ oçevydna.
Teorema dokazana.
Zameçanye. Pryvedennoe dokazatel\stvo teorem¥DD2 soxranqet sylu dlq
funkcyj x Lr∈ ∞ proyzvol\noj hladkosty v sluçae q ≥ 1 y qvlqetsq nov¥m
dokazatel\stvom neravenstva (1) A. A. Lyhuna.
Çerez Sn r, , n r, ∈N , oboznaçym mnoΩestvo 2π -peryodyçeskyx splajnov s
porqdka r defektaDD1 s uzlamy v toçkax i nπ / , i ∈Z.
Teorema&3. Pust\ n ∈N ; r = 2, k = 1 yly r = 3, k = 1, 2. Dlq lgboho
splajna s Sn r∈ , y lgboho q ∈[ , )0 1 ymeet mesto neravenstvo
s k
q
( ) ≤ n sk r k q
r
ϕ
ϕ
−
∞
∞. (30)
Neravenstvo (30) toçnoe y obrawaetsq v ravenstvo dlq funkcyj vyda
s t( ) = a tn rϕ , ( ) , a ∈R , n ∈N .
Neravenstvo (30) v sluçae q ≥ 1 dokazano ranee dlq vsex r k, ∈N , k ≤ r
V.DM. Tyxomyrov¥m [7] ( )q = ∞ y A. A. Lyhunom [8] ( [ , ))q ∈ ∞1 .
TeoremaDD3 v¥vodytsq yz teorem¥DD2 toçno tak Ωe, kak y v sluçae q ≥ 1
(sm., naprymer, [9], teoremaDD8.2.1).
1. Ligun A. A. Inequalities for upper bounds of functionals // Anal. Math. – 1976. – 2, # 1. –
P. 11 – 40.
2. Kofanov V. A. Sharp inequalities of Bernstein and Kolmohorov type // East J. Approxim. – 2005. –
11, # 2. – P. 131 – 145.
3. Kofanov V. A. O toçn¥x neravenstvax typa Bernßtejna dlq splajnov // Ukr. mat. Ωurn. –
2006. – 58, # 10. – S.D1357 – 1367.
4. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov.
– Kyev: Nauk. dumka, 1992. – 304 s.
5. Kolmohorov A. N. O neravenstvax meΩdu verxnymy hranqmy posledovatel\n¥x proyzvod-
n¥x funkcyy na beskoneçnom yntervale // Yzbr. trud¥. Matematyka, mexanyka. – M.: Nauka,
1985. – S.D252 – 263.
6. Xardy H. H., Lyttlvud D. E., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. – 456 s.
7. Tyxomyrov V. M. Popereçnyky mnoΩestv v funkcyonal\n¥x prostranstvax y teoryq nay-
luçßyx pryblyΩenyj // Uspexy mat. nauk. – 1960. – 15, # 3. – S.D81 – 120.
8. Lyhun A. A. Toçn¥e neravenstva dlq splajn-funkcyj y nayluçßye kvadraturn¥e formu-
l¥ dlq nekotor¥x klassov funkcyj // Mat. zametky. – 1976. – 19, # 6. – S.D913 – 926.
9. Kornejçuk N. P., Babenko V. F., Kofanov V. A., Pyçuhov S. A. Neravenstva dlq proyzvodn¥x
y yx pryloΩenyq. – Kyev: Nauk. dumka, 2003. – 590 s.
Poluçeno 05.02.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
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| id | umjimathkievua-article-3248 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:56Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/84/58720ac7fc57dc8307e61ac8e91f0384.pdf |
| spelling | umjimathkievua-article-32482020-03-18T19:49:15Z Inequalities for derivatives of functions in the spaces Lp Неравенства для производных функций в пространствах Lp Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved: $$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$ where $\varphi_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q \geq 1$ or for any $q > 0$ in the cases of $r = 2$ and $r = 3.$ As a corollary, well-known A. A. Ligun's inequality for functions $x \in L^{r}_{\infty}$ of the form $$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r}} ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q < \infty,$$ is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$ Отримано нову точну нерівність для локальних норм функцій $x \in L^{r}_{\infty,\infty}(\textbf{R}):$ $$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$ де $\varphi_r$ — ідеальний сплайн Ейлера, на проміжках $[a, b]$ монотонності $x$ для випадку $q \geq 1$, а також для довільних $q > 0$ у випадках $r = 2$ та $r = 3.$ Як наслідок, відому нерівність А. А. Лигуна для періодичних функцій $x \in L^{r}_{\infty}$ $$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r}} ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q < \infty,$$ Institute of Mathematics, NAS of Ukraine 2008-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3248 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 10 (2008); 1338 – 1349 Український математичний журнал; Том 60 № 10 (2008); 1338 – 1349 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3248/3242 https://umj.imath.kiev.ua/index.php/umj/article/view/3248/3243 Copyright (c) 2008 Kofanov V. A. |
| spellingShingle | Kofanov, V. A. Кофанов, В. А. Кофанов, В. А. Inequalities for derivatives of functions in the spaces Lp |
| title | Inequalities for derivatives of functions in the spaces Lp |
| title_alt | Неравенства для производных функций в пространствах Lp |
| title_full | Inequalities for derivatives of functions in the spaces Lp |
| title_fullStr | Inequalities for derivatives of functions in the spaces Lp |
| title_full_unstemmed | Inequalities for derivatives of functions in the spaces Lp |
| title_short | Inequalities for derivatives of functions in the spaces Lp |
| title_sort | inequalities for derivatives of functions in the spaces lp |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3248 |
| work_keys_str_mv | AT kofanovva inequalitiesforderivativesoffunctionsinthespaceslp AT kofanovva inequalitiesforderivativesoffunctionsinthespaceslp AT kofanovva inequalitiesforderivativesoffunctionsinthespaceslp AT kofanovva neravenstvadlâproizvodnyhfunkcijvprostranstvahlp AT kofanovva neravenstvadlâproizvodnyhfunkcijvprostranstvahlp AT kofanovva neravenstvadlâproizvodnyhfunkcijvprostranstvahlp |