Integral analog of one generalization of the Hardy inequality and its applications
Under certain conditions on continuous functions $μ, λ, a$, and $f$, we prove the inequality $$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$...
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| Date: | 2006 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509634994896896 |
|---|---|
| author | Mulyava, O. M. Мулява, О. М. |
| author_facet | Mulyava, O. M. Мулява, О. М. |
| author_sort | Mulyava, O. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:56:51Z |
| description | Under certain conditions on continuous functions $μ, λ, a$, and $f$, we prove the inequality
$$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$
and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence. |
| first_indexed | 2026-03-24T02:44:14Z |
| format | Article |
| fulltext |
UDK 517.53
O. M. Mulqva (Ky]v. nac. un-t xarç. texnolohij)
INTEHRAL|NYJ ANALOH ODNOHO UZAHAL|NENNQ
NERIVNOSTI HARDI TA JOHO ZASTOSUVANNQ
Under some conditions on continuous functions µ, λ, a, f the inequality
µ λ
λ
λ
µ λ( ) ( )
( ) ( )
( )
≤ ( ) ( ) ( )
∫
∫
∫ ∫
( )x x f
t a t dt
t dt
dx K x x f a x dx
x
x
y y
0
00 0
, y ≤ ∞,
is proved and its application to the study of the problem of belonging of Laplace integrals to the
convergence class is shown.
Za deqkyx umov na neperervni funkci] µ, λ, a i f dovedeno nerivnist\
µ λ
λ
λ
µ λ( ) ( )
( ) ( )
( )
≤ ( ) ( ) ( )
∫
∫
∫ ∫
( )x x f
t a t dt
t dt
dx K x x f a x dx
x
x
y y
0
00 0
, y ≤ ∞,
i vkazano na ]] zastosuvannq do vyvçennq naleΩnosti intehraliv Laplasa do klasu zbiΩnosti.
Nahada[mo klasyçnu nerivnist\ Hardi [1, s. 289]: dlq koΩnoho p > 1 i bud\-qko]
poslidovnosti ( an ) nevid’[mnyx çysel
1
111 1n
a
p
p
ak
k
n p
n
p
n
p
n== =
∑∑ ∑
≤
−
ω ω
, ω ≤ ∞.
Dlq doslidΩennq klasiv zbiΩnosti rqdiv Dirixle v [2] cg nerivnist\ uzahal\-
neno. Dovedeno, wo qkwo – ∞ ≤ A < an < B ≤ + ∞, poslidovnist\ ( λn ) [ dodat-
nog, poslidovnist\ ( µn ) — dodatnog i nezrostagçog, a funkciq f — dodatnog
na ( A, B ) i takog, wo f 1
/
p
, p > 1, — opukla na ( A, B ) funkciq, to
µ λ
λ
λ
µ λ
ω ω
n n
k kk
n
kk
n
n
p
n n n
n
f
a p
p
f a=
== =
∑
∑∑ ∑
≤
−
( )1
11 1
1
, ω ≤ ∞. (1)
U danij statti dovedeno nastupnyj intehral\nyj analoh nerivnosti (1) i vka-
zano na joho moΩlyvi zastosuvannq.
Teorema. Nexaj a ( x ), µ ( x ) i λ ( x ) — neperervni na ( 0, + ∞ ) funkci],
pryçomu – ∞ ≤ A < a ( x ) < B ≤ + ∞, λ ( x ) > 0 i µ ( x ) Ü µ ≥ 0 pry x → + ∞, krim
c\oho, dodatna na ( A, B ) funkciq f taka, wo funkciq f 1
/
p
, p > 1, [ opuklog
na ( A, B ). Todi
µ λ
λ
λ
µ λ( ) ( )
( ) ( )
( )
≤
−
( ) ( ) ( )∫
∫
∫ ∫ ( )x x f
t a t dt
t dt
dx
p
p
x x f a x dx
x
x
y p y
0
00 0
1
, y ≤ ∞. (2)
Dovedennq. Poznaçymo Λ ( x ) = λ( )∫ t dt
x
0
i A ( x ) = λ( ) ( )∫ t a t dt
x
0
. Todi dlq do-
vil\noho η > 0
A x
x
A x
x
x
x
A x A x
x x
x x
x
( + )
( + )
= ( )
( )
( )
( + )
+ ( + ) − ( )
( + ) − ( )
( + ) − ( )
( + )
η
η η
η
η
η
ηΛ Λ
Λ
Λ Λ Λ
Λ Λ
Λ
i
© O. M. MULQVA, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 9 1271
1272 O. M. MULQVA
Λ
Λ
Λ Λ
Λ
( )
( + )
+ ( + ) − ( )
( + )
x
x
x x
xη
η
η
= 1,
a oskil\ky funkciq f 1
/
p
[ opuklog, to
f
A x
x
f
A x
x
x
x
p p1 1/ /( + )
( + )
≤ ( )
( )
( )
( + )
η
η ηΛ Λ
Λ
Λ
+
+ f
A x A x
x x
x x
x
p1/ ( + ) − ( )
( + ) − ( )
( + ) − ( )
( + )
η
η
η
ηΛ Λ
Λ Λ
Λ
,
tobto
− ( + ) − ( )
( + ) − ( )
≤ ( )
( )
( )
( + ) − ( )
f
A x A x
x x
f
A x
x
x
x x
p p1 1/ /η
η ηΛ Λ Λ
Λ
Λ Λ
–
– f
A x
x
x
x x
p1/ ( + )
( + )
( + )
( + ) − ( )
η
η
η
ηΛ
Λ
Λ Λ
.
Tomu z 1 / q + 1 / p = 1 ma[mo
Q ( x ) : = f
A x
x
x x
( + )
( + )
( + ) − ( )( )η
η
η
Λ
Λ Λ –
– q f
A x A x
x x
f
A x
x
x xp q1 1/ /( + ) − ( )
( + ) − ( )
( + )
( + )
( + ) − ( )( )η
η
η
η
η
Λ Λ Λ
Λ Λ ≤
≤ f
A x
x
x x q f
A x
x
f
A x
x
xp q( + )
( + )
( + ) − ( ) +
( )
( )
( + )
( + )
( )( )η
η
η η
ηΛ
Λ Λ
Λ Λ
Λ1 1/ / –
– f
A x
x
f
A x
x
xp q1 1/ /( + )
( + )
( + )
( + )
( + )
η
η
η
η
η
Λ Λ
Λ =
= f
A x
x
x x q x
( + )
( + )
( + ) − ( ) − ( + )( )η
η
η η
Λ
Λ Λ Λ +
+ q f
A x
x
f
A x
x
xp q1 1/ /( )
( )
( )
( )
( )
Λ Λ
Λ .
Oskil\ky
1 1
p
a
q
bp q+ ≥ a b dlq vsix a ≥ 0 ta b ≥ 0 [2], to
Q ( x ) ≤ f
A x
x
q x x
( + )
( + )
( − ) ( + ) − ( )( )η
η
η
Λ
Λ Λ1 +
+ q
p
f
A x
x q
f
A x
x
x
1 1( )
( )
+ ( + )
( + )
( )
Λ Λ
Λη
η
=
= f
A x
x
q x
q
p
f
A x
x
x
( + )
( + )
( − ) ( + ) + ( )
( )
( )η
η
η
Λ
Λ
Λ
Λ1 =
=
1
1p
f
A x
x
x f
A x
x
x
−
( )
( )
( ) − ( + )
( + )
( + )
Λ
Λ
Λ
Λη
η
η .
Tomu dlq y > η
( − ) ( ) ( ) ≤ ( ) ( )
( )
( ) − ( + )
( + )
( + )
∫ ∫p x Q x dx x f
A x
x
x f
A x
x
x dx
y y
1
0 0
µ µ η
η
η
Λ
Λ
Λ
Λ ≤
≤ µ µ η η
η
η( ) ( )
( )
( ) − ( + ) ( + )
( + )
( + )∫ ∫x f
A x
x
x dx x f
A x
x
x dx
y y
Λ
Λ
Λ
Λ
0 0
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 9
INTEHRAL|NYJ ANALOH ODNOHO UZAHAL|NENNQ NERIVNOSTI HARDI … 1273
= µ µ
η
η
( ) ( )
( )
( ) − ( ) ( )
( )
( )∫ ∫
+
x f
A x
x
x dx x f
A x
x
x dx
y y
Λ
Λ
Λ
Λ
0
=
= µ µ
η η
( ) ( )
( )
( ) − ( ) ( )
( )
( )∫ ∫
+
x f
A x
x
x dx x f
A x
x
x dx
y
y
Λ
Λ
Λ
Λ
0
.
Za pravylom Lopitalq
lim lim
η
η
η η
µ
µ
µ η η
η
η
µ η η
η
→ + →
( ) ( )
( )
( )
( ) ( )
( )
( )
=
( ) ( )
( )
( )
( + ) ( + )
( + )
( +
∫
∫0
0
0
x f
A x
x
x dx
x f
A x
x
x dx
f
A
y f
A y
y
y
y
y
Λ
Λ
Λ
Λ
Λ
Λ
Λ
Λ ηη)
=
=
µ
µ
η
η
( ) ( )
( ) ( ) ( ) ( )
( )( )
( ) →
0 0
0
f a
y f A y y y/
lim
Λ Λ
Λ = 0.
Tomu η = η ( y ) moΩna vybraty tak, wob µ( ) ( )∫ x Q x dx
y
0
< 0, i, otΩe, na pidstavi
oznaçennq Q i nerivnosti Hel\dera ma[mo
µ η
η
η( ) ( + )
( + )
( + ) − ( )( )∫ x f
A x
x
x x dx
y
Λ
Λ Λ
0
≤
≤ q x f
A x A x
x x
f
A x
x
x x dxp q
y
µ η
η
η
η
η( ) ( + ) − ( )
( + ) − ( )
( + )
( + )
( + ) − ( )( )∫ 1 1
0
/ /
Λ Λ Λ
Λ Λ ≤
≤ q x f
A x A x
x x
x x dx
y p
µ η
η
η( ) ( + ) − ( )
( + ) − ( )
( + ) − ( )
( )∫ Λ Λ
Λ Λ
0
1/
×
× µ η
η
η( ) ( + )
( + )
( + ) − ( )
( )∫ x f
A x
x
x x dx
y q
Λ
Λ Λ
0
1/
,
zvidky
µ η
η
η( ) ( + )
( + )
( + ) − ( )
( )∫ x f
A x
x
x x dx
y p
Λ
Λ Λ
0
1/
≤
≤ q x f
A x A x
x x
x x dx
y p
µ η
η
η( ) ( + ) − ( )
( + ) − ( )
( + ) − ( )
( )∫ Λ Λ
Λ Λ
0
1/
,
tobto
µ
λ
λ
λ
η
η
η
( )
( ) ( )
( )
( )
+
+
+∫
∫
∫ ∫x f
t a t dt
t dt
t dt dx
x
x
y
x
x
0
0
0
≤
≤ q x f
t a t dt
t dt
t dt dxp x
x
x
x
y
x
x
µ
λ
λ
λ
η
η
η
( )
( ) ( )
( )
( )
+
+
+∫
∫
∫ ∫
0
.
Vykorystovugçy teoremu pro seredn[ i sprqmovugçy η do 0, zvidsy otrymu[mo
nerivnist\ (2).
Teoremu dovedeno.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 9
1274 O. M. MULQVA
Vybyragçy µ ( x ) = λ ( x ) = 1, z teoremy oderΩu[mo takyj naslidok.
Naslidok 1. Nexaj a ( x ) — neperervna na ( 0, + ∞ ) funkciq, – ∞ ≤ A <
< a ( x ) < B ≤ + ∞, a dodatna na ( A, B ) funkciq f taka, wo funkciq f 1
/
p
, p >
> 1, [ opuklog na ( A, B ). Todi
f
x
a t dt dx
p
p
f a x dx
xy p y
1
1
00 0
( )
≤
−
( )∫∫ ∫ ( ) , y ≤ ∞.
Umovy naslidku 1 z p = 2 zadovol\nq[ funkciq f ( x ) = e–
ρ
x
, ρ ∈ ( 0, + ∞ ).
Tomu ma[ misce nastupnyj naslidok.
Naslidok 2. Nexaj κ ( x ) — dodatna, neperervna i zrostagça na ( 0, + ∞ )
funkciq. Todi
exp exp{ }− ( ) ≤ − ( )
∫ ∫∫ρκ ρ κx dx
x
t dt dx
y xy
0 00
≤
≤ 4
0
exp{ }− ( )∫ ρκ x dx
y
, y ≤ ∞.
Prypustymo, wo funkciq ϕ [ dodatnog, neperervno dyferencijovnog na
( 0, + ∞ ) i takog, wo ϕ ( 0 ) = 1 i
1 1
x x
ln
ϕ( )
→ + ∞ pry x → + ∞. Todi
ϕ ( x ) eσ
x = exp ln−
( )
−
≤ −x
x x
e x1 1
ϕ
σ
dlq bud\-qkoho σ ∈ R i vsix x ≥ x0 ( σ ) i, otΩe, intehral Laplasa
F ( σ ) = ϕ σ( )
∞
∫ x e dxx
0
[ zbiΩnym dlq vsix σ ∈ R . Vykorystovugçy teoremu abo ]] naslidky, moΩna
vkazaty umovy na ϕ, za qkyx F naleΩyt\ do c\oho çy inßoho uzahal\nenoho
klasu zbiΩnosti [2]. Tut my zupynymos\ lyße na klasyçnomu klasi zbiΩnosti,
qkyj vyznaça[t\sq umovog
e F d−
∞
( )∫ ρσ σ σln
0
< + ∞, ρ = const > 0. (3)
Poklademo κ ( x ) = – ϕ′ ( x ) / ϕ ( x ) i prypustymo, wo κ ( x ) ↑ + ∞, x → + ∞. Nexaj
µ ( σ ) : = max :{ }( ) ≥ϕ σx e xx 0 . Todi ln µ ( σ ) : = max ln / :{ ( ) }− ( ) + ≥1 0ϕ σx x x , i
oskil\ky
d x
dx
ln /( )( )1 ϕ
= κ ( x ) ↑ + ∞, x → + ∞, to ln µ ( σ ) = ln ϕ ( ν ( σ ) ) + σ ν ( σ ),
de ν ( σ ) — [dyna toçka maksymumu funkci] − ( ) +( )ln /1 ϕ σx x , pryçomu
κ ( ν ( σ ) ) ≡ σ. Lehko pobaçyty, wo ν ( σ ) — nevid’[mna, neperervna i zrostagça
do + ∞ na [ 0, + ∞ ) funkciq.
Dlq δ > 0 ma[mo
F ( σ ) = ϕ µ σ δ µ σ δ
δ
σ δ δ δ( ) ≤ ( + ) = ( + )( + ) −
∞
−
∞
∫ ∫x e e dx e dxx x x
0 0
,
a qkwo ϕ ( x ) Ü 0, x → + ∞, to dlq σ ≥ 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 9
INTEHRAL|NYJ ANALOH ODNOHO UZAHAL|NENNQ NERIVNOSTI HARDI … 1275
F ( σ ) ≥ ϕ ϕ ν σ µ σσ
ν σ
ν σ
σ ν σ σ( ) ≥ ( ) = ( )
( )−
( )
( ( )− ) −∫ ( )x e dx e ex
1
1
,
tobto ln µ ( σ ) ≤ ln F ( σ ) + σ. Zvidsy vyplyva[, wo umova (3) rivnosyl\na umovi
e d−
∞
∫ ( )ρσ µ σ σ
0
ln < + ∞. (4)
Dali, dlq bud\-qkoho h > 0 ma[mo
ln µ ( σ + h ) = ln ϕ ( ν ( σ + h ) ) + ν ( σ + h ) ( σ + h ) ≤ ln µ ( σ ) + h ν ( σ + h )
i
ln µ ( σ ) = ln ϕ ( ν ( σ ) ) + ν ( σ ) ( σ + h ) – h ν ( σ ) ≤ ln µ ( σ + h ) – h ν ( σ ),
tobto ν ( σ ) ≤ ( ln µ ( σ + h ) – ln µ ( σ ) ) / h ≤ ν ( σ + h ). Taka Ω nerivnist\ vykonu[t\-
sq i u vypadku, koly h < 0. Sprqmovugçy h do 0, zvidsy otrymu[mo rivnist\
d
d
lnµ σ
σ
( )
= ν ( σ ). Vykorystovugçy ce spivvidnoßennq, nevaΩko pokazaty, wo
umova (4) rivnosyl\na umovi
e d−
∞
∫ ( )ρσν σ σ
0
< + ∞. (5)
Ale
e d d e td e
y y
t
y
− − − ( )
( )
( )
∫ ∫ ∫( ) = − ( ) ( ) = − ( )ρσ ρσ ρκ
ν
ν
ν σ σ
ρ
ν σ
ρ
0 0 0
1 1
.
Tomu, intehrugçy çastynamy, nevaΩko pokazaty, wo umova (5) rivnosyl\na umovi
e dxx− ( )
∞
∫ ρκ
0
< + ∞. (6)
Oskil\ky ϕ ( 0 ) = 1, to ln
1
0
ϕ
κ
( )
= ( )∫x
t dt
y
, i za naslidkom 2 umova (6) rivnosyl\-
na umovi
exp ln−
( )
∞
∫ ρ
ϕx x
dx
1
0
< + ∞.
OtΩe, dovedeno nastupnyj naslidok.
Naslidok 3. Nexaj funkciq ϕ [ dodatnog, neperervno dyferencijovnog
na ( 0, + ∞ ) i takog, wo ϕ ( 0 ) = 1, ϕ ( x ) Ü 0,
1 1
x x
ln
ϕ( )
→ + ∞ i – ϕ′ ( x ) / ϕ ( x ) ↑
↑ + ∞ pry x → + ∞. Todi umova (3) rivnosyl\na umovi ϕ ρ( )
∞
∫ x dxx/
0
< + ∞.
1. Xardy H. H., Lytlvud D. D., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. – 456 s.
2. Mulqva O. M. Pro klasy zbiΩnosti rqdiv Dirixle // Ukr. mat. Ωurn. – 1999. – 51, # 11. –
S.M1485 – 1494.
OderΩano 26.05.2005,
pislq doopracgvannq — 14.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 9
|
| id | umjimathkievua-article-3528 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:44:14Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/91/eb5020b0091ec0ecbe1814464a3cc391.pdf |
| spelling | umjimathkievua-article-35282020-03-18T19:56:51Z Integral analog of one generalization of the Hardy inequality and its applications Інтегральний аналог одного узагальнення нерівності Гарді та його застосування Mulyava, O. M. Мулява, О. М. Under certain conditions on continuous functions $μ, λ, a$, and $f$, we prove the inequality $$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$ and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence. За деяких умов на неперервні функції $μ, λ, a, f$ доведено нерівність $$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$ і вказано на її застосування до вивчення належності інтеїралів Лапласа до класу збіжності. Institute of Mathematics, NAS of Ukraine 2006-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3528 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 9 (2006); 1271–1275 Український математичний журнал; Том 58 № 9 (2006); 1271–1275 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3528/3793 https://umj.imath.kiev.ua/index.php/umj/article/view/3528/3794 Copyright (c) 2006 Mulyava O. M. |
| spellingShingle | Mulyava, O. M. Мулява, О. М. Integral analog of one generalization of the Hardy inequality and its applications |
| title | Integral analog of one generalization of the Hardy inequality and its applications |
| title_alt | Інтегральний аналог одного узагальнення нерівності Гарді та його застосування |
| title_full | Integral analog of one generalization of the Hardy inequality and its applications |
| title_fullStr | Integral analog of one generalization of the Hardy inequality and its applications |
| title_full_unstemmed | Integral analog of one generalization of the Hardy inequality and its applications |
| title_short | Integral analog of one generalization of the Hardy inequality and its applications |
| title_sort | integral analog of one generalization of the hardy inequality and its applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3528 |
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