Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”

We show that an equimeasurable rearrangement of any function satisfying the “reverse Jensen inequality” with respect to various multidimensional segments also satisfies the “reverse Jensen inequality” with the same constant.

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Date:	2005
Main Authors:	Korenovskii, A. A., Кореновский, А. А.
Format:	Article
Language:	Russian English
Published:	Institute of Mathematics, NAS of Ukraine 2005
Online Access:	https://umj.imath.kiev.ua/index.php/umj/article/view/3584
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Journal Title:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Korenovskii, A. A. Кореновский, А. А. Кореновский, А. А.
author_facet	Korenovskii, A. A. Кореновский, А. А. Кореновский, А. А.
author_sort	Korenovskii, A. A.
baseUrl_str	https://umj.imath.kiev.ua/index.php/umj/oai
collection	OJS
datestamp_date	2020-03-18T19:59:22Z
description	We show that an equimeasurable rearrangement of any function satisfying the “reverse Jensen inequality” with respect to various multidimensional segments also satisfies the “reverse Jensen inequality” with the same constant.
first_indexed	2026-03-24T02:45:17Z
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fulltext	UDK 517.5 A. A. Korenovskyj (Odes. nac. un-t, Yn-t matematyky, πkonomyky y mexanyky) OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU NERAVENSTVU JENSENA” * We consider the functions satisfying the “reverse Jensen inequality” with respect to various multidimensional segment. We show that the equimeasurable rearrangement of every function of this sort also satisfies the “reverse Jensen inequality” with the same constant. Pokazano, wo dlq bud\-qko] funkci], wo zadovol\nq[ „obernenu nerivnist\ {nsena” po vsilqkyx bahatovymirnyx sehmentax, ]] rivnovymirne perestavlennq takoΩ zadovol\nq[ „obernenu neriv- nist\ {nsena” z ti[g samog stalog. 1. Vvedenye. Pust\ Φ — klass poloΩytel\n¥x na ( 0, + ∞ ) , v¥pukl¥x vnyz funkcyj ϕ takyx, çto ϕ ( 0 ) = ϕ ( 0 + ) (ravnoe, b¥t\ moΩet, nulg yly besko- neçnosty); µ — absolgtno neprer¥vnaq mera, d µ ( x ) = w ( x ) d x , hde w — lo- kal\no summyruemaq, neotrycatel\naq vesovaq funkcyq. Dlq yzmerymoj na sehmente R0 ≡ [ , ]a bj jj d =∏ 1 ⊂ R d funkcyy f nevozrastagwej ravnoyzmery- moj perestanovkoj po otnoßenyg k mere µ naz¥vaetsq funkcyq f tµ [ ]( )↓ = sup inf ( ) , ( )e R e t x e f x ⊂ = ∈ 0 µ , 0 ≤ t ≤ µ ( R0 ) . Funkcyq fµ [ ]↓ ravnoyzmeryma s f v tom sm¥sle, çto dlq lgboho λ ∈ R spra- vedlyvo ravenstvo µ ( { x ∈ R0 : f ( x ) > λ } ) = t R f t∈ >{ }↓( , ( )] : ( )[ ]0 0µ λµ , hde symvolom \| ⋅ \| oboznaçena mera Lebeha. Analohyçno, neub¥vagwaq ravnoyz- merymaq perestanovka yzmerymoj na R0 funkcyy f opredelqetsq ravenstvom f tµ [ ]( )↑ = inf sup ( ) , ( )e R e t x e f x ⊂ = ∈0 µ , 0 ≤ t ≤ µ ( R0 ) . Fundamental\noe svojstvo perestanovok fµ [ ]↓ y fµ [ ]↑ , sledugwee neposredst- venno yz opredelenyq, sostoyt v tom, çto dlq lgboho t ∈ [ 0, µ ( R0 ) ] spravedly- v¥ ravenstva sup ( ) ( ) , ( )e R e t e f x d x ⊂ = ∫ 0 µ µ = f d t µ τ τ[ ]( )↓∫ 0 , inf ( ) ( ) , ( )e R e t e f x d x ⊂ = ∫ 0 µ µ = f d t µ τ τ[ ]( )↑∫ 0 , pryçem netrudno pokazat\, çto verxnqq y nyΩnqq hrany v lev¥x çastqx dosty- hagtsq. Krome toho, dlq neotrycatel\noj na R0 funkcyy f yz ravnoyzmery- mosty fµ [ ]↓ , fµ [ ]↑ y f sleduet, çto dlq lgboj ϕ ∈ Φ ϕ µ µ f t dt R [ ] ( ) ( )↓( )∫ 0 0 = ϕ µ µ f t dt R [ ] ( ) ( )↑( )∫ 0 0 = ϕ µf x d x R ( )( )∫ 0 . Pust\ ϕ ∈ Φ, funkcyq f neotrycatel\na na sehmente R0 ⊂ R d. Tohda dlq * Çastyçno podderΩana Hosudarstvenn¥m fondom fundamental\n¥x yssledovanyj Ukrayn¥ (hrant # F7/329-2001). © A. A. KORENOVSKYJ, 2005 158 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU …” 159 lgboho sehmenta R ⊂ R0 ymeet mesto neravenstvo Jensena [1, c. 182 – 184] ϕ µ µ1 ( ) ( ) ( ) R f x d x R ∫     ≤ 1 µ ϕ µ ( ) ( ) ( ) R f x d x R ( )∫ . (1) V dannoj rabote yzuçagtsq svojstva funkcyj f, udovletvorqgwyx „obratnomu neravenstvu Jensena” 1 µ ϕ µ ( ) ( ) ( ) R f x d x R ( )∫ ≤ B R f x d x R ϕ µ µ1 ( ) ( ) ( )∫     , R ⊂ R0, (2) hde postoqnnaq B > 1 ne zavysyt ot sehmenta R. V sluçae ϕ ( u ) = u p, p > 1, neravenstvo (2) naz¥vagt neravenstvom Herynha [2], yly „obratn¥m neravenstvom Hel\dera”, a pry ϕ ( u ) = u p− −1 1/( ) , p > 1 — Ap- uslovyem Makenxaupta [3]. Klass¥ funkcyj, udovletvorqgwyx uslovyqm He- rynha, Makenxaupta y blyzkym uslovyqm, ßyroko prymenqgtsq v teoryy kvazy- konformn¥x otobraΩenyj, dyfferencyal\n¥x uravnenyj s çastn¥my proyz- vodn¥my, v teoryy vesov¥x prostranstv y dr. Pry yzuçenyy svojstv funkcyj yz πtyx klassov çasto okaz¥vagtsq polezn¥my ocenky ravnoyzmerym¥x perestano- vok. Tak, v rabote [4] s pomow\g ocenky nevozrastagwej perestanovky dano bo- lee prostoe (po sravnenyg s yzvestn¥my ranee) dokazatel\stvo osnovnoho svoj- stva funkcyy yz klassa Herynha — pov¥ßenyq pokazatelq summyruemosty. V rabotax [5, 6] analohyçnoe svojstvo dlq funkcyy, udovletvorqgwej uslovyg Makenxaupta, takΩe poluçeno s pomow\g ocenky ravnoyzmerymoj perestanov- ky. Pry nekotor¥x dopolnytel\n¥x predpoloΩenyqx na funkcyg ϕ ∈ Φ v ra- bote [7] poluçena ocenka perestanovky funkcyy, udovletvorqgwej „obratnomu neravenstvu Jensena” (2), v kotorom vmesto sehmentov rassmatryvagtsq vsevoz- moΩn¥e kub¥. V πtom sluçae vopros o toçnosty podobn¥x ocenok, kak pravylo, trudn¥j. Avtoru neyzvestn¥ rabot¥, v kotor¥x b¥ly b¥ poluçen¥ toçn¥e ocenky ravnoyzmerym¥x perestanovok funkcyj, udovletvorqgwyx uslovyg (2) „po mnohomern¥m kubam” daΩe dlq kakyx-lybo specyal\noho vyda funkcyj ϕ. V prostranstve razmernosty d = 1 toçnaq ocenka ravnoyzmerymoj perestanov- ky funkcyy, udovletvorqgwej „obratnomu neravenstvu Jensena” (2), poluçena v [8, 9]. S pomow\g πtoj ocenky v [8, 9] najden¥ predel\n¥e pokazately summy- ruemosty funkcyj, udovletvorqgwyx uslovyg Herynha y Makenxaupta, a v ra- botax [10, 11] ustanovlena toçnaq svqz\ πtyx klassov meΩdu soboj. Dlq mono- tonnoj funkcyy yz klassa Herynha ranee toçn¥j pokazatel\ summyruemosty b¥l najden v [12, 13]. Toçn¥j pokazatel\ summyruemosty funkcyy, udovletvo- rqgwej uslovyg Herynha „po mnohomern¥m sehmentam”, v¥çyslen v rabote [14]. Ytak, opysanye πkstremal\n¥x svojstv funkcyj, udovletvorqgwyx „obrat- nomu neravenstvu Jensena” (2), moΩno uprostyt\ pry nalyçyy toçn¥x ocenok ravnoyzmerym¥x perestanovok funkcyj yz sootvetstvugwyx klassov. V svog oçered\, ocenky perestanovok bazyrugtsq, kak pravylo, na prymenenyy tak naz¥vaem¥x lemm „o pokr¥tyy”. Tradycyonno dlq poluçenyq takyx ocenok ys- pol\zovalas\ lemma Kal\derona – Zyhmunda [15], no v odnomernom sluçae ocen- ky, osnovann¥e na prymenenyy πtoj lemm¥, okaz¥vagtsq zav¥ßenn¥my. Pry d = 1 bolee toçn¥m varyantom lemm¥ Kal\derona – Zyhmunda qvlqetsq lemma F.QRyssa „o vosxodqwem solnce” [16; 1, c. 352; 17]. Ymenno prymenenye πtoj lemm¥ v [8] dalo vozmoΩnost\ pry d = 1 poluçyt\ toçnug ocenku peresta- novky funkcyy, udovletvorqgwej uslovyg (2). V mnohomernom sluçae analoh lemm¥ „o vosxodqwem solnce” ne ymeet mesta, esly vmesto odnomern¥x ot- rezkov rassmatryvat\ mnohomern¥e kub¥ (sm. [18]). Esly Ωe vmesto kubov ys- pol\zovat\ vsevozmoΩn¥e mnohomern¥e sehment¥, to sootvetstvugwyj analoh lemm¥ F.QRyssa „o vosxodqwem solnce” ostaetsq spravedlyv¥m (sm. nyΩe lem- ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 160 A. A. KORENOVSKYJ m¥Q1 y 2) y, takym obrazom, modyfycyruq rassuΩdenyq yz [8], pryxodym k sle- dugwej teoreme, kotoraq sostavlqet osnovnoj rezul\tat dannoj rabot¥. Teorema)1. Pust\ ϕ ∈ Φ, neotrycatel\naq na sehmente R0 ⊂ R d funk- cyq f udovletvorqet uslovyg (2). Tohda dlq lgboho otrezka I ⊂ [ 0, µ ( R0 ) ] v¥polnqgtsq neravenstva 1 I f t dt I ∫ ↓( )ϕ µ [ ]( ) ≤ B I f t dt I ϕ µ 1 ∫ ↓     [ ]( ) , (3) 1 I f t dt I ∫ ↑( )ϕ µ [ ]( ) ≤ B I f t dt I ϕ µ 1 ∫ ↑     [ ]( ) . (4) Zameçanye. Dokazatel\stvo teorem¥, pryvedennoe nyΩe, bolee podrobnoe y bolee prostoe, neΩely dokazatel\stvo sootvetstvugwej ocenky v rabote [8] dlq d = 1. Krome toho, v otlyçye ot [8] teoremaQ1 budet dokazana v bolee ob- wem vesovom sluçae. ∏to sdelano ne s cel\g prostoho obobwenyq, a potomu, çto dlq vesov¥x klassov funkcyj, udovletvorqgwyx uslovyg Herynha y Ma- kenxaupta, lehko moΩno ustanavlyvat\ svqz\ meΩdu πtymy klassamy (sm. [19]). 2. Vspomohatel\n¥e utverΩdenyq y dokazatel\stvo teorem¥)1 . Kak ot- meçeno v¥ße, klgçevug rol\ pry dokazatel\stve teorem¥Q1 yhragt sledugwye dve lemm¥, kotor¥e qvlqgtsq mnohomern¥my analohamy yzvestnoj lemm¥ F.QRyssa „o vosxodqwem solnce”. Lemma)1 [20, 21]. Pust\ funkcyq f summyruema po mere µ na sehmente R0 y çyslo A ≥ ( ( )) ( ) ( )µ µR f x d x R0 1 0 − ∫ . Tohda suwestvuet ne bolee çem sçet- n¥j nabor sehmentov Rj ⊂ R0, j = 1, 2, … , vnutrennosty kotor¥x poparno ne peresekagtsq, takyx, çto 1 µ µ ( ) ( ) ( ) R f x d x j Rj ∫ = A, j = 1, 2, … , f ( x ) ≤ A dlq µ-poçty vsex x ∈ R Rj j 0 1 \ ≥     ∪ . Lehko vydet\, çto lemmuQ1 moΩno sformulyrovat\ v sledugwem πkvyva- lentnom vyde. Lemma)2. Pust\ funkcyq f summyruema po mere µ na sehmente R0 y çyslo A ≤ ( ( )) ( ) ( )µ µR f x d x R0 1 0 − ∫ . Tohda suwestvuet ne bolee çem sçetn¥j nabor sehmentov Rj ⊂ R0, j = 1, 2, … , vnutrennosty kotor¥x poparno ne pe- resekagtsq, takyx, çto 1 µ µ ( ) ( ) ( ) R f x d x j Rj ∫ = A, j = 1, 2, … , f ( x ) ≥ A dlq µ-poçty vsex x ∈ R Rj j 0 1 \ ≥     ∪ . Lemma)3 [8]. Pust\ neotrycatel\naq, summyruemaq funkcyq g monoton- na na otrezke [ a, b ] y otrezok [ α, β ] ⊂ [ a, b ] takoj, çto 1 b a g t dt a b − ∫ ( ) = 1 β α α β − ∫ g t dt( ) . ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU …” 161 Tohda dlq lgboj ϕ ∈ Φ v¥polnqetsq neravenstvo 1 β α ϕ α β − ∫ ( ( ))g t dt ≤ 1 b a g t dt a b − ∫ ϕ ( ( )) . Sledugwaq lemma otraΩaet prostoe svojstvo v¥pukloj funkcyy, kotoroe nam ponadobytsq v dal\nejßem. Lemma)4 [8]. Pust\ çysla 0 ≤ γ1, γ2 ≤ 1, a ≥ b ≥ c ≥ d > 0 takov¥, çto γ γ1 11a d+ −( ) = γ γ2 21b c+ −( ) . Tohda dlq lgboj ϕ ∈ Φ v¥polnqetsq neraven- stvo γ ϕ γ ϕ1 11( ) ( ) ( )b c+ − ≤ γ ϕ γ ϕ2 21( ) ( ) ( )a d+ − . Osnovu dokazatel\stva teorem¥Q1 sostavlqgt dve sledugwye lemm¥. Lemma)5. Pust\ neotrycatel\naq na mnoΩestve E E∪ ˆ funkcyq f yme- et sledugwye svojstva: 1 µ µ ( ) ( ) ( ) E f x d x E ∫ = 1 µ µ ( ˆ ) ( ) ( ) ˆE f x d x E ∫ ≡ A, (5) f ( x ) ≤ A, x ∉ E E∩ ˆ , (6) f ( x ) ≤ f ( y ) , x ∈ ˆ \E E , y ∈ E . (7) Tohda dlq lgboj ϕ ∈ Φ v¥polnqetsq neravenstvo 1 µ ϕ µ ( ) ( ) ( ) E f x d x E ( )∫ ≤ 1 µ ϕ µ ( ˆ ) ( ) ( ) ˆE f x d x E ( )∫ . (8) Dokazatel\stvo. Rassmatryvaem netryvyal\n¥j sluçaj, kohda f ne qvlq- etsq µ -πkvyvalentnoj toΩdestvennoj postoqnnoj A. PokaΩem snaçala, çto µ( ˆ )E ≤ µ ( E ) . (9) Dejstvytel\no, yz (5) y (6) sleduet E E A f x d x \ ˆ ( ( )) ( )∫ − µ = ˆ \ ( ( )) ( ) E E A f x d x∫ − µ . (10) Yspol\zuq uslovye (7), naxodym c takoe, çto f ( x ) ≤ c ≤ f ( y ) , x ∈ ˆ \E E , y ∈ E . Tohda poluçaem E E A f x d x \ ˆ ( ( )) ( )∫ − µ ≤ ( ) ˆ( \ )A c E E− µ , ˆ \ ( ( )) ( ) E E A f x d x∫ − µ ≥ ( ) ˆ( \ )A c E E− µ . Poskol\ku, oçevydno, c < A, yz πtyx dvux neravenstv y yz (10) sleduet µ( \ )Ê E ≤ µ( \ )ˆE E , a πto ravnosyl\no (9). Postroym mnoΩestva E ′ y E ″ takye, çto ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 162 A. A. KORENOVSKYJ ′ ′′E E∪ = E E\ ˆ , ′ ′′E E∩ = ∅, µ( )′E = µ( \ )Ê E y f ( x ) ≤ f ( y ) pry lgb¥x x ∈ E ″, y ∈ E ′ . Zadadym natural\noe k y razob\em mnoΩestva E ′, E ″ y ˆ \E E na poparno neperesekagwyesq podmnoΩestva sle- dugwym obrazom. Oboznaçym g t( )( )1 = ( )[ ]( )f E t′ ↓ µ , 0 ≤ t ≤ µ ( E ′ ) , g t( )( )2 = ( )[ ]( )f E t′′ ↓ µ , 0 ≤ t ≤ µ ( E ″ ) , g t( )( )3 = f E E t( \ )ˆ ( ) [ ]( ) ↓ µ , 0 ≤ t ≤ µ( \ )Ê E , y poloΩym α0, ( ) k j = g j( )( )0 , j = 1, 2, 3, τ0, ( ) k j = 0, j = 1, 2. PredpoloΩym, çto uΩe postroen¥ çysla τ0 1 , ( ) k < τ1 1 , ( ) k < … < τs k, ( )1 , τ0 2 , ( ) k < τ1 2 , ( ) k < … < τs k, ( )2 , α0 1 , ( ) k ≥ α1 1 , ( ) k ≥ … ≥ αs k, ( )1 ≥ α0 2 , ( ) k ≥ α1 2 , ( ) k ≥ … ≥ αs k, ( )2 ≥ α0 3 , ( ) k ≥ α1 3 , ( ) k ≥ … ≥ αs k, ( )3 y poparno neperesekagwyesq mnoΩestva ′El k, ⊂ E ′, ′′El k, ⊂ E ″ , ˆ ,El k ⊂ ˆ \E E , l = 1, 2, … , s, takye, çto max ,, ( ) , ( )α αl k l k k 1 1 1 1 − −    ≤ f ( x ) ≤ αl k−1 1 , ( ) , x ∈ ′El k, , max ,, ( ) , ( )α αl k l k k 2 1 2 1 − −    ≤ f ( x ) ≤ αl k−1 2 , ( ) , x ∈ ′′El k, , αl k, ( )3 ≤ f ( x ) ≤ αl k−1 3 , ( ) x ∈ ˆ ,El k , µ( ),′El k = µ( )ˆ ,El k = τ τl k l k, ( ) , ( )1 1 1− − , (11) µ( ),′′El k = τ τl k l k, ( ) , ( )2 1 2− − , f x d x f x d x E El k l k ( ) ( ) ( ) ( ) , , ˆ µ µ ′ ∫ ∫− = ′′ ∫ − El k A f x d x , ( ( )) ( )µ . (12) Esly µ ( E ′ ) = µ( ),′ = ∑ El k l s 1 , to process razbyenyq mnoΩestv E ′, E ″ y ˆ \E E na πtom zakançyvaetsq. V pro- tyvnom sluçae budem stroyt\ αs k j +1, ( ) , τs k j +1, ( ) , ′+Es k1, , ′′+Es k1, y ˆ ,Es k+1 po takoj sxeme. Rassmatryvaem stroho vozrastagwye neprer¥vn¥e funkcyy η ( τ ) = τ τ s k g t g t dt , ( ) ( ) ( )( ) ( ) 1 1 3∫ −( ) , τs k, ( )1 ≤ τ ≤ µ ( E ′ ) , y ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU …” 163 ζ ( ξ ) = τ ξ s k A g t dt , ( ) ( )( ) 2 2∫ −( ) , τs k, ( )2 ≤ ξ ≤ µ ( E ″ ) . Qsno, çto η τ( ), ( ) s k 1 = ζ τ( ), ( ) s k 2 = 0 y η µ( ( ))′E = ζ µ( ( ))′′E . V sylu strohoj mono- tonnosty funkcyy ζ ( ξ ) dlq kaΩdoho τ τ µ∈ ′[ ( )], ( ) ,s k E1 najdetsq edynstvennoe znaçenye ξ = ξ ( τ ) takoe, çto ζ ( ξ ( τ ) ) = η ( τ ), (13) pryçem funkcyq ξ ( τ ) neprer¥vna. Oboznaçym τs k+1 1 , ( ) = sup , : max ( ), ( ), ( ) ( ) , ( ) ( ) ( ) , ( ) ( )( ) ( ) ( ) ( )τ τ µ τ τ τ ξ τ∈ ′( ] − −[ ] <{ }s k s k s kE g g g g k 1 1 1 1 2 2 2 1 , τs k+1 2 , ( ) = ξ τ( ), ( ) s k+1 1 . Otmetym, çto verxnqq hran\ beretsq po nepustomu mnoΩestvu, poskol\ku ne- vozrastagwaq perestanovka neprer¥vna sprava. PoloΩym αs k+1 1 , ( ) = g s k ( ) , ( )( )1 1 1τ + , αs k+1 2 , ( ) = g s k ( ) , ( )( )2 1 2τ + , αs k+1 3 , ( ) = g s k ( ) , ( )( )3 1 1τ + . Tohda moΩno postroyt\ takye mnoΩestva ′+Es k1, ⊂ ′ ′    = E El k l s \ , 1 ∪ , ′′+Es k1, ⊂ ′′ ′′    = E El k l s \ , 1 ∪ , ˆ ,Es k+1 ⊂ ( \ )\ˆ ˆ ,E E El k l s =    1 ∪ , çto max ,, ( ) , ( )α αs k s k k+ −   1 1 1 1 ≤ f ( x ) ≤ αs k, ( )1 , x ∈ ′+Es k1, , (14) max ,, ( ) , ( )α αs k s k k+ −   1 2 2 1 ≤ f ( x ) ≤ αs k, ( )2 , x ∈ ′′+Es k1, , (15) αs k+1 3 , ( ) ≤ f ( x ) ≤ αs k, ( )3 , x ∈ ˆ ,Es k+1 . Pry πtom ravenstvo (13) oznaçaet, çto f x d x Es k ( ) ( ) , µ ′+ ∫ 1 – f x d x Es k ( ) ( ) ˆ , µ + ∫ 1 = ( ( )) ( ) , A f x d x Es k − ′′+ ∫ µ 1 . Esly okaΩetsq, çto µ( )′E = µ( ),′ = + ∑ El k l s 1 1 , to yz (10) poluçym, çto y µ( )′′E = µ( ),′′ = + ∑ El k l s 1 1 , µ( \ )Ê E = µ( )ˆ ,El k l s = + ∑ 1 1 , y poπtomu process razbyenyq mnoΩestv E ′, E ″ y ˆ \E E na πtom zaverßytsq. V protyvnom sluçae v¥polnqetsq neravenstvo max , , ( ) , ( ) j s k j s k j = +−( ) 12 1α α ≥ 1 k , ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 164 A. A. KORENOVSKYJ yz kotoroho sleduet, çto kolyçestvo ßahov s ne moΩet vozrastat\ neohrany- çenno, t. e. najdetsq takoe sk, dlq kotoroho E ′ = ′ = ∑ El k l sk , 1 , E ″ = ′′ = ∑ El k l sk , 1 , ˆ \E E = ˆ ,El k l sk = ∑ 1 . Oboznaçym bl k, = 1 µ µ ( ) ( ) ( ) , , ′ ′ ∫E f x d x l k El k , cl k, = 1 µ µ ( ) ( ) ( ) , , ′′ ′′ ∫E f x d x l k El k , dl k, = 1 µ µ ( ˆ ) ( ) ( ) , ˆ , E f x d x l k El k ∫ , l = 1, … , sk . Tohda ravenstvo (12) prynymaet vyd b E c El k l k l k l k, , , ,( ) ( )µ µ′ + ′′ = d E A El k l k l k, , ,( ) ( )ˆµ µ+ ′′ , a s uçetom (11) otsgda ymeem µ µ µ µ µ µ ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , ′ ′ + ′′ + ′′ ′ + ′′ E E E b E E E c l k l k l k l k l k l k l k l k = = µ µ µ µ µ µ ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ , , , , , , , E E E d E E E A l k l k l k l k l k l k l k+ ′′ + ′′ + ′′ . Poskol\ku A ≥ bl k, ≥ cl k, ≥ dl k, , prymenqq lemmuQ4 s a = A, b = bl k, , c = = cl k, , d = dl k, y γ1 = γ2 = µ µ µ( )/( ( ) ( ))ˆ ˆ , , ,E E El k l k l k+ ′′ , poluçaem µ ϕ µ ϕ( ) ( ) ( ) ( ), , , ,′ + ′′E b E cl k l k l k l k ≤ µ ϕ µ ϕ( ) ( ) ( ) ( )ˆ , , ,E d E Al k l k l k+ ′′ . Prymenqq ko vtoromu slahaemomu sprava neravenstvo Jensena (1), ymeem µ ϕ µ ϕ( ) ( ) ( ) ( ), , , ,′ + ′′E b E cl k l k l k l k ≤ ≤ ϕ µ µ ϕf x d x E A E l k l k ( ) ( ) ˆ , , ( ) ( )( ) + ′′∫ , l = 1, … , sk . Poπtomu ϕ µ µ ϕ µ ϕf x d x E b E c E E l s l k l k l k l k k ( ) ( ) ˆ , , , ,( ) ( ) ( ) ( )( ) + ′ + ′′( )∫ ∑ =∩ 1 ≤ ≤ ϕ µ ϕ µ µ ϕf x d x f x d x E A E E E E ( ) ( ) ( ) ( ) ˆ ˆ \ ( ) ( )( ) + ( ) + ′′∫ ∫ ∩ = = ϕ µ µ ϕf x d x E A E ( ) ( ) ˆ ( ) ( )( ) + ′′∫ = = µ µ ϕ µ µ µ ϕ µ µ ϕ( ) ( ˆ ) ( ) ( ) ( ) ( ˆ ) ( ) ( ) ˆ ˆ ( ) ( )E E f x d x E E f x d x E A E E ( ) + −     ( ) + ′′∫ ∫1 . (16) No poskol\ku ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU …” 165 µ µ( ) ( ˆ )E E− = µ µ µ( ) ( ) ( \ )ˆ′ + ′′ −E E E E = µ( )′′E , yz neravenstva Jensena (1) y yz (5) sleduet 1 −     ( ) + ′′∫µ µ ϕ µ µ ϕ( ) ( ˆ ) ( ) ( ) ˆ ( ) ( )E E f x d x E A E ≤ ≤ µ µ µ ϕ µ µ µ ϕ µ( ) ( ) ( ) ( ) ˆ ( ) ˆ ( ) ( ) ˆ ( ) ( ) ˆ ˆ E E E f x d x E E f x d x E E −( ) ( ) + ′′ ( )∫ ∫1 1 = = 1 µ ϕ µ µ µ µ ( ) ( ) ( )ˆ ( ) ( ) ˆ ( ) ˆE f x d x E E E E ( ) − + ′′( )∫ = 0. Poπtomu yz (16) ymeem 1 1µ ϕ µ µ ϕ µ ϕ ( ) ( ) ( ) ˆ , , , ,( ) ( ) ( ) ( ) E f x d x E b E c E E l s l k l k l k l k k ( ) + ′ + ′′( )        ∫ ∑ =∩ ≤ ≤ 1 µ ϕ µ ( )ˆ ( ) ( ) ˆE f x d x E ( )∫ . (17) Oboznaçym f xk ( ) = f x x b x c x E E l s l k E l k E k l k l k ( ) ( ) ( ) ( )ˆ , ,, , χ χ χ∩ + +( ) = ′ ′′∑ 1 , x ∈ E . Tohda neravenstvo (17) moΩem perepysat\ tak: 1 µ ϕ µ ( ) ( ) ( ) E f x d xk E ( )∫ ≤ 1 µ ϕ µ ( )ˆ ( ) ( ) ˆE f x d x E ( )∫ . (18) Poskol\ku yz (14) y (15) sleduet, çto dlq µ -poçty vsex x ∈ E v¥polnqetsq ne- ravenstvo f x f xk( ) ( )− ≤ 1 k , v sylu neprer¥vnosty v¥pukloj funkcyy ϕ poluçaem, çto posledovatel\nost\ funkcyj ϕ fk( ) sxodytsq k ϕ f( ) µ -poçty vsgdu. Poπtomu, prymenqq k (18) teoremu Fatu, poluçaem (8). Lemma dokazana. Lemma)6. Pust\ neotrycatel\naq na mnoΩestve E E∪ ˆ funkcyq f yme- et sledugwye svojstva: 1 µ µ ( ) ( ) ( ) E f x d x E ∫ = 1 µ µ ( )ˆ ( ) ( ) ˆE f x d x E ∫ ≡ A, f ( x ) ≥ A, x ∉ E E∩ ˆ , f ( x ) ≥ f ( y ) , x ∈ ˆ \E E , y ∈ E . Tohda dlq lgboj ϕ ∈ Φ ymeet mesto neravenstvo 1 µ ϕ µ ( ) ( ) ( ) E f x d x E ( )∫ ≤ 1 µ ϕ µ ( )ˆ ( ) ( ) ˆE f x d x E ( )∫ . ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 166 A. A. KORENOVSKYJ Dokazatel\stvo πtoj lemm¥ analohyçno dokazatel\stvu lemm¥Q5, y m¥ eho opuskaem. Dokazatel\stvo teorem¥)1. Poskol\ku neravenstvo f tµ [ ]( )↓ = f R tµ µ[ ] ( )↑ −( )0 spravedlyvo vo vsex toçkax neprer¥vnosty perestanovok, t. e. poçty vsgdu na [ ], ( )0 0µ R , neravenstva (3) y (4) ravnosyl\n¥. Zafyksyruem otrezok I ⊂ [ ], ( )0 0µ R . Esly 1 I f t dt I µ [ ]( )↓∫ ≥ 1 0 0 µ µ ( ) ( ) ( ) R f x d x R ∫ , to najdetsq takoe T ∈ [ ], ( )0 0µ R , çto 1 I f t dt I µ [ ]( )↓∫ = 1 0T f t dt T ∫ ↓ µ [ ]( ) . V πtom sluçae oboznaçym J = [ 0, T ] . Esly Ωe 1 I f t dt I µ [ ]( )↓∫ < 1 0 0 µ µ ( ) ( ) ( ) R f x d x R ∫ , to najdem takoe T ∈ [ ], ( )0 0µ R , dlq kotoroho 1 I f t dt I µ [ ]( )↓∫ = 1 0 0 T f t dt R T R µ µ µ ( ) ( ) [ ]( ) − ∫ ↓ . Tohda oboznaçym J = [ ]( ) , ( )µ µR T R0 0− . V lgbom sluçae ymeem 1 I f t dt I µ [ ]( )↓∫ = 1 J f t dt J µ [ ]( )↓∫ , a yz monotonnosty perestanovky fµ [ ]↓ sleduet, çto J ⊃ I. Polahaq v lemmeQ3 [ a, b ] = J, [ α, β ] = I, g = fµ [ ]↓ , poluçaem neravenstvo 1 I f t dt I ϕ µ [ ]( )↓( )∫ ≤ 1 J f t dt J ϕ µ [ ]( )↓( )∫ . Takym obrazom, neravenstvo (3) dostatoçno dokazat\ lyß\ dlq otrezka J. Yn¥- my slovamy, (3) y (4) qvlqgtsq sledstvyqmy takyx dvux neravenstv: 1 0T f t dt T ∫ ↓( )ϕ µ [ ]( ) ≤ B T f t dt T ϕ µ 1 0 ∫ ↓     [ ]( ) , 0 ≤ T ≤ µ ( R0 ) , (19) 1 0T f t dt T ∫ ↑( )ϕ µ [ ]( ) ≤ B T f t dt T ϕ µ 1 0 ∫ ↑     [ ]( ) , 0 ≤ T ≤ µ ( R0 ) . (20) Dlq dokazatel\stva (19) y (20) zafyksyruem T ∈ [ ], ( )0 0µ R y oboznaçym A[ ]↓ = 1 0T f t dt T ∫ ↓ µ [ ]( ) ≥ 1 0 0 µ µ ( ) ( ) ( ) R f x d x R ∫ , (21) ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU …” 167 A[ ]↑ = 1 0T f t dt T ∫ ↑ µ [ ]( ) ≤ 1 0 0 µ µ ( ) ( ) ( ) R f x d x R ∫ . Yspol\zovav lemm¥Q1 yQ2, postroym dva nabora sehmentov Rj [ ]↓ ⊂ R0 y Rk [ ]↑ ⊂ ⊂ R0 takyx, çto vnutrennosty sehmentov Rj [ ]↓ poparno ne peresekagtsq y vnutrennosty sehmentov Rk [ ]↑ poparno ne peresekagtsq, 1 µ µ ( )[ ] ( ) ( ) [ ]R f x d x j Rj ↓ ↓ ∫ = A[ ]↓ , j = 1, 2, … , (22) f ( x ) ≤ A[ ]↓ dlq µ -poçty vsex x ∈ R R j j0 1 \ [ ] ≥ ↓     ∪ , 1 µ µ ( )[ ] ( ) ( ) [ ]R f x d x k Rk ↑ ↑ ∫ = A[ ]↑ , k = 1, 2, … , (23) f ( x ) ≥ A[ ]↑ dlq µ -poçty vsex x ∈ R R k k0 1 \ [ ] ≥ ↑     ∪ . Oboznaçym ˆ[ ]E ↓ = j jR≥ ↓ 1∪ [ ] , ˆ[ ]E ↑ = k kR≥ ↑ 1∩ [ ]. Tohda µ x R f x A E∈ >{ }( )↓ ↓ 0 : ( ) ˆ[ ] [ ]\ = 0, (24) µ x R f x A E∈ <{ }( )↑ ↑ 0 : ( ) ˆ[ ] [ ]\ = 0. Esly m¥ dokaΩem neravenstva 1 0T f t dt T ∫ ↓( )ϕ µ [ ]( ) ≤ 1 µ ϕ µ ( )ˆ ( ) ( )[ ] ˆ [ ]E f x d x E ↓ ↓ ( )∫ , (25) 1 0T f t dt T ∫ ↑( )ϕ µ [ ]( ) ≤ 1 µ ϕ µ ( )ˆ ( ) ( )[ ] ˆ [ ]E f x d x E ↑ ↑ ( )∫ , (26) to yz nyx srazu Ωe poluçym (19) y (20). V samom dele, yz (25), (2) y (22) sleduet 1 0T f t dt T ∫ ↓( )ϕ µ [ ]( ) ≤ j j j R R f x d x j ≥ − ≥ ∑ ∑ ∫↓ ↓ ( )    ( ) 1 1 1 µ ϕ µ[ ] ( ) ( ) [ ] ≤ ≤ sup ( ) ( ) ( )[ ] [ ]j j R R f x d x j ≥ ↓ ↓ ( )∫ 1 1 µ ϕ µ ≤ ≤ B R f x d x j j Rj sup ( ) ( ) ( )[ ] [ ]≥ ↓ ↓ ∫        1 1ϕ µ µ = B Aϕ( )[ ]↓ , t. e. (19). Analohyçno, neravenstvo (20) sleduet yz (26), (2) y (23). Takym obra- zom, ostaetsq dokazat\ (25) y (26). Dlq πtoho postroym mnoΩestva E[ ]↓ y E[ ]↑ takye, çto ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 168 A. A. KORENOVSKYJ µ ( )[ ]E ↓ = µ ( )[ ]E ↑ = T y f ( x ) ≥ f Tµ [ ]( )↓ , x ∈ E[ ]↓ , f ( x ) ≤ f Tµ [ ]( )↑ , x ∈ E[ ]↑ . PokaΩem, çto mnoΩestva E = E[ ]↓ y Ê = ˆ[ ]E ↓ udovletvorqgt uslovyqm lem- m¥Q5. Dejstvytel\no, yz (21) y (22) sleduet ravenstvo 1 µ µ ( ) ( ) ( )[ ] [ ]E f x d x E ↓ ↓ ∫ = 1 0T f t dt T ∫ ↓ µ [ ]( ) = A[ ]↓ = = j j j R R f x d x j ≥ − ≥ ∑ ∑ ∫↓ ↓ ( )   1 1 1 µ µ[ ] ( ) ( ) [ ] = 1 µ µ ( ˆ ) ( ) ( )[ ] ˆ [ ]E f x d x E ↓ ↓ ∫ , t. e. (5) pry A = A[ ]↓ . Dalee, yz (24) ymeem, çto vklgçenye x R f x A∈ >{ }↓ 0 : ( ) [ ] ⊂ ˆ[ ]E ↓ spravedlyvo s toçnost\g do mnoΩestva µ-mer¥ nul\, a vklgçenye x R f x A∈ >{ }↓ 0 : ( ) [ ] ⊂ E[ ]↓ sleduet yz opredelenyq ravnoyzmerymoj perestanovky fµ [ ]↓ . Qsno, çto (6) sle- duet yz πtyx dvux vklgçenyj. Nakonec, neravenstvo (7) qvlqetsq prost¥m svoj- stvom perestanovky fµ [ ]↓ . Analohyçno pokaz¥vaem, çto mnoΩestva E = E[ ]↑ y Ê = ˆ[ ]E ↑ udovletvorqgt uslovyqm lemm¥Q6. Prymenqq lemm¥Q5 yQ6, poluçaem 1 µ ϕ µ ( ) ( ) ( )[ ] [ ]E f x d x E ↓ ↓ ( )∫ ≤ 1 µ ϕ µ ( ˆ ) ( ) ( )[ ] ˆ [ ]E f x d x E ↓ ↓ ( )∫ , 1 µ ϕ µ ( ) ( ) ( )[ ] [ ]E f x d x E ↑ ↑ ( )∫ ≤ 1 µ ϕ µ ( ˆ ) ( ) ( )[ ] ˆ [ ]E f x d x E ↑ ↑ ( )∫ , a πty dva neravenstva ravnosyl\n¥ (25) y (26) sootvetstvenno. Teorema dokazana. 1. Xardy H. H., Lyttl\vud D. E., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. – 456Qs. 2. Gehring F. W. The Lp-integrability of the partial derivatives of a quasiconformal mapping // Acta math. – 1973. – 130. – P. 265 – 277. 3. Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function // Trans. Amer. Math. Soc. – 1972. – 165. – P. 207 – 226. 4. Franciosi M., Moscariello G. Higher integrability results // Manuscr. math. – 1985. – 52, # 1-3. – P. 151 – 170. 5. Wik I. On Muckenhoupt classes of weight functions // Dep. Math. Univ. Umea [publ.]. – 1987. – # 3. – P. 1 – 13. 6. Wik I. On Muckenhoupt classes of weight functions // Stud. Math. – 1989. – 94, # 3. – P. 245 – 255. ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 OCENKA PERESTANOVKY FUNKCYY, UDOVLETVORQGWEJ „OBRATNOMU …” 169 7. Sbordone C. Rearrangement of functions and reverse Jensen inequalities // Proc. Symp. Pure Math. – 1986. – 45, # 2. – P. 325 – 329. 8. Korenovskyj A. A. O toçnom prodolΩenyy obratnoho neravenstva Hel\dera y uslovyq Makenxaupta // Mat. zametky. – 1992. – 52, v¥p.Q6. – S.Q32 – 44. 9. Korenovskyj A. A. Obratnoe neravenstvo Hel\dera, uslovye Makenxaupta y ravnoyzmerym¥e perestanovky funkcyj // Dokl. AN SSSR. – 1992. – 323, #Q2. – S.Q229 – 232. 10. Malaksyano N. A. O toçn¥x vloΩenyqx klassov Herynha v klass¥ Makenxaupta // Mat. zametky. – 2001. – 70, v¥p.Q5. – S.Q742 – 750. 11. Malaksiano N. A. The precise embeddings of the one-dimensional Muckenhoupt classes in Gehring classes // Acta Sci. Math. (Szeged). – 2002. – 68. – P. 237 – 248. 12. Sbordone C. Rearrangement of functions and reverse Hölder inequalities // Res. Notes Math. – 1983. – 125. – P. 139 – 148. 13. D’Apuzzo L., Sbordone C. Reverse Hölder inequalities. A sharp result // Rend. mat. – 1990. – 10, Ser. VII. – P. 357 – 366. 14. Kinnunen J. Sharp result on reverse Hölder inequalities // Ann. Acad. Sci. Fenn., Ser. A I. Math. Diss. – 1994. – 95. – P. 1 – 34. 15. Calderón A. P., Zygmund A. On the existence of certain singular integrals // Acta math. – 1952. – 88. – P. 85 – 139. 16. Riesz F. Sur un téorème de maximum de Mm Hardy et Littlewood // J. London Math. Soc. – 1932. – 7. – P. 10 – 13. 17. Klemes I. A mean oscillation inequality // Proc. Amer. Math. Soc. – 1985. – 93 , # 3. – P. 497 – 500. 18. Korenovskyj A. A. Lemma Ryssa „o vosxodqwem solnce” dlq mnohyx peremenn¥x y neravenstvo DΩona – Nyrenberha // Mat. zametky. – 2005. – 77, # 1. – S. 53 – 66. 19. Popoli A. Weighted reverse Holder inequalities // Rend. Accad. sci. fis. e mat. – 1995. – 62 . – P. 187 – 212. 20. Korenovskyy A. A., Lerner A. K., Stokolos A. M. On multidimensional F. Reisz’s “rising sun” lemma. – 2003. – 3 p. – (Preprint / arxiv: math. CA; # 0308211). 21. Korenovskyy A. A., Lerner A. K., Stokolos A. M. On a multidimensional form of F. Reisz’s “rising sun” lemma // Proc. Amer. Math. Soc. – 2005. – 133, # 5. – P. 1437 – 1440. Poluçeno 03.10.2003 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
id	umjimathkievua-article-3584
institution	Ukrains’kyi Matematychnyi Zhurnal
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language	rus English
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publishDate	2005
publisher	Institute of Mathematics, NAS of Ukraine
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spelling	umjimathkievua-article-35842020-03-18T19:59:22Z Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality” Оценка перестановки функции, удовлетворяющей „обратному неравенству Йенсена" Korenovskii, A. A. Кореновский, А. А. Кореновский, А. А. We show that an equimeasurable rearrangement of any function satisfying the “reverse Jensen inequality” with respect to various multidimensional segments also satisfies the “reverse Jensen inequality” with the same constant. Показано, що для будь-якої функції, що задовольняє „обернену нерівність Єнсена" по всіляких багатовимірних сегментах, її рівновимірне переставлення також задовольняє „обернену нерівність Єнсена" з тією самою сталою. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3584 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 158–169 Український математичний журнал; Том 57 № 2 (2005); 158–169 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3584/3900 https://umj.imath.kiev.ua/index.php/umj/article/view/3584/3901 Copyright (c) 2005 Korenovskii A. A.
spellingShingle	Korenovskii, A. A. Кореновский, А. А. Кореновский, А. А. Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”
title	Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”
title_alt	Оценка перестановки функции, удовлетворяющей „обратному неравенству Йенсена"
title_full	Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”
title_fullStr	Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”
title_full_unstemmed	Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”
title_short	Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”
title_sort	estimate for a rearrangement of a function satisfying the “reverse jensen inequality”
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3584
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Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”

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