Nerivnist Koshi Shvarca Koshi Shvarca angl Cauchy Schwarz inequality angl Cauchy Schwarz inequality nerivnist sho zv yazuye normu ta skalyarnij dobutok vektoriv vektornogo prostoru Ekvivalentno nerivnosti trikutnika dlya normi v prostori zi skalyarnim dobutkom Znahodit zastosuvannya v linijnij algebri dlya vektoriv v matematichnomu analizi dlya neskinchennih ryadiv ta integruvannya dobutkiv ta v teoriyi jmovirnostej pri zastosuvanni do variaciyi ta kovariaciyi Nerivnist dlya sum bulo opublikovano Ogyustenom Koshi 1821 tomu cej vipadok nazivayut Nerivnist Koshi a vidpovidna nerivnist dlya integraliv bula vpershe sformulovana Viktorom Bunyakovskim 1859 ta vdruge vidkrita Germanom Shvarcem 1888 Zmist 1 Formulyuvannya 1 1 Zagalnij vipadok 1 2 Chastinni vipadki 1 2 1 Linijnij prostir UNIQ postMath 00000009 QINU 1 2 2 Linijnij prostir UNIQ postMath 0000000F QINU 2 Dovedennya 2 1 Zagalnij vipadok 2 2 Chastinnij vipadok 2 2 1 Linijnij prostir UNIQ postMath 0000001C QINU 3 Najvidomishi zastosuvannya nerivnosti Koshi Bunyakovskogo 3 1 Nerivnist trikutnika 3 2 Matematichni olimpiadi 4 Dzherela 5 Div takozhFormulyuvannya RedaguvatiZagalnij vipadok Redaguvati Dlya dovilnih vektoriv x displaystyle x nbsp y displaystyle y nbsp iz pregilbertovogo prostoru vikonuyetsya nastupna nerivnist x y 2 x x y y displaystyle langle x y rangle 2 leq langle x x rangle cdot langle y y rangle nbsp de displaystyle langle cdot cdot rangle nbsp operaciya skalyarnogo dobutku a displaystyle cdot nbsp modul chisla Yaksho oznachiti normu to nerivnist mozhna zapisati yak x y x y displaystyle langle x y rangle leq x cdot y nbsp Prichomu rivnist vikonuyetsya lishe u vipadku koli vektori x displaystyle x nbsp y displaystyle y nbsp linijno zalezhni Chastinni vipadki Redaguvati Linijnij prostir R n displaystyle mathbb R n nbsp Redaguvati Skalyarnij dobutok vektoriv x x 1 x 2 x n displaystyle x x 1 x 2 ldots x n nbsp i y y 1 y 2 y n displaystyle y y 1 y 2 ldots y n nbsp oznachimo za formuloyu x y i 1 n x i y i displaystyle langle x y rangle sum limits i 1 n x i y i nbsp todi otrimayemo sho dlya dijsnih chisel x 1 x 2 x n y 1 y 2 y n displaystyle x 1 x 2 ldots x n y 1 y 2 ldots y n nbsp vikonuyetsya nerivnist i 1 n x i y i 2 i 1 n x i 2 i 1 n y i 2 displaystyle left sum i 1 n x i y i right 2 leq left sum i 1 n x i 2 right left sum i 1 n y i 2 right nbsp u zadanij formi nerivnist Koshi Shvarca chasto vikoristovuyetsya na matematichnih olimpiadah Linijnij prostir C a b displaystyle C a b nbsp Redaguvati C a b displaystyle C a b nbsp linijnij prostir neperervnih na vidrizku C a b displaystyle C a b nbsp funkcij Skalyarnij dobutok dlya funkcij f x g x C a b displaystyle f x g x in C a b nbsp oznachimo cherez f x g x a b f x g x d x displaystyle langle f x g x rangle int limits a b f x g x dx nbsp to vikonuvatimetsya nerivnist a b f x g x d x 2 a b f x 2 d x a b g x 2 d x displaystyle left int limits a b f x g x dx right 2 leq int limits a b left f x right 2 dx cdot int limits a b left g x right 2 dx nbsp Dovedennya RedaguvatiZagalnij vipadok Redaguvati Dlya dovilnogo l R displaystyle lambda in mathbb R nbsp Rozglyanemo skalyarnij kvadrat vektora x l y displaystyle x lambda y nbsp 0 x l y x l y x x 2 l x y l 2 y y displaystyle 0 leq langle x lambda y x lambda y rangle langle x x rangle 2 lambda langle x y rangle lambda 2 langle y y rangle nbsp Otrimuyemo kvadratichnu nerivnist l 2 y 2 2 l x y x 2 0 displaystyle lambda 2 y 2 2 lambda langle x y rangle x 2 geq 0 nbsp dlya vsih l R displaystyle lambda in mathbb R nbsp Ce mozhlivo todi i tilki todi koli yiyi diskriminant 4 x y 2 4 x 2 y 2 displaystyle 4 langle x y rangle 2 4 x 2 y 2 nbsp ne bilshij vid nulya Zvidki otrimuyemo x y x y displaystyle langle x y rangle leq x cdot y nbsp Chastinnij vipadok Redaguvati Linijnij prostir R n displaystyle mathbb R n nbsp Redaguvati V linijnomu prostori R n displaystyle mathbb R n nbsp z vvedenim skalyarnim dobutkom x y i 1 n x i y i displaystyle langle x y rangle sum limits i 1 n x i y i nbsp nerivnist Koshi Bunyakovskogo mozhna dovesti i po inshomu zokrema tak i 1 n j 1 n x i y j x j y i 2 i 1 n x i 2 j 1 n y j 2 j 1 n x j 2 i 1 n y i 2 2 i 1 n x i y i j 1 n x j y j displaystyle sum limits i 1 n sum limits j 1 n left x i y j x j y i right 2 sum i 1 n x i 2 sum j 1 n y j 2 sum j 1 n x j 2 sum i 1 n y i 2 2 sum i 1 n x i y i sum j 1 n x j y j nbsp abo pislya zvedennya odnakovih dodankiv 1 2 i 1 n j 1 n x i y j x j y i 2 i 1 n x i 2 i 1 n y i 2 i 1 n x i y i 2 displaystyle frac 1 2 sum i 1 n sum j 1 n left x i y j x j y i right 2 sum i 1 n x i 2 sum i 1 n y i 2 left sum i 1 n x i y i right 2 nbsp Oskilki liva chastina ostannoyi totozhnosti zavzhdi ye nevid yemnoyu bo ye sumoyu kvadrativ to prava takozh prijmaye nevid yemni znachennya zvidki negajno sliduye nerivnist Koshi Shvarca v linijnomu prostori R n displaystyle mathbb R n nbsp i 1 n x i 2 i 1 n y i 2 i 1 n x i y i 2 0 displaystyle sum i 1 n x i 2 sum i 1 n y i 2 left sum i 1 n x i y i right 2 geq 0 nbsp Najvidomishi zastosuvannya nerivnosti Koshi Bunyakovskogo RedaguvatiNerivnist trikutnika Redaguvati x y 2 x y x y x 2 x y y x y 2 x 2 2 x y y 2 x 2 2 x y y 2 x y 2 displaystyle begin aligned x y 2 amp langle x y x y rangle amp x 2 langle x y rangle langle y x rangle y 2 amp leq x 2 2 langle x y rangle y 2 amp leq x 2 2 x y y 2 amp left x y right 2 end aligned nbsp dobuvshi korin z obidvoh chastin otrimayemo nerivnist trikutnika Matematichni olimpiadi Redaguvati Na matematichnih olimpiadah chasto vikoristovuyut naslidok z nerivnosti Koshi Bunyakovskogo dlya linijnogo prostoru R n displaystyle mathbb R n nbsp dlya dodatnih dijsnih a 1 a 2 a n b 1 b 2 b n displaystyle a 1 a 2 ldots a n b 1 b 2 ldots b n nbsp a 1 2 b 1 a 2 2 b 2 a n 2 b n a 1 a 2 a n 2 b 1 b 2 b n displaystyle dfrac a 1 2 b 1 dfrac a 2 2 b 2 ldots dfrac a n 2 b n geq dfrac a 1 a 2 ldots a n 2 b 1 b 2 ldots b n nbsp Nerivnist negajno sliduye z nerivnosti Koshi Shvarca yaksho poklasti x i a i 2 b i y i b i displaystyle x i sqrt dfrac a i 2 b i quad y i sqrt b i nbsp Zokrema danu nerivnist mozhna vikoristati dlya dovedennya nerivnosti Nesbita z nerivnostej Koshi Shvarca i troh kvadrativ otrimuyemo a 2 a b c b 2 b a c c 2 c a b a b c 2 2 a b b c a c 3 2 displaystyle dfrac a 2 a b c dfrac b 2 b a c dfrac c 2 c a b geq dfrac a b c 2 2 ab bc ac geq dfrac 3 2 nbsp z chogo negajno sliduye nerivnist Nesbita Dzherela RedaguvatiE Bekkenbah R Bellman 1965 Neravenstva Moskva Mir V I Andrijchuk B V Zabavskij 2008 Linijna algebra Lviv Vidavnichij centr LNU imeni Ivana Franka ISBN 978 966 613 623 0 Div takozh Redaguvati nbsp Portal Matematika Skalyarnij dobutok Otrimano z https uk wikipedia org w index php title Nerivnist Koshi Bunyakovskogo amp oldid 36840650
