Inegalitatea lui Holder
Posted: Thu Feb 26, 2009 9:33 pm
Pentru tripletele \( (a,b,c),\: (m,n,p), \: (x,y,z) \) de numere pozitive avem urmatoarea inegalitate:
Teorema(Holder):\( \left(a^{3}+b^{3}+c^{3}\right)\left(m^{3}+n^{3}+p^{3}\right)\left(x^{3}+y^{3}+z^{3}\right)\ge\left(amx+bny+cpz\right)^{3} \).
Mai general:
Pentru \( m \) secvente de numere reale pozitive \( \left(a_{1,1},a_{1,2},\ldots,a_{,1,n}\right),\:\left(a_{2,1},a_{2,2},\ldots,a_{2,n}\right),\ldots,\left(a_{m,1},a_{m,2},\ldots,a_{m,n}\right) \) avem ca:
\( \prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i,j}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{j=1}^{n}a_{i,j}}\right) \).
Teorema(Holder):\( \left(a^{3}+b^{3}+c^{3}\right)\left(m^{3}+n^{3}+p^{3}\right)\left(x^{3}+y^{3}+z^{3}\right)\ge\left(amx+bny+cpz\right)^{3} \).
Mai general:
Pentru \( m \) secvente de numere reale pozitive \( \left(a_{1,1},a_{1,2},\ldots,a_{,1,n}\right),\:\left(a_{2,1},a_{2,2},\ldots,a_{2,n}\right),\ldots,\left(a_{m,1},a_{m,2},\ldots,a_{m,n}\right) \) avem ca:
\( \prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i,j}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{j=1}^{n}a_{i,j}}\right) \).