# Young’s inequality

ចំពោះចំនួនវិជ្ជមាន $a_1 ; a_2 ; ... ; a_n$ និង $p_1 ; p_2 ; ... ; p_n > 1 ; \forall{n\in N^{*}}$

ហើយផ្ទៀងផ្ទាត់: $\displaystyle \frac{1}{p_1} + \frac{1}{p_2} + ... + \frac{1}{p_n} = 1$ គេបាន :

$\displaystyle \frac{{a_1}^{p_1}}{p_1} + \frac{{a_2}^{p_2}}{p_2} + ... + \frac{{a_n}^{p_n}}{p_n}\geq a_1a_2 ... a_n$

## សំរាយបញ្ជាក់

របៀបទី 1

តាង $f(x) = lnx ; x > 0$

$\displaystyle f'(x) = \frac{1}{x}$

$\displaystyle f''(x) = -\frac{1}{x^2} < 0$

នាំអោយ $f(x)$ ជាអនុគមន៍ផតលើ $(o ; +\infty)$

ដោយ $\displaystyle \frac{1}{p_1} + \frac{1}{p_2} + ... + \frac{1}{p_n} = 1$

តាមវិសមភាព Jensen គេបាន:

$\displaystyle \frac{1}{p_1}.f({a_1}^{p_1}) + \frac{1}{p_2}.f({a_2}^{p_2}) + ... + \frac{1}{p_n}.f({a_n}^{p_n})$

$\displaystyle \leq f\biggl(\frac{1}{p_1}.{a_1}^{p_1} + \frac{1}{p_2}.{a_2}^{p_2} + ... + \frac{1}{p_n}.{a_n}^{p_n}\biggl)$

$\displaystyle \frac{1}{p_1}.ln({a_1}^{p_1}) + \frac{1}{p_2}.ln({a_2}^{p_2}) + ... + \frac{1}{p_n}.ln({a_n}^{p_n})$

$\displaystyle \leq ln\biggl(\frac{1}{p_1}.{a_1}^{p_1} + \frac{1}{p_2}.{a_2}^{p_2} + ... + \frac{1}{p_n}.{a_n}^{p_n}\biggl)$

$\displaystyle ln(a_1a_2 ... a_n)\leq ln\biggl(\frac{1}{p_1}.{a_1}^{p_1} + \frac{1}{p_2}.{a_2}^{p_2} + ... + \frac{1}{p_n}.{a_n}^{p_n}\biggl)$

$\displaystyle \frac{1}{p_1}.{a_1}^{p_1} + \frac{1}{p_2}.{a_2}^{p_2} + ... + \frac{1}{p_n}.{a_n}^{p_n}\geq a_1a_2 ... a_n$

## សូមមើលផងដែរ

• ### វិសមភាព Polya

ត្រឡប់មកទំព័រ វិសមភាព Holder

បន្តទៅទំព័រ វិសមភាព Minkowski