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Prob 6 - ans 4

# Prob 6 - ans 4 - A Note on some inequalities A basic...

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A Note on some inequalities. A basic inequality is Young’s inequality : If a , b are nonnegative numbers and 1 p < then ab a p p + b q q , 1 p + 1 q = 1 . This inequality can be proven by elementary means (when b 6 = 0 let x = a p - 1 /b and then ﬁnd the minimum of the function f ( x ) = x/p + ( p - 1) x - 1 / ( p - 1) /p ). One consequence of Young’s inequality is H¨older’s inequality . For vectors this inequality says n X j =1 x j y j ≤ k x k p k y k q , 1 p + 1 q = 1 . To prove this we use Young’s inequality | x j | k x k p | y j | k y k p 1 p | x j | | x j | p + 1 q | y j | | y j | q . When we sum over i we have H¨older’s inequality for vectors. Similarly H¨older’s inequalityfor functions reads Z Ω f ( x ) g ( x ) dx ≤ k f k p k g k q , 1 p + 1 q = 1 . Here k f k p = p s Z Ω | f ( x ) | p dx. To prove this inequality we do essentially the same thing:

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Prob 6 - ans 4 - A Note on some inequalities A basic...

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