Concave function - Generic proof f(x)<=2f(x/2)<=3f(x/3)<=⋯≤Nf(x/N)

JetssheetaDumcb

JetssheetaDumcb

Answered question

2022-10-31

Concave function - Generic proof
f ( x ) 2 f ( x 2 ) 3 f ( x 3 ) N f ( x N )

Answer & Explanation

Warkallent8

Warkallent8

Beginner2022-11-01Added 16 answers

If f is concave with f ( 0 ) 0 and 0 < a b then
f ( a ) b a b 0 f ( 0 ) + a 0 b 0 f ( b ) a b f ( b ) f ( a ) a f ( b ) b
Setting a = x / v and b = x / u gives
u f ( x u ) v f ( x v )
for 0 < u v and x > 0.

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