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compare (a+b)/2 with \mu (a,b) i.e. mean value of a and b with respect to g

Source: Spanish Mathematical Olympiad 1986 P4

August 2, 2018

functionmeanalgebra

Problem Statement

Denote by

m(a,b)

the arithmetic mean of positive real numbers

a,b

. Given a positive real function

g

having positive derivatives of the first and second order, define

\mu (a,b)

the mean value of

a

and

b

with respect to

g

2g(\mu (a,b)) = g(a)+g(b)

. Decide which of the two mean values

m

and

\mu

is larger.