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Spain Contests
Spain Mathematical Olympiad
1986 Spain Mathematical Olympiad
4
4
Part of
1986 Spain Mathematical Olympiad
Problems
(1)
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
8/2/2018
Denote by
m
(
a
,
b
)
m(a,b)
m
(
a
,
b
)
the arithmetic mean of positive real numbers
a
,
b
a,b
a
,
b
. Given a positive real function
g
g
g
having positive derivatives of the first and second order, define
μ
(
a
,
b
)
\mu (a,b)
μ
(
a
,
b
)
the mean value of
a
a
a
and
b
b
b
with respect to
g
g
g
by
2
g
(
μ
(
a
,
b
)
)
=
g
(
a
)
+
g
(
b
)
2g(\mu (a,b)) = g(a)+g(b)
2
g
(
μ
(
a
,
b
))
=
g
(
a
)
+
g
(
b
)
. Decide which of the two mean values
m
m
m
and
μ
\mu
μ
is larger.
function
mean
algebra