Young's Inequality
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Form for Hölder exponents
If are non-negative reals, and
are positive reals that satisfy
, then the following inequality holds for all possible values of
and
.
with equality iff
Form for definite integrals
Suppose is a strictly increasing and continuous function on the interval
where
is a positive real number, and also
.
Then the following inequality holds for all
and
with equality iff
.
Proof
The logarithm is concave and we know that , so by Jensen's Inequality, we have
Young's Inequality then follows by exponentiation of both sides.