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Inequalities
My favorite topic, saved for last.
Trivial Inequality
For any real ,
, with equality iff
.
Arithmetic Mean/Geometric Mean Inequality
For any set of real numbers ,
with equality iff
.
Cauchy-Schwarz Inequality
For any real numbers and
, the following holds:
Cauchy-Schwarz Variation
For any real numbers and positive real numbers
, the following holds:
.
Power Mean Inequality
Take a set of functions .
Note that does not exist. The geometric mean is
.
For non-negative real numbers
, the following holds:
for reals
.
, if is the quadratic mean,
is the arithmetic mean,
the geometric mean, and
the harmonic mean.
RSM-AM-GM-HM Inequality
For any positive real numbers :
with equality iff .
Chebyshev's Inequality
Given real numbers and
, we have
.
Minkowski's Inequality
Given real numbers and
, the following holds:
Nesbitt's Inequality
For all positive real numbers ,
and
, the following holds:
.
Schur's inequality
Given positive real numbers and real
, the following holds:
.
Jensen's Inequality
For a convex function and real numbers
and
, the following holds:
Holder's Inequality
For positive real numbers , the following holds:
Muirhead's Inequality
For a sequence that majorizes a sequence
, then given a set of positive integers
, the following holds:
Rearrangement Inequality
For any multi sets and
,
is maximized when
is greater than or equal to exactly
of the other members of
, then
is also greater than or equal to exactly
of the other members of
.
Newton's Inequality
For non-negative real numbers and
the following holds:
,
with equality exactly iff all are equivalent.
MacLaurin's Inequality
For non-negative real numbers , and
such that
, for
the following holds:
with equality iff all are equivalent.
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