Ref: Titu Andreescu and Zumin Feng.
Problem:
Let x, y, z be positive real numbers such that . Determine the minimum value of
Solution:
An application of Cauchy–Schwarz inequality makes this as a one step problem. Nevertheless, we present a proof which involves only the easier inequality for real numbers x and y by setting first
and
and second
and
.
Clearly, z is a real number in the interval . Hence, there is an angle a such that
. Then,
, or
. For an angle b, we have
. Hence, we can set
, and
for some b, it suffices to find the minimum value of
or
Expanding the right hand side gives
Equality holds when and
, which implies that
and
. Because
, equality holds when
and
, that is,
,
,
.
More later.
Nalin Pithwa