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Taiwan Quiz 2014:
Prove that for positive reals $a$, $b$, $c$ we have
$$3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$$
Applying Power-mean inequality (weighted) in $$a_1={abc} , a_2={\frac{a^3+b^3+c^3}{3}}$$ with $r=1$, $s=\dfrac{1}{3}$ and weights $\dfrac{8}{9}, \dfrac{1}{9}$ ...
In mathematics, generalized means (or power mean, or Hölder mean) are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).
== Definition ==
If p is a non-zero real number, and
x
1
,
…
,
x
n
{\displaystyle x_{1},\dots ,x_{n}}
are positive real numbers, then the generalized mean or power mean with exponent p of th...
0
2
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.
== Motivation ==
Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring R has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules is more general and gives useful information. Because of this, one often studies...
