TITLE: Examples of inequality implied by equality. 
QUESTION [8 upvotes]: It is well known Cauchy's inequality is implied by Lagrange's identity. Bohr's inequality  $|a -b|^2 \le p|a|^2 +q|b|^2$, where $\frac{1}{p}+\frac{1}{q}=1$, is implied by $|a -b|^2 +|\sqrt{p/q}a+\sqrt{q/p}b|^2= p|a|^2 +q|b|^2$. L.K Hua's determinant inequality $\det(I-A^*A)\cdot \det(I-B^*B)\le |\det(I-A^*B)|^2$ is implied by Hua's matrix equality $(I-B^*B)-(I-B^*A)(I-A^*A)^{-1}(I-A^*B)=-(A-B)^*(I-AA^*)(A-B)$. What other examples can be found?

REPLY [3 votes]: Two examples due to Hurwitz. 

The AM-GM inequality.
For the function $f=f(x_1,x_2,\dots,x_n)$ let $Pf(x_1,x_2,\dots,x_n)$ denote the sum of $f$ over the $n!$ quantities that result from all possible $n!$ permutations of the $x_i$. Then
$$\frac{x_1^n+x_2^n+\dots+x_n^n}{n}-x_1x_2\dots x_n=\frac{1}{2\ n!}(\phi_1+\phi_2+
\dots \phi_n),$$
where
$$\phi_k=P[(x_1^{n-k}-x_2^{n-k})(x_1-x_2)x_3x_4\dots x_{k+1}]=P[(x_1-x_2)^2(x_1^{n-k-1}+\dots x_2^{n-k-1})x_3x_4\dots x_{k+1}]\geq0.$$
The proof can be found in Inequalities by Beckenbach and Bellman. 
The isoperimetric inequality. Let the boundary of $\Omega\subset \mathbb R^2$ be a rectifiable Jordan curve $\partial \Omega=\{((x(s),y(s))|\ s\in[0,2\pi))\}$. Then 
$$L^2-4\pi A=2\pi^2\sum\limits_{n=1}^{\infty}\left[(na_n-d_n)^2+(nb_n+c_n)^2+
(n^2-1)(c_n^2+d_n^2)\right],$$
where
$$x(s)=\sum\limits_{n=0}^{\infty}(a_n\cos ns+b_n\sin ns),\quad  y(s)=\sum\limits_{n=0}^{\infty}(c_n\cos ns+d_n\sin ns).$$