I need to proof a property of an object of statistic.
I would have finished successfully if it was not for me to use an upper bound of a power of a binomial that i found in my book that I can't proof :(
$|x+y|^k ≤ 2^{k-1}(|x|^k+|y|^k)$
I've tried differents roads but unseccesfuly, my "best results" was
$2^{k-1} ≤ \frac{x^k\sum\limits_{i=1}^k{{{k}\choose{i}}\frac{y^i}{x^i}}}{x^k + y^k}$
That i can't simpplify so... Help please.
Assume $k \geq 1$, $x\geq 0, y \geq 0$, Take $$ f(x) = x^k. $$
$f$ is convex, hence by convexity, for $0\leq t \leq 1$, $$ f(a t + b (1-t)) \leq f(a)t+f(b)(1-t) $$
Now take $a=x, b=y, t=1/2$ and simplify will get $$ (\frac{x+y}{2})^k \leq2^{-1}(x^k+y^k) $$
Simplifying lead to the inequality. It's now not hard to extend to the case with $x,y$ possibly negative by noting $|x+y|\leq |x| + |y|$.