Upper bound for the even power of a trinomial

Question

It's easy to get the following estimate for a trinomial to the power of $2$, $$ (a + b + c)^2 \leq 3a^2 + 3b^2 + 3c^2. $$ Is there any analogous result for even powers, this is, for the more general case $(a + b + c)^{2n}$, with $n\in\mathbb{N}$?

Answer 1

Depends on what you mean by "analogous", best I can think of is using the Cauchy–Schwarz inequality :

|<u,v>|² $\leq$ <u,u><v,v>

It applies to any pre-hilbert space , so you can produce "analogous" results by varying either the dimension of the space or the definition of inner product before appliying the inequality to well chosen vectors.

For example, take the euclidian space of dimension 3 with the usual definition of inner product. Let u=(x,y,z) and v=(1,1,1)

Applying the inequality gives : (x+y+z)² $\leq$ 3.(x²+y²+z²)

Other spaces will give you other inequalities.

Hope it helps :)