An exercise in my textbook asked to prove the following inequality, valid for all $a,b,c,d \in R $
$(a/2 + b/3 + c/12 + d/12)^4 \leq a^4/2 + b^4/3 + c^4/12 + d^4/12$
There is a straightforward proof using Convex Functions:
- $f(x) = x^4$ is a convex function satisfying $f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1 - \lambda)f(y)$ for all $x,y \in R$ and $\lambda \in [0,1]$
- Since $1/2 + 1/3 + 1/12 + 1/12 = 1$, we can use the convexity property to obtain the inequality.
Since this question was on the chapter about Convex Functions, I was able to find the solution quickly. However, had I seen the problem in a "standalone" manner I would probably take longer to solve it, and at least spend a lot of muscle opening up the left hand term :)
My question is: What would be other ways to obtain this same result? What if someone had shown me this problem back when I was in eight grade?