$a_n=\min\limits_{x+y=1}(x^n+y^n)=\frac{1}{2^{n-1}}$

Question

We must prove :$\min\limits_{x+y=1}(x^n+y^n)=\frac{1}{2^{n-1}} $ for all $n \in \mathbb{N}_{>0}$

And to prove this we can use the inequality: $\frac{x+y}{2}\leq (\frac{x^n+y^n}{2})^{\frac{1}{n}}$ where equality is satisfied if $x=y=1/2$

My question is how we can prove inquality: $\frac{x+y}{2}\leq (\frac{x^n+y^n}{2})^{\frac{1}{n}}$ ?

Answer 1

This is just convexity of the function $f(x)=x^{n}$. The inequality simply says $f(\frac {x+y} 2) \leq \frac {f(x)+f(y)} 2$.