What are the minimums of $x^y+y^x$ and $x^y+y^z+z^x$?

Question

What are the minima of $x^y+y^x$ and $x^y+y^z+z^x$ on $\{x,y,z\}\subset\mathbb R^+ $?

I was able to show that $x^y+y^x > 1$, but is this the best possible bound? Thanks.

Answer 1

We'll prove that $$\inf_{\{x,y,z\}\subset(0,+\infty)}(x^y+y^z+z^x)=1.$$ Indeed, we can assume that $\{x,y,z\}\subset(0,1)$.

Now, by Bernoulli $$\left(\frac{1}{x}\right)^y=\left(1+\frac{1}{x}-1\right)^y<1+y\left(\frac{1}{x}-1\right)=\frac{x+y-xy}{x}.$$ Thus, $$\sum_{cyc}x^y>\sum_{cyc}\frac{x}{x+y-xy}>\sum_{cyc}\frac{x}{x+y}> \sum_{cyc}\frac{x}{x+y+z}=1$$ and since for $y=0$ and $x\rightarrow+\infty$ we get $\sum\limits_{cyc}x^y\rightarrow1$, we are done!