Infimum and supremum of $\frac{x}{y^2+1}+\frac{y}{x^2+1}$ for $x+y=1$ and $x,y\in{\mathbb{R}^{+}}$

Question

I've been trying to find the infimum and supremum of $$\frac{x}{y^2+1}+\frac{y}{x^2+1}$$ for $x+y=1$ and $x,y\in{\mathbb{R}^{+}}$

My only idea was to represent $x$ and $y$ as $\sin^2{z}$ and $\cos^2{z}$ but that didn't help me a lot. I'd like if someone could provide some sort of a hint for this. Thanks.

Answer 1

It can be shown that when $x + y = 1$, $f(x, y) = \dfrac{x}{y^2 + 1} + \dfrac{y}{x^2 + 1}$ reduces to $g(u) = \dfrac{2 - 3 u}{2 - 2 u + u^2}$, where $u = x y \in [0, 1 / 4]$. Can you try to find bounds for $g(u)$ in this situation?