Find the inequality with the best possible $k= constant$ (with the condition $x^{2}+ y^{2}\leq k$). 
Find the inequality with the best possible $constant$


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*Given two non-negative numbers $x, y$ so that $x^{2}+ y^{2}\leq \frac{2}{7}$. Prove that
$$\frac{1}{1+ x^{2}}+ \frac{1}{1+ y^{2}}+ \frac{1}{1+ xy}\leq \frac{3}{1+ \left ( \frac{x+ y}{2} \right )^{2}}$$
where $constant= \frac{2}{7}$ is the best possible.

*Given two non-negative numbers $x, y$ so that $x^{2}+ y^{2}\leq \frac{2}{5}$. Prove that
$$\frac{1}{\sqrt{1+ x^{2}}}+ \frac{1}{\sqrt{1+ y^{2}}}+ \frac{1}{\sqrt{1+ xy}}\leq \frac{3}{\sqrt{1+ \left ( \frac{x+ y}{2} \right )^{2}}}$$
where $constant= \frac{2}{5}$ is the best possible.

They are my two examples. I'm looking forward to seeing more many inequalities alike. Thanks for all your nice comments.

For example.

Let $x$ and $y$ be non-negative numbers such that $x^2+y^2\leq\frac{2}{15}.$ Prove that:
$$\frac{1}{1+ x^{2}}+ \frac{1}{1+ y^{2}}+ \frac{1.5}{1+ xy}\leq \frac{3.5}{1+ \left ( \frac{x+ y}{2} \right )^{2}},$$
where $\frac{2}{15}$ is a best such constant.

It's interesting that:

For any non-negatives $x$ and $y$ the following inequality is true.
$$\frac{1}{1+ x^{2}}+ \frac{1}{1+ y^{2}}+ \frac{2}{1+ xy}\geq \frac{4}{1+ \left ( \frac{x+ y}{2} \right )^{2}}$$

By the way, the last inequality is true for any reals $x$ and $y$ such that $xy+1>0.$
