How to write this in big O notation?

Question

I have a function $f(m,n)$ for which there exists a constant $\alpha<2$ such that, for fixed $m$, as $n\rightarrow\infty$, we have $f(m,n)\leq\alpha\sqrt{m/n}$, and for fixed $n$, as $m\rightarrow\infty$, we also have $f(m,n)\leq\alpha\sqrt{m/n}$. Is there a proper concise way to express this using big O notation?

Answer 1

Perhaps you meant to say that $\alpha$ depends on $m$ in the first case and $n$ in the latter, but it doesn't matter since $\alpha \lt 2$, we have $f(m,n) \le 2 \cdot \sqrt{\frac{m}{n}}$ or $f(m,n) = O(\sqrt{\frac{m}{n}})$.