Putnam 1961 A2

For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$ , define $f$ to be linearly bounded if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.