Real Analysis: Prove that there exists some positive real number M with the property that f(x, y) ^5 ≤ Mg(x, y)^ 3

Overview

Let f: R^2 → R and Let g: R^2 → R be continuous real-valued functions on R ^2 with the properties that f(kx, ky) = k^3 f(x, y) and g(kx, ky) = k^5 g(x, y) for all real numbers x and y, and for all non-negative real numbers k. Suppose also that f(x, y) and g(x, y) are strictly positive real numbers whenever (x, y) ̸= (0, 0). Prove that there exists some positive real number M with the property that f(x, y)^5 ≤ Mg(x, y)^3 for all real numbers x and y.