Sublinear Inequality

Question

For $f(x,y)=x^2/y$, is it possible to prove that $$f(x_1,y_1)-f(x_2,y_2)\le Cf(x_1-x_2,y_1-y_2)$$ or $$|f(x_1,y_1)-f(x_2,y_2)|\le C|f(x_1-x_2,y_1-y_2)|$$ with $0< x_k,y_k \le M<\infty$, $k=1,2$, where $C$ is a positive constant? Thanks for any suggestions.

Answer 1

The following is true \begin{eqnarray*} 0 \leq (x_1 y_2 - x_2 y_1)^2. \end{eqnarray*} Rearrange & create some terms \begin{eqnarray*} &x_1^2y_1y_2 - x_1^2y_2^2 &-x_2^2y_1^2 +x_2^2y_1y_2 &\leq x_1^2y_1y_2 - 2x_1x_2y_1y_2 +x_2^2y_1y_2, \\ &x_1^2 y_2(y_1-y_2)&-x_2^2y_1(y_1-y_2) &\leq (x_1-x_2)^2 y_1 y_2. \end{eqnarray*} Now assume $y_1 > y_2 >0 $,so we can divide by $y_1y_2(y_1-y_2)$ and we have \begin{eqnarray*} \frac{x_1^2}{y_1} -\frac{x_2^2}{y_2} \leq \frac{(x_1-x_2)^2 }{y_1- y_2} . \end{eqnarray*}