Inequality for locally Lipschitz continuous function.

Question

Assume that function is locally Lipschitz continuous, i.e. $$|f(x) - f(y)| \leq L |x-y| \left(1+|x|^s+|y|^s\right)$$ for all $x \in R^m$ and some $L,s>0$. Then it should be true that $$\left|f(x) - f(y)\right| \leq M \left|x-y\right| \left(1+|x|^s\right) + M|x-y|^{s+1}.$$ However I cannot see a way to prove it. Any help appreciated.

Answer 1

Case I: suppose that $|y| < 2|x-y|$. Then $$|f(x) - f(y)| \le L|x-y|(1+|x|^s) + 2^s L|x-y|^{1+s}.$$ Case II: suppose that $|y| \ge 2|x-y|$. Then $$|y| \le |x| + |x-y| \le |x| + \frac 12 |y|$$ so that $$|y| \le 2|x|$$ and thus $$(1 + |x|^s + |y|^s) \le (1 + 2^s)(1 + |x|^s)$$ and $$|f(x) - f(y)| \le (1+2^s)L |x-y| (1 + |x|^s).$$