# Lipschitz-continuity of a particular function

I have the following question. Let $g_1,\ldots,g_k: \mathbb{R}^n\rightarrow \mathbb{R}$ be Lipschitz continuous (with respective constants $L_1,\ldots,L_k>0$). How can I proove the Lipschitz-continuity of the function $h(x):=\left(\sum_{i=1}^k |g_i(x)|^d\right)^{\frac{1}{d}}$ for $d>1$?

I don't know how to proceed here in order to use the Lipschitz condition for the functions $g_i$: $|h(x)-h(y)|=\left|\left(\sum_{i=1}^k |g_i(x)|^d\right)^{\frac{1}{d}}-\left(\sum_{i=1}^k |g_i(y)|^d\right)^{\frac{1}{d}} \right|$

• Are you aware of $p$-norms? Of the Minkowski inequality? – Omnomnomnom Jun 18 '16 at 11:00
• Hello, of course I am aware of the Minkowski inequality. Unfortunately, I did not manage to apply it here (since I tried to merge the two sums somehow which was not successfull). But now, I saw the trick $g_i(x)=g_i(x)-g_i(y)+g_i(y)$. Afterwards, the Minkowski inequality can be applied and the new Lipschitz constant should result to $\left( \sum_{i=1}^k L_i^d\right)^{\frac{1}{d}}$, right? – Phil Jun 18 '16 at 11:38
• That's right. In particular, we have $$\left|\left(\sum_{i=1}^k |g_i(x)|^d\right)^{\frac{1}{d}}-\left(\sum_{i=1}^k |g_i(y)|^d\right)^{\frac{1}{d}} \right| \leq \left(\sum_{i=1}^k |g_i(x) - g_i(y)|^d\right)^{\frac{1}{d}}$$ By the triangle inequality – Omnomnomnom Jun 18 '16 at 12:16
• Ok, thank you very much. Then this problem is solved! – Phil Jun 18 '16 at 12:36
• great! If you want, you can write up an answer and accept it to resolve this question post – Omnomnomnom Jun 18 '16 at 14:07

On the basis of the previous comments I was able to find the answer. As mentioned there, the trick $g(x)=g(x)-g(y)+g(y)$ and the triangle inequality lead to the following: $|h(x)-h(y)|=\left|\left(\sum_{i=1}^k |g_i(x)|^d\right)^{\frac{1}{d}}-\left(\sum_{i=1}^k |g_i(y)|^d\right)^{\frac{1}{d}} \right| \leq \left(\sum_{i=1}^k |g_i(x) - g_i(y)|^d\right)^{\frac{1}{d}} \leq \left(\sum_{i=1}^k L_i|x-y|^d\right)^{\frac{1}{d}} = \left( \sum_{i=1}^k L_i^d\right)^{\frac{1}{d}}\cdot |x-y|$.