Given real number $x_1, x_2, \ldots, x_n$ such that $x_1 + x_2 + \ldots +x_n>0$, must this other inequality be true?

Given real numbers $x_1, x_2, \ldots, x_n$ such that $$x_1 + x_2 + \ldots + x_n >0$$ with the condition that $\max\{|x_i|\}_{i=1}^n = \max\{x_i\}_{i=1}^n$, must it be true that $${x_1}^m + {x_2}^m + \ldots + {x_n}^m > 0$$ for all natural numbers $m$? If so, can you prove it?

I don't know how to approach the problem. Is there any generalization of the power-mean inequality to real numbers that might help? The inequality clearly holds true for all even $m$ from the Trivial Inequality, but the case for odd $m$ feels difficult.

Thanks, I appreciate any help!

• Well, no: Consider $x_1=3,\ x_2=x_3=x_4=\frac12,\ x_5=-1,\ x_6=-3$. Then the sum for odd values of $m$ is $\frac{3}{2^m}-1$. – user228113 Oct 30 '16 at 15:48