A question regarding inequality

let $0 < x < 1$ and $r,i_1,...,i_n > 0$.$$\frac{x^{i_1}+x^{i_2}+ \ldots +x^{i_n}}{n}\leq x^r$$for some x.

Can we say that the inequality holds as x decreases?

Note that it may not hold for all $0 < x < 1$