show that: for any positive numbers $k$ and $N$, have $$\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\dfrac{1}{q}$$
where $\displaystyle\sum_{q\le N}$ is meaning no more than $N$ prime power q summation(including $q=1$),and Let $\omega{(n)}$ denote the number of distinct prime factors of a positive integer $n$
maybe this problem background is K-th mean value Estimate of Number of prime divisors of integer see http://www.doc88.com/p-703867145586.html
Thank you,This is 2014 china TST test problem , maybe is old reslut?
Thank you