An upper bound on the logarithm of factorial

Question

Does the following inequality hold:

$$ \ln(x!) \leq x \cdot \ln^{c}(x) $$

where $0<c<1$ is a real number?

I tried to use stirling approximation but failed.

Answer 1

This inequality is not true. Indeed, it is equivalent to $$ \frac{1}{n}\sum_{k=2}^n \ln k\leq \ln^c n. $$ Note that $$ \frac{1}{n}\sum_{k=2}^n\ln k\geq \frac{1}{n}\sum_{k=n/2}^n\ln k\geq\frac{n}{4}\frac{1}{n}\ln (n/2). $$ If the original inequality was true, we could deduce $$ \frac{\ln (n/2)}{4}\leq \ln^c n, $$ which is impossible for $0<c<1$.