Let $2 \leq q \leq 4$ and $N<T^{1/2}$.
On page 130 of Ten lectures on the interface between analytic number theory and harmonic analysis, we suppose we have
$\int_{0}^T \left| \sum_{n=1}^Nn^{-it} \right|^4 dt\ll TN^2 (\log N)^3.$
Then why does Holder's inequality allows us to conclude
$\int_{0}^T \left| \sum_{n=1}^Nn^{-it} \right|^q dt\ll TN^{q/2} (\log N)^{3q/4}.$
You can probably apply Holder's inequality to the product of the function $1$ and the integrand, but rather than working out the exponents I prefer to use Jensen's inequality:
Since $q\leq 4$, it follows from Jensen's inequality that $$ \Big[\frac{1}{T}\int_0^T\Big|\sum_{n=1}^Nn^{-it}\Big|^q\;dt\Big]^{\frac{1}{q}}\leq \Big[\frac{1}{T}\int_0^T\Big|\sum_{n=1}^Nn^{-it}\Big|^4\;dt\Big]^{\frac{1}{4}}\ll N^{\frac{1}{2}}(\log N)^{\frac{3}{4}}$$
Raising both sides to the $q$th power and multiplying by $T$ then yields the desired result.