Bounding the maximum of a sequence of continuous functions using integrals

I have absolutely non clue on how to solve this one.

First, recall that

$$\lim_{p \rightarrow +\infty }\left ({\int\limits_a^b |f(x)|^{p}dx)} \right) ^{\frac{1}{p}} = \max_{x \in [a,b]} |f(x)|$$

Have $$f_n:[a,b] \rightarrow \mathbb{R}$$ there exist constants $$\gamma > 1$$, $$\beta > 0$$, independent of $$n$$, where

$$\left ({\int\limits_a^b |f_n(x)|^{\gamma p}dx)} \right) ^{\frac{1}{\gamma}} \leq p \cdot\beta^{\frac{1}{p}} \cdot \left ({\int\limits_a^b |f_n(x)|^{p}dx)} \right) ^{1 -\frac{1}{p}}$$

for any $$p > 1$$. Then, there exists a constant $$C$$, independent of $$n$$ such that

$$\max_{x \in [a,b]} |f_n(x)| \leq C \left ( { \left({\int\limits_a^b |f_n(x)|^{2}dx} \right) ^{\frac{1}{2}} + 1}\right)$$

There is a hint that we should at least consider the case of $$p = p' \cdot \gamma$$