Suppose $(X, \mu)$ is a measure space with finite measure and $f: X \rightarrow \mathbb{R}$ a measurable function and there exists a $c > 0$ such that $\int_{X}e^{c|f|}\, d\mu < \infty$. Why does there exist a constant $C > 0$ such that $\|f\|_{L^{p}(X)} \leq Cp$ for all $1 \leq p < \infty$?
Using the fact that for $n = 1, 2, 3, \ldots$, $c^{n}|f(x)|^{n}/n! \leq e^{c|f(x)|}$, I can show for these $n$, $\|f\|_{L^{n}(X)} \leq Cn$ for some $C > 0$. However, I can't seem to be able to prove this for, say $p \in (n, n + 1)$.