Looking at wikipedia I noted the following inequality:
If $f$ and $g$ are Riemann-integrable on $[a,b]$ then $f \circ g$ $$\left( \int_{a}^{b} (f \circ g)(x) dx \right)^2 \leq \left( \int_{a}^{b} f(x)^2 dx \right) \left( \int_{a}^{b} g(x)^2 dx \right)$$
NOTE: In fact the inequality was $$\left( \int_{a}^{b} (f g)(x) dx \right)^2 \leq \left( \int_{a}^{b} f(x)^2 dx \right) \left( \int_{a}^{b} g(x)^2 dx \right)$$ so I confused the product for the composition. The following continues with the notion of composition.
I've toyed with the following (putative) generalization:
$$\left( \int_{a}^{b} \left( f_1 \circ f_2 \cdots \circ f_n \right)(x) dx \right)^n \leq \prod_{j=1}^{n} \left( \int_{a}^{b} |f_j(x)|^n dx \right)$$
but I may have found a counterexample. Consider choosing $f_j(x) = x^2$ for all $j \in \{ 1,\cdots,n\}$, then
$$\left( \int_a^b \left( \left( \left(x^2 \right)^2 \right)^{\vdots} \right)^2 dx \right)^n \leq \prod_{j=1}^{n} \left( \int_{a}^{b} |x^2|^n dx \right)$$
$$\iff$$
$$\left( \int_a^b x^{2^n} dx \right)^n \leq \prod_{j=1}^{n} \left( \int_{a}^{b} x^{2n} dx \right)$$
$$\iff$$
$$\left( \frac{x^{2^n+1}}{2^n+1} \Big|_{a}^{b} \right)^n \leq \prod_{j=1}^{n} \left(\frac{x^{2n+1}}{2n+1} \Big|_{a}^{b} \right)$$
$$\iff$$
$$\left( \frac{b^{2^n+1}}{2^n+1} - \frac{a^{2^n+1}}{2^n+1} \right)^n \leq \prod_{j=1}^{n} \left(\frac{b^{2n+1}}{2n+1} - \frac{a^{2n+1}}{2n+1}\right)$$
$$\iff$$
$$\left(\frac{1}{2^n+1} \right)^n \left( b^{2^n+1} - a^{2^n+1} \right)^n \leq \left(\frac{1}{2n+1}\right)^n \left(b^{2n+1} - a^{2n+1}\right)^n$$
Perhaps comparisons on certain exponential parts on the left with certain linear parts of the right would provide a more general refutation, but consider a simple counter example of $(a,b,n) = (1,2,3)$.
$$\left(\frac{1}{2^3+1} \right)^3 \left( 2^{2^3+1} - 1^{2^3+1} \right)^3 \leq \left(\frac{1}{2(3)+1}\right)^3 \left(2^{2(3)+1} - 1^{2(3)+1}\right)^3$$
$$\iff$$
$$\frac{133432831}{729} \leq \frac{2048383}{343}$$
$$\iff$$
$$183035.4334705075 \leq 5971.962099125363$$
which is clearly false. Was I successful in refuting the inequality?