Hölder Inequality: Let $\left\{a_{1}, a_{2}, \cdots, a_{n} \right\},$ $\left\{b_{1}, b_{2}, \cdots, b_{n}\right\} ,$ $ \cdots\cdots,$ $ \left\{l_{1}, l_{2}, \cdots, l_{n}\right\}$ be $l $ sets of positive real numbers and $\alpha, \beta, \cdots,\lambda$ be positive rational numbers such that $\alpha+\beta+\cdots+\lambda=1 .$ Then$$ a_{1}^{\alpha} b_{1}^{\beta} \cdots l_{1}^{\lambda}+a_{2}^{\alpha} b_{2}^{\beta} \cdots l_{2}^{\lambda}+\cdots + a_{n}^{\alpha} b_{n}^{\beta} \cdots l_{n}^{\lambda}\leq \left( a_{1}+a_2+\cdots+a_{n} \right)^{\alpha} \left( b_{1}+b_2+\cdots +b_{n}\right)^{\beta} \cdots\left(l_{1}+l_{2}+\cdots+l_{n}\right)^{\lambda}$$The equality occurs when $ a_i,b_i, \cdots,l_i $ are proportional.
Now what should be the integral version of this. I guessed that
Let $f_1,f_2,\cdots, f_n$ are n functions who are positive in the region $[a,b] $. And $p_1, p_2, \cdots,p_n$ be positive rational numbers such that $p_1+p_2+\cdots+p_n=1 .$ Then$$\prod_{k=1}^n \left[\left(\int_a^b f_k (x)dx \right)^{p_k}\right]\geq \int_a^b\left[ \prod_{k=1}^n\left(f_k (x)\right)^{p_k}\right]dx$$
I proved this in this way:-
\begin{align*} & \sum_{k=1}^n \left[\frac{p_kf_k}{\displaystyle{\int_a^b f_k (x)dx}}\right]\geq \prod_{k=1}^n \left[\left(\frac{f_k (x)}{\displaystyle{\int_a^b f_k (x)dx}} \right)^{p_k}\right]\\ \implies & \int_a^b \left[\sum_{k=1}^n \left[\frac{p_kf_k(x)}{\displaystyle{\int_a^b f_k (x)dx}}\right] \right]dx \geq \int_a^b \left[ \prod_{k=1}^n \left[\left(\frac{f_k (x)}{\displaystyle{\int_a^b f_k (x)dx}} \right)^{p_k}\right]\right]dx\\ \implies & \sum_{k=1}^n \left[\frac{\displaystyle{p_k\int_a^bf_k(x)dx}}{\displaystyle{\int_a^b f_k (x)dx}}\right]\geq \frac{\displaystyle{\int_a^b \left[ \prod\limits_{k=1}^n \left( f_k (x)\right)^{p_k}\right]dx }}{\displaystyle{\prod\limits_{k=1}^n \left[\left(\int_a^b f_k (x)dx \right)^{p_k}\right]}}\\ \implies & \prod\limits_{k=1}^n \left[\left(\int_a^b f_k (x)dx \right)^{p_k}\right] \geq \int_a^b\left[ \prod_{k=1}^n\left(f_k (x)\right)^{p_k}\right]dx \end{align*}
This works great but i am not sure about the statement of the integral version of inequality. What if one of the integration gives negative value or what if any function gives negative value at some points in the region $[a,b] $.
Please tell me what is the actual statement of the integral version of Hölder Inequality. If the statement is different from my guess then tell me the proof.
