The is the standard version of Young's inequality $$ab \leq \frac{a^p}{p}+\frac{b^q}{q}$$ for $a,b,p,q >0$ and $$\frac{1}{p}+\frac {1}{q}=1$$
But there is another formula called generalization of Young inequality. Why it's a generalization?
let $f$ denote a real-valued, continuous and strictly increasing function on $[0, c]$ with $c > 0$ and $f(0) = 0$.
Let $f^{−1}$ denote the inverse function of $f$. Then, for all $a ∈ [0, c]$ and $b ∈ [0, f(c)]$,
$$ab \le \int_{0}^{a} f(x)dx + \int_{0}^{b}f^{-1}(x)dx$$
$$\frac 1p + \frac 1q=1 \implies (p-1)(q-1)=1$$
Now let $$f(x)=x^{p-1}$$ $$f^{-1}(x)=x^{\frac 1{p-1}}=x^{q-1}$$