Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if possible - to get from $f$ to $g$ or vice versa :
$$ \max \int_{-\infty}^{\infty} f(x) dx = \min \int_{-\infty}^{\infty} g(x) dx $$
Where both sides converge , and a given condition for the other ( if the condition was for $f$ , then we get one for $g$. Or vice versa ).
I considered Cauchy-Schwartz , contour-integration and substitution but without succes.
How to handle this ?