= Derivative growth outside a finite-measure set
{title2=$\phi'\le\phi^2\quad\text{outside a finite-measure set}$}
For a nonnegative increasing continuously differentiable function not identically zero, choose $x_0$ with $\phi(x_0)>0$. On $I=\{x\ge x_0:\phi'>\phi^2\}$, $1<\phi'/\phi^2$. Thus $|I|\le\int_{x_0}^\infty\phi'/\phi^2\le1/\phi(x_0)$. Adding the finite initial interval controls all exceptions. For the zero function there are none. Taking logarithms turns <derivative> growth into logarithmic auxiliary-function growth in value-distribution estimates.
Back to article page
