Derivative growth outside a finite-measure set (${\varphi }^{\prime }\le {\varphi }^2\text{outside a finite-measure set}$)

For a nonnegative increasing continuously differentiable function not identically zero, choose $x_0$ with $\varphi (x_0)>0$. On $I=\{x\ge x_0:{\varphi }^{\prime }>{\varphi }^2\}$, $1<{\varphi }^{\prime }/{\varphi }^2$. Thus $|I|\le {\int }_{x_0}^{\infty }{\varphi }^{\prime }/{\varphi }^2\le 1/\varphi (x_0)$. 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.