The Hardy-Stein-Spencer identities for meromorphic functions (Q1364949)

Let \(n(f=w,D)\) be the number of roots in \(D\) of the equation \(f(z) =w\), and let for a function \(f\) analytic in \(|z|<1\) \[ I_\lambda (r,f)= {1\over 2\pi} \int^{2\pi}_0 \bigr |f(re^{i \theta}) \bigr|^\lambda d\theta, \] \(0<r<1\), \(\lambda>0\). Theorem. Suppose that \(f\) is meromorphic in \(|z|\leq r\) and has no pole on \(|z|=r\). Then we have for \(\lambda>0\) \[ r{d\over dr} I_\lambda (r,f)= {\lambda^2 \over 2\pi} \iint_C {n\bigl(f=w, |z|\leq r\bigr) -n\bigl(f= \infty, |z|\leq r\bigr) \over|w|^{2- \lambda}} dudv, \] \(w= u+ iv \). Furthermore, if \(f\) is analytic in \(r_k< |z|<r_{k+1}\), \(r> r_{k+1}\), and has no pole on \(|z|=r_k\), and \(|z|= r_{k+1}\), then \(r{d \over dr} I_\lambda(r,f)\) is continuous and nondecreasing on \((r_k; r_{k+1})\) from \(-\infty\) to \(+\infty\).