Meromorphic functions of finite order with maximal deficiency sum (Q1190048)

In 1929 F. Nevanlinna conjectured that any meromorphic function of finite order \(\rho\) with maximal deficiency sum \(\sum_ a \delta(a,f)=2\) must have the following properties: 1) \(2\rho\) is an integer \(\geq 2\); 2) \(\delta(a,f)=p(a)/\rho\), where \(p(a)\) is a non-negative integer \((a\in\widehat\mathbb{C}\) arbitrary); 3) all deficient values are asymptotic. This conjecture was settled by \textit{D. Drasin} in 1987 in a voluminous paper [Acta Math. 158, 1-94 (1987; Zbl 0622.30028)]. The present authors give a new and short proof. In fact, two properties are added: 4) \(T(r,f)\sim r^ \rho\ell_ 1(r)\) and \[ \sum_{\{a:\delta(a,f)>0\}}\log{1\over|(f- a)(re^{i\theta})|}\sim \pi r^ \rho\ell_ 1(r)|\cos\rho(\theta-\ell_ 2(r))|,\leqno 5) \] uniformly with respect to \(\theta\), outside a union of small disks; \(\ell_ 1\) and \(\ell_ 2\) are slowly varying continuous functions. Moreover, the constants \(a\) may be replaced by small functions and the assumption \(\rho<\infty\) by \(\lambda<\infty\) (lower order). The proof is purely potential-theoretic and based on the following theorem, which seems to be of own interest: Theorem 2. Let \((u_ k)^ \omega_{k=1}\), \(\omega\leq\infty\), be non-negative \(\delta\)-subharmonic functions such that \(\sum^ \omega_{k=1} u_ k=\bigvee^ \omega_{k=1} u_ k\), whose Riesz charges \(\mu_ k\) are uniformly bounded: \(\mu_ k\leq\mu\) for some measure \(\mu\) and all \(k\). Then \(\sum^ \omega_{k=1} \mu_ k\leq 2\bigvee^ \omega_{k=1} \mu_ k\) holds.