Borel exceptional values of meromorphic functions (Q1775380)

The authors discuss Borel exceptional values of transcendental meromorphic functions and their relations to Picard and Nevanlinna exceptional values. They improve several results due to \textit{H.~S.~Gopalakrishna} and \textit{S.~S.~Bhoosnurmath} [Ann. Pol. Math. 32, 83--93 (1976; Zbl 0282.30028), Rev. Roum. Math. Pures Appl. 23, 721--726 (1978; Zbl 0391.30019)] and \textit{S.~K.~Singh} and \textit{H.~S.~Gopalakrishna} [Math. Ann. 191, 121--142 (1971; Zbl 0207.07401)]. To be more precise, let \(f\) be a nonconstant meromorphic function in the complex plane and \(a \in \mathbb{C} \cup \{\infty\}\). We use the usual notations of Nevanlinna theory such as \(T(r,f)\) (characteristic function), \(N(r,a,f)\), \(\overline{N}(r,a,f)\) (counting functions), \(\delta(a,f)\) (Nevanlinna deficiency) and \(\Theta(a,f)\) (ramification index). If \(\delta(a,f)>0\), then \(a\) is called a deficient value or Nevanlinna exceptional value of \(f\). For a positive integer \(k\) or \(k=\infty\) denote by \(\overline{N}_k(r,a,f)\) the counting function of distinct \(a\)-points of \(f\) of multiplicity at most \(k\) and by by \(N_k(r,a,f)\) the counting function of \(a\)-points of \(f\), where an \(a\)-point of multiplicity \(\nu\) is counted \(\nu\)-times if \(\nu \leq k\) and \((k+1)\)-times if \(\nu>k\). Note that \(\overline{N}_\infty(r,a,f)=\overline{N}(r,a,f)\). Let \(\rho\), \(0 \leq \rho \leq \infty\) be the order of growth of \(f\), and let \(\rho(a,f)\), \(\overline{\rho}(a,f)\) and \(\overline{\rho}_k(a,f)\) be the orders of the functions \(N(r,a,f)\), \(\overline{N}(r,a,f)\) and \(\overline{N}_k(r,a,f)\), respectively. If \(f\) is of order \(\rho\), and if \(\rho(a,f)<\rho\), then \(a\) is called a \textit{Borel exceptional value} of \(f\) (for short we write \(a \in B(f)\)). Furthermore, \(a\) is called a Borel exceptional value of \(f\) for distinct zeros (\(a \in \overline{B}(f)\)), if \(\overline{\rho}(a,f)<\rho\), and a Borel exceptional value of \(f\) for distinct zeros of multiplicity at most \(k\) (\(a \in \overline{B}_k(f)\)), if \(\overline{\rho}_k(a,f)<\rho\). If \(f\) is transcendental, then every Picard exceptional value is a Borel exceptional value. A classical theorem of E.~Borel states that a meromorphic function of finite order has at most two Borel exceptional values. Generalizations and refinements of this result are due to G.~Valiron, the above mentioned authors and others. Finally, set \[ \delta_k(a,f) = 1-\limsup_{r\to\infty}{\frac{N_k(r,a,f)}{T(r,f)}}\,. \] Obviously, \(\delta(a,f) \leq \delta_k(a,f) \leq \delta_{k-1}(a,f) \leq \dotsb \leq \delta_1(a,f) \leq \Theta(a,f) \leq 1\). Now, the main results read as follows. Theorem 1. Let \(f\) be a meromorphic function of order \(\rho\). If there exist distinct elements \(a_1,\dotsc,a_p\), \(b_1,\dotsc,b_q\), \(c_1,\dotsc,c_s\) in \(\overline{B}_k\), \(\overline{B}_l\), \(\overline{B}_m\), respectively (where \(p\), \(q\), \(s\), \(k\), \(l\) and \(m\) are positive integers), then \[ \frac{pk}{1+k} + \frac{ql}{1+l} + \frac{sm}{1+m} + \frac{1}{1+k} \sum_{j=1}^p \delta_k(a_j,f) + \frac{1}{1+l} \sum_{j=1}^q \delta_l(b_j,f) + \frac{1}{1+m} \sum_{j=1}^s \delta_m(c_j,f) \leq 2\,. \] Theorem 2. Let \(f\) be a meromorphic function of order \(\rho\). If there exist \(a \in \mathbb{C} \cup \{\infty\}\) and positive integers \(k\) and \(q\) such that \[ (1+k)\Theta(a,f) + \sum_{b \neq a} \delta(b,f) > 2-k(q-1)\,, \] then \(\overline{B}_k(f) \setminus \{a\}\) contains at most \(q\) elements. Theorem 3. Let \(f\) be a meromorphic function of order \(\rho\). If \(0\), \(\infty \in \overline{B}(f)\), then for any homogeneous differential polynomial \(P\) generated by \(f\) the set \(\overline{B}_1(P) \setminus \{0\}\) is empty. If \(P\) is generated by \(f'\), then the assertion holds if \(a\), \(\infty \in \overline{B}(f)\) for some \(a \in \mathbb{C}\).