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On deviations, defects and asymptotic functions of meromorphic functions. (English) Zbl 1241.30009
Let \(f\) be a meromorphic function and denote by \(\pi A(r,f)\) the spherical area (area on the Riemann sphere) of \(f(\{ z:|z|\leq r\})\). Set \(\mathcal{L} (r,\infty,f)=\max _{|z|=r} \log^+ |f(z)|\) and \(\mathcal{L}(r,a,f)=\mathcal{L}(r, \infty, 1/(f-a))\), where \(a\in \mathbb{C}\).
In 1994, W. Bergweiler and H. Bock proved [J. Anal. Math. 64, 327–336 (1994; Zbl 0828.30013)] that for a meromorphic function of infinite lower order, \[ \liminf _{r\to \infty} \frac{\mathcal{L}(r,\infty, f)}{rT'_-(r,f)}\leq \pi, \] where \(T'_-(r,f)\) is the left derivative of the Nevanlinna characteristic function. In connection with this result, A. Eremenko [Complex Variables, Theory Appl. 34, No. 1–2, 83–97 (1997; Zbl 0905.30025)] introduced the quantity \[ b(a,f)= \liminf _{r\to \infty} \frac{\mathcal{L}(r,\infty, f)}{A(r,f)}. \] Since \(rT'_-(r,f)=A(r,f)+O(1)\) \((r\to\infty)\), the theorem of Bergweiler and Bock implies that \(b(a,f)\leq \pi\) for each \(a\in \mathbb{C}\). In [Eremenko, loc. cit.] an analogue of the deficiency relation for magnitudes of deviations \(b(a,f)\) was obtained. A counterpart for the unit disk was introduced in [the second author and I. G. Nikolenko, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 2002, No. 2, 25–28 (2002; Zbl 1074.30519)]:
Theorem K. For a meromorphic function \(f\) such that the set \(\{a\in \overline{\mathbb{C}}: b(a,f)>0\}\) contains more than one point we have \[ \sum_{a\in \mathbb{C}} b(a,f)\leq 2\pi. \] In a series of works, e.g. [the second author, Sb. Math. 189, No. 6, 875–899 (1998); translation from Mat. Sb. 189, No. 6, 59–84 (1998; Zbl 0942.30019); the authors, Math. Notes 85, No. 1, 20–33 (2009); translation from Mat. Zametki 85, No. 1, 22–35 (2009; Zbl 1177.30033)], the authors have systematically studied properties of the quantity \(b(a,f)\).
In particular, they proved the following theorem:
Theorem M. Let \(f\) be an entire function of lower order \(\lambda>0\) and denote by \( M\) the set of all rational functions. Then the set \(\Omega=\{ q\in M: b(q,f)>0\}\) is at most countable and \[ \sum_{q\in M} b(a,f)\leq \begin{cases} \frac{\pi}{\sin \pi\lambda} & \text{ if } 0<\lambda\leq 0.5, \\ \pi & \text{ if } 0.5<\lambda \leq \infty .\end{cases} \]
In the present paper the authors try to extend the results to meromorphic functions.
Theorem 1. Let \(f\) be a meromorphic function of lower order \(\lambda\), \(0<\lambda\leq \infty\), with \(N(r,\infty, f)=o(T(r,f))\) (\(r\to\infty\)), and denote by \(M\) the set of all rational functions. Then the set \(\Omega=\{ q\in M: b(q,f)>0\}\) is at most countable and \[ \sum_{q\in M} b(q,f)\leq \begin{cases} \frac{\pi}{\sin \pi\lambda} & \text{ if } 0<\lambda\leq 0.5, \\ \pi & \text{ if } 0.5<\lambda \leq \infty .\end{cases} \]
Theorem 2. Let \(f\) be a meromorphic function of lower order \(\lambda\), \(0<\lambda\leq \infty\), let also \(P_d\) be a set of all polynomials of degree at most \(d\). Then \[ \sum_{q\in P_d} b(q,f)\leq \begin{cases} {\pi}(d+2) \sqrt{\Delta(2-\Delta)} & \text{ if } \lambda \geq 0.5 \text{ or } 0<\lambda< 0.5 \text{ and } \sin \frac {\pi \lambda}2 \geq \frac {\Delta}{2},\\ \pi(d+2) (\Delta \mathop{ctg} (\pi\lambda) +\mathop{tg} \frac {\pi \lambda}2) & \text{ if } 0<\lambda <0.5 \text{ and } \sin \frac {\pi \lambda}2< \frac {\Delta}{2}, \end{cases} \] where \(\Delta=\Delta(0,f^{(d+1)})\) is the Valiron deficiency of \(f^{(d+1)}\) at the origin.
As a consequence, results on asymptotic polynomials are obtained.
The proofs are based on a combination of Petrenko’s method and techniques of Bearnstein’s \(*\)-function and Pólya peaks for functions of infinite lower order introduced by Bergweiler and Bock.
MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable, general theory
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