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An extension of Picard’s theorem for meromorphic functions of small hyper-order. (English) Zbl 1173.30017

Let \(a\in\widehat C\), let \(f\) be a meromorphic function, and denote \(f^{-1} (\{a\})=\{z\in\mathbb{C}:f(z)=a\}\), where \(\{\cdot\}\) denotes a multiset which takes into account multiplicities of its elements. The hyper-order of \(f\) is defined by \[ \zeta(f)=\limsup_{r\to\infty}\frac{\log\log T(r,f)}{\log r}, \] where \(T(r, f)\) is the Nevanlinna characteristic function. One says that the pre-image of \(a\) is forward invariant with respect to the function \(\tau\) if \(\tau(f^{-1} (\{a\}))\subset f^{-1} (\{a\})\). The author obtains a version of the second fundamental theorem of Nevanlinna. Let \(\omega(z)=cz^n+p_{n-1}z^{n-1}+\cdots+ p_0\) and \(\varphi(z)=cz^n+q_{n-1}z^{n-1}+\cdots+q_0\) be non-constant polynomials. Let \(q\geq 2\) and let \(a_1,\dots,a_q\) be distinct constants. If \(f\circ\omega \not\equiv f\circ\varphi\) and \(\zeta(f)<1/n^2\), then \[ m(r,f\circ\varphi)+ \sum^q_{k=1}m\left(r,\frac{1}{f\circ\varphi-a_k}\right)\leq 2T(r,f\circ\varphi)-N_\omega(r,f\circ\varphi)+o(T(r,f\circ \varphi)), \] where \[ N_\omega(r,f\circ \varphi)=2N(r,f\circ\omega-f\circ \varphi)+N\left(r,\frac{1}{f\circ\omega-f\circ \varphi}\right) \] and \(r\) lies outside of an exceptional set of logarithmic measure. As a corollary of this result it is shown that if \(n\in\mathbb{N}\) and three distinct values of a meromorphic function \(f\) with \(\zeta(f)<1/n^2\) have a forward invariant pre-image with respect to a fixed branch of the algebraic function \(\tau(z) =z+\alpha_{n-1}z^{1-1/n}+\cdots+\alpha_1z^{1/n}+\alpha_0\) with constant coefficients, then \(f\circ\tau\equiv f\). The author gives an example that shows that the growth condition \(\zeta(f)<1/n^2\) cannot be deleted in the above result.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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