## 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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### References:

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