On asymptotic curves of entire functions of finite order

For any $\rho$, $ 0\leqslant\rho\leqslant\infty$, there exists an entire function of order $\rho$ such that for any asymptotic curve $\Gamma$ on which $f\to\infty$ the relation $l(r,\Gamma)=O(r)$, $r\to\infty$, does not hold, where $l(r,\Gamma)$ is the length of that part of $\Gamma$ contained in the disc $\{z:|z|\leqslant r\}$. The same is true of asymptotic curves on which $f\to a\ne\infty$ under the natural restriction that $1/2\leqslant\rho\leqslant\infty$. This disproves a well-known conjecture of Hayman and Erdösh. Several closely related results are obtained.
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