Common Borel radius of an algebroid function and its derivative

TL;DR

This paper proves that under certain growth conditions, an algebroid function and its derivative share the same Borel radius, extending Valiron's conjecture from meromorphic functions to algebroid functions.

Contribution

It extends Valiron's conjecture by establishing the equality of Borel radii for algebroid functions and their derivatives under specific conditions.

Findings

The Borel radii of the algebroid function and its derivative coincide.

The result applies to functions with infinite characteristic function growth relative to log(1/(1-r)).

The work generalizes known results from meromorphic to algebroid functions.

Abstract

In this article, by comparing the characteristic functions, we prove that for any $\nu$-valued algebroid function $w(z)$ defined in the unit disk with $\limsup_{r\to1-}T(r,w)/\log\frac{1}{1-r}=\infty$ and the hyper order $\rho_2(w)=0$, the distribution of the Borel radius of $w(z)$ and $w'(z)$ is the same. This is the extension of G. Valiron's conjecture for the meromorphic functions defined in $\widehat{\mathbb{C}}$.