On the growth of algebroid functions of finite order (Q2772935)

This is an interesting paper extending to algebroid functions a previous theorem due to J.~Miles and D.~Shea for meromorphic functions. Given a meromorphic function \(f\) such that NEWLINE\[NEWLINE \mu^{*}:=\inf\{\rho>0\mid \liminf_{r,A\to\infty}T(Ar) /(A^{\rho}T(r))=0\}<\infty, NEWLINE\]NEWLINE \textit{J. Miles} and \textit{D. Shea} proved in [Duke Math. J. 43, 171-186 (1976; Zbl 0333.30017)] that NEWLINE\[NEWLINE \limsup_{r\to\infty}\big(N(r,0;f)+N(r,\infty;f)\big)/m_2 (r,f)\geq\sup_{\mu^{*}\leq\rho\leq\lambda^{*}}C(\rho), NEWLINE\]NEWLINE where \(m_2(r,f):=(\frac{1}{2\pi}\int_{0}^{2\pi} (\log|f(re^{i\theta})|)^2 d\theta)^{1/2}\), \(C(\lambda):=\frac{|\sin \pi\lambda|}{\pi\lambda}\{2/(1+(\sin(2\pi\lambda))/ (2\pi\lambda)) \}^{1/2}\), and \(\lambda^{*}\) has been defined as \(\mu^{*}\) with \(\limsup\) replacing \(\liminf\). Let now \(f(z)\) be an algebroid transcendental function defined by an irreducible equation \(F(z,f)=A_0(z)f^n+A_1(z)f^{n-1}+\cdots+A_n(z)=0\), where \(A_0,\ldots,A_n\) are entire functions with no common zeros. Let further \(a_0,\ldots a_{n-1}\) be polynomials and \(a_n=\infty\), define \(g_j(z):=F(z,a_j(z))\), \(f_j(z):=g_j(z)/A_0(z)\) and \(N_j(r):=N(r,0;g_j)+N(r,0;A_0)\) for \(j=1,\ldots n\). Now, the following conclusion will be proved for an algebroid function \(f\) such that \(\mu^{*}<\infty\): NEWLINE\[NEWLINE \limsup_{r\to\infty}\Biggl(n\sum_{j=0}^{n}N(r,a_j;f)\Biggr)\Biggl/ m_n(r,f) \geq\sup_{\mu^{*}\leq\rho\leq\lambda^{*}}C(\lambda), NEWLINE\]NEWLINE where \(m_n(r,f):=(\frac{1}{2\pi}\int_{0}^{2\pi} \sum_{j=1}^{n}(\log|f_j(re^{i\theta})|)^2 d\theta)^{1/2}\). A simple example shows that this inequality is sharp.