Nevanlinna constant and its analogues for entire and meromorphic functions of finite nonintegral order (Q2724905)

The authors prove the following result: Let \(f\) be meromorphic with finite nonintegral order \(\lambda\). If \(a\) and \(b\) are two distinct meromorphic functions which are small with respect to \(f\), then for any real number \(k>0\), the counting functions of \(f\) for \(a\) and \(b\) satisfy NEWLINE\[NEWLINE\limsup_{r\to +\infty}\frac{n(r,f=a)+n(r,f=b)}{U(kr,f)} \geq (q+1-\lambda)(\lambda-q)/c_1(q)k^{\lambda},NEWLINE\]NEWLINE where \(U(r,f)\) is a type function of the Nevanlinna characteristic function \(T(r,f)\) of \(f\), \(q=[\lambda]\), and NEWLINE\[NEWLINEc_1(q)= \begin{cases} 1 & q=0, \\ 2(q+1)[2+\log(q+1)] &q>0. \end{cases}NEWLINE\]NEWLINE In particular, if \(f\) is entire and if \(a\) is finite constant, they obtain NEWLINE\[NEWLINE\limsup_{r\to +\infty}\frac{n(r,f=a)}{U(r,f)} \geq (q+1-\lambda)(\lambda-q)/c_1(q),NEWLINE\]NEWLINE where \(U(r,f)\) is a type function of the logarithmic function \(\log M(r,f)\) of the maximum modulus function \(M(r,f)\) of \(f\). This implies the results due to \textit{G. Valiron} [Lecture on the general theory of integral functions (1924; JFM 50.0254.01) Chelsea, 1949 (reprint)] and S. M. Shah [Arch. Rat. Mech. Anal. 26, 179-187 (1967; Zbl 0165.08702)].