On results of Gopalakrishna-Bhoosnurmath and Singh-Dhar (Q1125508)

Let \(f\) be a meromorphic function in the complex plane, and \(a\in \not\subset\cup \{\infty\}\). Denote by \(\overline E(a,k,f)\) the set of distinct zeros of \(f(z)-a\) with multiplicities \(\leq k\). Using the Nevanlinna calculus, the author obtains results of the following nature. Suppose \(f\) and \(g\) are two meromorphic functions with \(\overline E(a_j,k_j,f) =\overline E(a_j,k_j,g)\) for \(j=1,2,\dots,q\) where \(a_1,a_2, \dots,a_q\) are distinct elements in \(\not\subset \cup\{\infty\}\) and \(k_1,k_2, \dots,k_q\) are positive integers or infinity with \(k_1\geq k_2\geq \cdots\geq k_q\). Let \(S(q)= \sum^q_{j=3} (k_j/(k_j+1))\). (i) If \(S(q)>2\) then \(f\equiv g\). (ii) If \(S(q)=2\), then outside a set of \(r\) of finite measure, \(\lim_{r\to \infty}(T(r,f)/T(r,g))=1\). The theorems extend work of \textit{H. Gopalakrishna} and \textit{S. Bhoosnurmath} [Math. Scand. 39, 125-130 (1976; Zbl 0341.30023)] and [Tamkang, J. Math. 16, No. 4, 49-57 (1985; Zbl 0596.30045)], \textit{A. Singh} and \textit{R. Dhar} [Indian J. Pure Appl. Math. 26, No.~7, 697-703 (1995; Zbl 0845.30019)], and \textit{G. Gundersen} [J. Lond. Math. Soc. 20, 456-466 (1979; Zbl 0428.30017)].