Results on logarithmic Borel exceptional values of meromorphic functions with their difference operators (Q2166146)

The logarithmic order of a meromorphic function \(f\) is defined as \(\rho_{\log}(f) = \limsup_{r \to \infty}\frac{\log^{+} T(r, f)}{\log\log r}\). \textit{P. T. Y. Chern} [Trans. Am. Math. Soc. 358, No. 2, 473--489 (2006; Zbl 1079.30038)] proved the following theorem: If \(f\) is a transcendental meromorphic function with finite logarithmic order \(\rho_{\log}(f)\), then for each positive integer \(k\) \[\limsup_{r \to \infty}\frac{\log^{+}\{n(r, 0; f) + \overline n(r, 1; f^{(k)})\}}{\log\log r} = \rho_{\log}(f) - 1,\] where \(n(r, 0; f)\) denotes the number of zeros of \(f\) in \(\mid z \mid \leq r\), counting with multiplicities, and \(\overline n(r, 1; f^{(k)})\) denotes the reduced number of distinct \(1\)-points of \(f^{(k)}\) in \(\mid z \mid \leq r\). \par In the paper the authors study the problem similar to Chern's theorem for \(q\)-difference and shift difference operators in the complex plane.