A theorem on the growth of entire functions on asymptotic paths and its application to the oscillation theory of \(w''+Aw=0\)
It is a well-known conjecture that the convergence exponent \(\lambda (E)\) for the zeros of \(E=w_ 1w_ 2\), where \(w_ 1\), \(w_ 2\) are linearly independent solutions of \(w'' + A(z) w = 0\), \(A(z)\) transcendental entire, becomes infinite, provided \(\rho (A)\) is finite and non-integer. Results into this direction appear in \textit{S. B. Bank} and \textit{I. Laine} [Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026)], \textit{C.-Z. Huang} [Kodai Math. J. 14, 313-319 (1991; Zbl 0754.34009)], \textit{J. Rossi} [Proc. Am. Math. Soc. 97, 61-66 (1986; Zbl 0596.30047)] and \textit{L.-C. Shen} [Kexue Tongbao 30, 1581-1585 (1985; Zbl 0636.34003)]. The present paper contains a variant to these results, namely either \(\rho (E) = + \infty\) or \({N \over \mu (A)} + {1 \over \rho (E)} \leq 2\), provided \(\mu (A) < \infty\) and for a \(K \geq {1 \over 2}\), the set \(\{z:| A(z) | > K\}\) has at least \(N\) components. The proof relies heavily on a theorem about the growth of an entire function on asymptotic paths, generalizing an earlier result due to \textit{J. Rossi} and \textit{A. Weitsman} in [Comment. Math. Helv. 60, 1-14 (1985; Zbl 0569.30021)].
scientific article; zbMATH DE number 558888
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