Complex oscillation and removable sets (Q1300034)

Let \(A\) be a transcendental entire function of finite order, and let \(E\) be the product of linearly independent solutions of \(w''+A(z)w=0\). Suppose that \(K>1\) and \(M\) are positive constants, suppose there exists a positive sequence \(r_m\) such that for each large positive integer \(m\) the number of zeros of \(E\) in the annulus \(\{z:K^{-1}r_m<| z|<Kr\}\) is at most \((r_m)^M\). The author proves that \(\log T(K^{-1}r_m,E)=O(\log r_m)\), \(m\to\infty\) and assumes in addition that \(\limsup_{m\to\infty}(\log r_{m+1}/\log r_m)<+\infty\), then \(E\) has finite order. The approach of this paper is more direct than that of [J. Math. Anal. Appl. 214, No. 1, 233-244 (1997; Zbl 0893.34004)] and improves its results.