On the zeros of solutions of the equation \(y^{(2k + 1)} + p(z)y = 0\) (Q1874776)

The author proves several theorems related to the distribution of zeros of equation (1) \(y^{(2k+1)} + p(z)y = 0\). E.g., they establish the following theorem: Let \(p(z)\) be an analytic function, and let \[ \mid p(z)\mid \leq 2^{-2k - 1}(2k + 1)\pi^{2k - 2}\mid z\mid^{-3}(1 - \mid z\mid)^{-2}, \mid z\mid<1. \] Then equation (1) with \(y(0) = y'(0) =\cdots= y^{(k -1)}(0)= 0\) does not have a nontrivial solution with a \((k + 1)\)-multiple zero in the annulus \(D_0\). This theorem generalizes a result due \textit{W. J. Kim} [J. Math. Anal. Appl. 25, 189-208 (1969; Zbl 0186.41101)].