A note on the order of magnitude of certain Titchmarsh-Weyl m-functions (Q923187)

Consider the linear second-order differential equation \((*)\quad - (py')'+qy=\lambda wy,\) \(0\leq x<\infty\), where p, q, w are functions and \(\lambda\) is a complex parameter. The Titchmarsh-Weyl m-function m(\(\lambda\)) is defined to be either the unique limit-point or a fixed point on the boundary of the limit disc to which converges the Weyl disc D(X,\(\lambda\)) as \(X\to \infty\). Some bounds for \(| m(\lambda)|\) as \(| \lambda | \to \infty\) in the upper half plane are given. If N is an integer, denote by \(S_{N,\epsilon}=\{\lambda\); \(0<\epsilon <\arg (\lambda)<(2^ N-1)^{-1}(\pi -\epsilon)\}\) and by \(\rho_ j(x)=\int^{x}_{0}w(t)\rho_{j-1}(t)^ 2dt,\) \(j=2,...,N\), where \(\rho_ 1(x)=| \int^{x}_{0}p(t)^{-1}dt|.\) The main result is: For \(\lambda \in S_{N,\epsilon}\) with \(| \lambda |\) sufficiently large, all \(m\in D(c(\lambda),\lambda)\) satisfy \[ (i)\quad | m|^{-1}\geq 2^{-1} \sin (\epsilon)| \lambda | \int^{c(\lambda)}_{0}w(t)dt, \] \[ (ii)\quad | m| \geq 2^{- 1} \sin (\epsilon)| \lambda |^{2^ N- 1}\int^{c(\lambda)}_{0}w(t)\rho_ N(t)^ 2 dt. \]