Abstract
Given nonnegative integers \(t<s\) , pairs ( \(f^{(t)}(z)\) , \(f^{(s)}(z)\) ) of derivatives of finite-order meromorphic functions \(f(z)\) for which \(f^{(t)}(z)\) is neither a rational function, a rational function of an exponential \(e^{\alpha z}\) , nor an elliptic function are considered. For positive integers \(n\) and \(H\) and a positive number \(R\) , let \(B(n,H,R)\) be the set of points \(z\) in the disk \(|z|\le R\) for which \(f^{(t)}(z)\) and \(f^{(s)}(z)\) are algebraic numbers of degree at most \(n\) and height at most \(H\) (and, moreover, \(|f^{(t)}(z)|\) and \(|f^{(s)}(z)|\) are not very large). An upper bound for the number of points in \(B(n,H,R)\) is obtained for almost all, in a certain sense, \((n,H,R)\) .
