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^ 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| 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).
A. Ya. Yanchenko (Mon,) studied this question.