The geometry of level sets plays an important role in the analysis of various function-theoretic problems. Often, however, the level sets are so complicated that one must either choose sets associated with special levels (see [3], [4]) or resort to the approximation of level sets by shorter curves (see[l, pp. 550-553]). In [3] and [4], there are some weak estimates on the length e(r, R) of the sets {z: l f (z ) I = R, Izt < r} associated with a function [ meromorphic in the plane. For such an / we do not know whether the quantity ca(r, R) can be bounded in terms of Nevanlinna's characteristic function T without reference to exceptional levels. But the following is implicit in the results in [3, pp. 121-123] and [4, p. 44]: I / [ is meromorphic in lzl-<2r and f (0 )= 1, then each subinterval [a,/3] of (O, oo) contains a set I of measure ([3a)/2 such that the inequality