517.925.7 Assume that the coefficients and solutions of the equation f (n) +pn−1(z)f (n−1) +. . .+ps+1(z)f (s+1) + . . . + p0(z)f = 0 have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients pj, j = s + 1, . . . , n − 1, increase slower (in terms of the Nevanlinna characteristics) than ps(z). It is proved that this equation has at most s linearly independent solutions of finite order.By A b , we denote the ring of all functions analytic in G = {z : r 0 6 |z| < +1} with unique singular point 1. For functions f 2 A b , the point 1 can be either a logarithmic singularity or an algebraic branching point of order n − 1 if n branches of the function f are connected at 1 (or, in particular, a branching point of order zero if f is a single-valued function holomorphic in G). Since the ring A b is entire (without divisors of zero), it can be embedded into a field [1, pp. 53, 59]. By M b , we denote the least field such thatFor a function f 2 M b , it is also convenient to use the notation f (z), z 2 G.If f 2 M b , then, with the exception of the branching point at 1, the role of singular points of the function f can be played solely by the poles isolated on the Riemannian surface of the analytic function f (z), z 2 G.LetIn what follows, only for the sake of definiteness, we assume that the function f has a logarithmic singularity at 1 because, for finite-valued (single-valued) and infinite-valued functions, the definitions and notation of the Nevanlinna characteristics T (r, f ), S ↵,β (r, f ) are noticeably different [2, pp. 23, 37].We choose arbitrary ↵, β, −1 < ↵ < β < +1. By f (z), z 2 g ↵β = {z = re i✓ : ↵ 6 ✓ 6 β, r 0 6 r < +1}, we denote a single-valued branch of the function f 2 M b in the angular domain g ↵,β on a Riemann surface of the analytic function f (z), z 2 G. (For a more detailed definition of a single-valued branch and the definitions of arithmetic operations over multivalued functions, see, e.g., [3, p. 478].) The Nevanlinna characteristics of the branch f (z), z 2 g ↵β , are determined as follows [2, p. 40]