SMAJDOR, A., and W. SMAJDOR Math. Zeitschr. 98, 235--242 (1967) On the Existence and Uniqueness of Analytic Solutions of a Linear Functional Equation A. SMAJDORand W. SMAJDOR Received May 14, 1966 Introduction The subject of the present paper is the problem of the existence and uni- queness of analytic solutions of the functional equation (1) ~oIf(z)] = g(z) ~o (~) + F(z), where f, g and F are known analytic complex-valued functions of a complex variable in a neighbourhood of z=0. We assume that (2) f(0) =0, f'(O) = s and Is] = 1. Thus the functionf(z) is representable by a power series (3) f(z)=sz+ ~ a.z" n=2 convergent in a neighbourhood of the origin. Particular cases of this problem have been treated among others by V. GANAVATHY IYER [2] and D.VAmA [10], (wheref(z)=c~z and F(z)=-O), G. A. PFEIFFER[6, 7], H. CREMER[1], C. L. SIE- GEL [8, 9], G. JULIA [3] and N. PASTID~S [5]. In the first part we discuss the problem of the existence and uniqueness of analytic solutions for the homogeneous linear equation (4) cp[f(z)] = g(z) ~o (z), wheref and g are known analytic functions, andf fulfils (2). In the second part of this paper we shall study solutions of Eq. (1) making use of the results con- cerning Eq. (4). I Let us introduce the following sets (here C is the set of complex numbers and N the set of positive integers): A={s: sEC, IsI= I, V.~NS"+ I} , S={s: sEA,~Ks>OV.~Nlog[s"--li<=Kslogn}, n_>2 V={s: sEA, 3Ms>O'~n~_M ~ q.<e"},