MULTIVALENT LINEARLY ACCESSIBLE FUNCTIONS AND CLOSE-TO-CONVEX FUNCTIONS ABDALLAH LYZZAIK [Received 19 April 1979] 1. Introduction Let & = {z: |z | < 1}, and let/: 08 -> # be a regular function such that /(z) = b l z + b 2 z 2 + ... + b n z" + .... Let S be the class of functions/univalent in 0& such that b y — \. Functions of this class are called normalized univalent functions, f is said to be p-valent if it admits each value at most p times, and some value exactly p times. In this paper we generalize the concept of a linearly accessible domain as defined and developed by Biernacki [2]. Then we introduce a class of multivalent functions based on a purely geometric characterization. We study the relationship between this class and the following classes: close-to-convex functions of order p of Livingston [8], and weakly close-to-convex functions of order p of Styer [16]. Our most interesting result is that any close-to-convex function of order p enjoys a representation as a composition of a polynomial of degree p and a function in S. This result, besides its geometric significance, has been used by Styer and the author in [10] to strengthen Goodman's conjecture for a bound on the modulus of the nth coefficient of a p-valent function as a linear combination of the moduli of the first p coefficients [4]. 2. Definitions and known results This section is devoted to the definitions and results that we will use. DEFINITION 2.1. A domain is linearly accessible in the strict sense, or simply linearly accessible, if its complement in # is a union of semi-lines that are pairwise disjoint with the exception that the origin of one semi-line may lie on another semi-line. For this definition see Biernacki [2]. There he proved the following proposition: PROPOSITION 2.1. Let (/„) be a sequence of univalent functions that converge to f uniformly on compact subsets of @. If for each /„, / n (0) = 0, f' n (0) > 0, and f n {S8) is a linearly accessible domain, then f(0$) is linearly accessible. DEFINITION 2.2. Let S a (p) be the class of functions / regular in <%, with p zeros (counting multiplicity) which satisfy > 0 This paper forms, for the most part, a part of the author's doctoral dissertation which was written at the University of Cincinnati, Cincinnati, Ohio 45221, under the direction of Professor David Styer. Proc. London Math. Soc. (3), 44 (1982), 178-192.