EXTREMAL PROPERTIES OF CLOSE-TO-CONVEX FUNCTIONS (UDC 517.53) I. A. Aleksandrov and V. Ya. Gutlyanskii Translated from Sibirskii Matematicheskii Zhurnal, Vol. 7, No. 1, pp. 8-22, January-February, 1966 Original article submitted January 19, 1965 INTRODUCTION According to the definition proposed by W. Kaplan [1], a function ](z) = z + c2z 2 +... + c,~z n +... (1) holomorphic in the unit disc E ={z : [z I < 1}, is close-to-convex if there is a univalent function 90(z) mapping E into a convex region such that Re[ if(z)]>0 (2) ~'(z) for any point z of E. The set of close-to-convex functions forms a class L including as subclasses the class S* of univalent starlike functions, the class SO of univalent convex functions, the class V of univalent functions of hounded rotation,* and the class L0 introduced by I. E. Bazelevich consisting of the function f(z) whose derivatives can be expressed in the form f '(z) =h(z)" p (z), where h(z) E P,J" r (z) E S ~ In its turn L is a subclass of the class S of univalent holomorphic functions f(z) in E with an expansion abou~ the origin of the form (1). Z. Lewandowski [2, 3] proved a theorem concerning the equivalence of the class L to the class of linearly accessible functions of M. Biernacki [4]. In a recent paper [5] Biernacki and Lewandowski presented a new and shorter proof of this result. The extremal properties of dose-to-convex functions have been studied by L Krzyz [6], E. Szczepankiewicz and J. Zamorski [7], Ogawa [8], and other authors. As Krzyz has noted however [9], the results in [6-8] refer not to the whole class L but to L0. In the present article we develop a variational method for solving the extremal problem in the class L of close-to-convex functions. The basic variational formulas for these functions are derived inSection I. We analyze the boundary functions in the classes L, S*, S ~ P, and Lo relative to the functionals I(/) = J ( Uot, vol , . . . , u~,~, v ~,~ ; ... ; C~om, y o r e , . . . , U,~,~r~, V ) Ca) and 1(/) ----- ](uii, vii,..., un,~, vn,i; .... , u,~, vim,. .., u mm, v~,~m )" Here J is an arbitrary fixed function depending analytically on u~h = fs)(zh), vs~ = fs)(z~), (s = O, 1 ..... nh; k = t. 2,..., m), where z t, z 2..... z m (m ; 1, 2 .... ) are distinct fixed points in E and f(~ = f(z). (4) *I.e., functions w=f(z) for which l arg f'(z) [ < ~r/2 in E. ?I.e., h(z) is holomorphic in E, Reh(z)> 0 in E, and h(0)=1.