The Coefficients of Multivalent Close-to-Convex Functions

Abstract

2t(n + p) ! . .(1.1) \on\ = D tí (p + t)\(p -/)!(» -P -l)!(w2 -t2) Inequality(1.1) reduces to the well-known Bieberbach conjecture when p -l.The conjecture was proven by Goodman and Robertson [3] for a function in S(p), in case all its coefficients are real and by Robertson [7], in case di = a2= • • ■ =ap_2 = 0, the remaining coefficients being complex.The author [5] proved (1.1) for n = p + l for functions in 3C(p), no restrictions being made on the coefficients.In this paper, we will prove (1.1) for functions of the class X,(p) for the case <ïi = a2= • • ■ =aj,_2 = 0, the remaining coefficients being complex.The case p -2 of our proof gives (1.1) for the entire class 3C(2).Inequality (1.1) is known to be true for the class 3C(1) [ó].