Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 337–350 LOGARITHMIC COEFFICIENTS OF UNIVALENT FUNCTIONS Daniel Girela Universidad de M´ alaga, An´ alisis Matem´ atico, Facultad de Ciencias E-29071 M´ alaga, Spain; girela@anamat.cie.uma.es Abstract. We prove that if n ≥ 2 there exists a close-to convex function f in S whose n -th logarithmic coefficient γ n satisfies |γ n | > 1/n . Also, we prove some results related to a conjecture of Milin on the logarithmic coefficients of functions in the class S and give some applications of them to obtain upper bounds on the integral means of these functions. 1. Introduction and statement of results Let S be the class of functions f analytic and univalent in the unit disc Δ= {z ∈ C : |z | < 1} with f (0) = 0, f ′ (0) = 1. Let S ⋆ denote the subset of S consisting of those functions f in S for which f (Δ) is starlike with respect to 0. It is well known (see [6] or [20]) that if f is analytic in Δ, with f (0) = 0 , f ′ (0) = 1, then f ∈ S ⋆ if and only if Re ( zf ′ (z )/f (z ) ) > 0, for all z in Δ. Finally, we let C denote the set of those functions f in S for which there exists a real number α and a function g in S ⋆ such that Re zf ′ (z ) e iα g(z ) > 0, z ∈ Δ. The elements of C are called close-to-convex functions. Clearly, S ⋆ ⊂ C . Associated with each f in S is a well defined logarithmic function log f (z ) z =2 ∞  n=1 γ n z n , z ∈ Δ. The numbers γ n are called the logarithmic coefficients of f . Thus the Koebe function k(z )= z (1 − z ) −2 has logarithmic coefficients γ n =1/n . If f (z )= z + ∑ ∞ n=2 a n z n ∈ S then γ 1 = 1 2 a 2 . Hence, since |a 2 |≤ 2, |γ 1 |≤ 1. 1991 Mathematics Subject Classification: Primary 30C45, 30C50, 30C55. This research has been supported in part by a grant from “El Ministerio de Educaci´ on y Cultura, Spain” PB97-1081 and by a grant from “La Junta de Andaluc´ ıa”.