PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 7, July 2007, Pages 2051–2054 S 0002-9939(07)08660-1 Article electronically published on March 2, 2007 A SHARP INEQUALITY FOR THE LOGARITHMIC COEFFICIENTS OF UNIVALENT FUNCTIONS OLIVER ROTH (Communicated by Juha M. Heinonen) Dedicated to the memory of Professor Nikolaos Danikas Abstract. We prove the sharp inequality ∞  k=1  k k +1  2 |c k (f )| 2 ≤ 4 ∞  k=1  k k +1  2 1 k 2 = 2 π 2 − 12 3 for the logarithmic coefficients c k (f ) of a normalized univalent function f in the unit disk. 1. Introduction Let S denote the set of univalent functions f in the unit disk D := {z ∈ C : |z| < 1} normalized by f (0) = 0 and f ′ (0) = 1. The logarithmic coefficients c k (f ) of a function f ∈S are defined 1 by log f (z) z = ∞  k=1 c k (f )z k . The celebrated de Branges’ inequalities (the former Milin conjecture) for univalent functions state that (1.1) n  k=1 (n − k + 1)k |c k (f )| 2 ≤ 4 n  k=1 n +1 − k k , n =1, 2, 3,..., with equality if and only if f (z)= z (1 − e iθ z) 2 , θ ∈ R; see, for instance, [5, 6]. In addition to providing a proof of the Bieberbach conjecture the de Branges’ inequalities are also the source of many other interesting inequalities involving logarithmic coefficients of univalent functions such as (1.2) ∞  k=1 |c k (f )| 2 ≤ 4 ∞  k=1 1 k 2 = 2 π 2 3 ; Received by the editors September 13, 2005 and, in revised form, January 31, 2006. 2000 Mathematics Subject Classification. Primary 30C50; Secondary 30A10. Key words and phrases. Univalent functions, logarithmic coefficients, de Branges’ weight functions. 1 We use the notation employed in [6], which differs from the notation in [3, 1, 4]. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 2051 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use