Logarithmic coefficients and a coefficient conjecture for univalent functions

Let U ( λ ) denote the family of analytic functions f ( z ), f ( 0 ) = 0 = f ′ ( 0 ) - 1 , in the unit disk D , which satisfy the condition | ( z / f ( z ) ) 2 f ′ ( z ) - 1 | < λ for some 0 < λ ≤ 1 . The logarithmic coefficients γ n of f are defined by the formula log ( f ( z ) / z ) = 2 ∑ n = 1 ∞ γ n z n . In a recent paper, the present authors proposed a conjecture that if f ∈ U ( λ ) for some 0 < λ ≤ 1 , then | a n | ≤ ∑ k = 0 n - 1 λ k for n ≥ 2 and provided a new proof for the case n = 2 . One of the aims of this article is to present a proof of this conjecture for n = 3 , 4 and an elegant proof of the inequality for n = 2 , with equality for f ( z ) = z / [ ( 1 + z ) ( 1 + λ z ) ] . In addition, the authors prove the following sharp inequality for f ∈ U ( λ ) : ∑ n = 1 ∞ | γ n | 2 ≤ 1 4 π 2 6 + 2 Li 2 ( λ ) + Li 2 ( λ 2 ) , where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S .