ON LOGARITHMIC COEFFICIENTS OF SOME CLOSE-TO-CONVEX FUNCTIONS

The logarithmic coefficients gamma(n) of an analytic and univalent function f in the unit disk D = {z is an element of C : vertical bar z vertical bar < 1} with the normalization f(0) = 0 = f'(0) - 1 are defined by log f(z)/z = 2 Sigma(infinity)(n=1) gamma(n)z(n). Recently, D. K. Thomas [Proc. Amer. Math. Soc. 144 (2016), 1681-1687] proved that vertical bar gamma(3)vertical bar <= 7/12 for functions in a subclass of close-to-convex functions (with argument 0) and claimed that the estimate is sharp by providing a form of an extremal function. In the present paper, we point out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument 0). We also determine a sharp upper bound of vertical bar gamma(3)vertical bar for close-to-convex functions (with argument 0) with respect to the Koebe function.