Estimates of the initial coefficients in the classes of univalent functions

Let S be the class of functions, regular and univalent in the circle ¦Z¦ < 1. Assume that D_n (), n=2,3,..., are defined by the expansion, ¦z¦<1,-11. In the paper one obtains sharp estimates for D₄() in the class S_R of functions from S with real coefficients C₂,C₃,... for all –1</1. In particular, as a consequence, one obtains sharp estimates for the coefficients C_3k+1 in the class S^K/R of K-symmetric functions from S_R for all K=2,3,... For k=2, the last result strengthens a result of Leeman.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 166–183, 1983.