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1. Let Siby) be the class of univalent bounded functions in the unit disc with the series development /(z) = S byz\ 0

This class has often been studied in the equivalent form of normalized univalent functions F{z) = fe1_1/(2) = z+ Ï tv\ (2) a, = A, \Fiz)\ib;K We shall study estimates for the coefficient aA in dependence on the first coefficient by .¡For this purpose an inequality of the Grunsky-Nehari type [2], [3] will be used. Such inequalities were first applied by V. Singh [4] in studying the coefficient a4 of bounded univalent functions with real coefficients. The method of Singh, which follows the argument of Charzynski and Schiffer [1] in the case of unbounded univalent functions, can be extended to the case of nonreal coefficients. Though much harder to apply here than in the real case, this method yields sharp bounds for a4 in an interval of ¿^-values near by = 0 and in an interval near by = 1. 2. We consider the univalent and bounded function (3) F(z) = V/(z2) and apply to it the Grunsky-Nehari inequalities for such functions. We use specifically the estimate (4) 13 / 3 \ a4-2a2a3 +-a\ + x\a2 + x2(a3-a2) + 2xt la3-a\ I Ú |(1-b\)-\by\a2\2 + 2|x1|2(l-by) + |x2|2(l-b2y)-2byRe{xyä2}.