LetSbe the class of functionsf(z) =z+c2z2+ … analytic and univalent in the diskLetbe forf(z) ∈S. One considers means of the formThe choicexkwhich realizes extremal properties in the classSof the functionis of main interest, especially the casexk=n-k+ 1 (k= 1,…,), which corresponds to Milin's conjecture:In 1984 L. de Branges proved this conjecture. It is known that the famous Bieberbach conjecture, Robertson's conjecture and some other conjectures for the coefficients of univalent functions hold ifInis nonpositive for alln. In this paper we use the properties ofInto obtain some inequalities for the functions of classS. In these inequalities the equality is attained by the functionKx(z).