Typically-real functions with assigned zeros

is said to be typically-real of order p, if in (1.1) the coefficients bn are all real and if either (I) f(z) is regular in I zl I 1 and 3f(eie) changes sign 2p times as z = ei9 traverses the boundary of the unit circle, or (II) f(z) is regular in I zI < 1 and if there is a p < 1 such that for each r in p <r < 1, af(rei9) changes sign 2p times as z = rei9 traverses the circle z = r. This set of functions is denoted by T(p). The name typically-real was first suggested by Rogosinski [61 who studied these functions in the case p= 1. The more general set of functions T(p) was first introduced by Robertson [5; 4], and in a recent paper by Robertson and Goodman [3] the sharp upper bound for IbnI in terms of Ib, * , Ibp| was obtained, namely for n=p +1, p+2,