Typically-real Functions with Assigned Zeros
نویسنده
چکیده
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 |a| =S1 and 3/(ei9) changes sign 2p times as z = eie traverses the boundary of the unit circle, or (II) f(z) is regular in | z\ < 1 and if there is a p < 1 such that for each r in p<r<l, $f(reie) changes sign 2p times as z = reie traverses the circle \z\ =r. This set of functions is denoted by T(p). The name typically-real was first suggested by Rogosinski [6]1 who studied these functions in the case p = i. 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 \bn\ in terms of \bi\, • • ■ , \bp\ was obtained, namely for n = p + 1, P + 2, . . * 2k(n + p)\ . . (1.2) Í, =Z -\bk\. 11 ti (p+k)\(pk)\(n-pl)\(n2k2)[ '
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