A coefficient inequality for functions of positive real part with an application to multivalent functions (Q1067038)

Let \(P(z)=c_ 0+c_ 1z+..\). be analytic and Re P(z)\(>0\) for z in the open unit disk \(\Delta\). Earlier [Proc. Am. Math. Soc. 21, 545-552 (1965; Zbl 0186.399)], the author showed that \(| c_ n/c_ 0-c_ 1c_{n-1}/c^ 2_ 0| \leq 2.\) In the present paper he generalizes this in a very powerful way (too cumbersome to reproduce here). Corollary to his result is the inequality \[ | \sum^{p}_{k=t}d_{k- t}c_{n-k}| \leq 2,\quad when\quad 1/P(z)=d_ 0+d_ 1z+.... \] He uses the last inequality to establish Goodman's conjecture for a class of p-valent functions with real coefficients; this extends his earlier work [Trans. Am. Math. Soc. 115, 161-179 (1965; Zbl 0154.081)] on the subject, which appears to be the best to date.