Weakly close-to-convex functions (Q581702)

Goodman has conjectured that if \(f(z)=z^ q+a_{q+1}z^{q+1}+..\). is analytic and at most p-valent in the unit disc and if \(f(\beta_ j)=0\), \(0<| \beta_ j| <1\), \(j=1,...,s\), then the coefficients of the Taylor expansion of f(z) are dominated in absolute value by those of \[ F(z)=z^ q(1-z)^{-2p}(1+z)^{2(p-s-q)}\prod (1+z/| \beta_ j|)(1+| \beta_ j| z) \] (if there are no \(\beta_ j\), set \(s=0\) and take the product as 1). The conjecture has been proved for p- valent starlike functions and p-valent weakly starlike functions. In this paper the author proves that the conjecture holds for p-valent weakly close to convex functions, using the results of Lyzzaik and the proof of the Bieberbach conjecture by DeBrange. Several integral mean problems are also considered.