On a subclass of Bazilevich functions (Q5903055)

Let S be the class of regular, normalized univalent functions with power series expansion \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) for \(z\in D\), where \(D=\{z:\) \(| z| <1\}\). Let B(\(\alpha)\) denote the class of normalized Bazilevich functions of type \(\alpha >0\) with respect to the starlike function g. This means that there exists a normalized starlike function g in S such that for \(z\in D\), Re\(\frac{zf'(z)}{f(z)^{1-\alpha}g(z)^{\alpha}}>0\). Let \(B_ 1(\alpha)\) denote the subset of B(\(\alpha)\) obtained by taking the starlike function \(g(z)=z\). In this paper the author obtains distortion bounds for \(| f(z)|\) and \(| f'(z)|\) when \(f\in B_ 1(\alpha)\). These upper and lower bounds approach the known bounds for S asymptotically as \(\alpha\) \(\to 0\). The author also proves a sharp coefficient estimate for \(| a_ n|\) when \(f\in B_ 1(1/N)\) and N is a positive integer.