Some remarks on Bazilevič functions
The Bazilevich functions of type \(\gamma =\alpha +i\beta\), where \(\alpha\geq 0\) and \(-\infty <\beta <\infty\) were introduced in [\textit{I. E. Bazilevich}, Mat. Sb., Nov. Ser. 37(79), 471-476 (1955; Zbl 0065.310)]. For \(\alpha\geq 0\) and \(\beta\geq 0\), let K(\(\alpha\),\(\beta)\) denote the set of functions g which are analytic and non-zero in \(U=\{z: | z| <1\}\) such that \(g(0)=1\) and \[ -\alpha \pi +(\alpha - \beta)(\theta_ 2-\theta_ 1)\leq \arg g(re^{i\theta_ 2})-\arg g(re^{i\theta_ 1})\leq (\alpha -\beta)(\theta_ 2-\theta_ 1)+\beta \pi \] for \(\theta_ 1<\theta_ 2<\theta_ 1+2\pi\) and \(0<r<1\). This family was introduced by the author [J. Anal. Math. 34, 204-239 (1978; Zbl 0405.30005)], and Bazilevich functions can be characterized as solutions in U to the differential equation \(f'(z)[f(z)/z]^{\gamma - 1}=g(z)\) where \(f(0)=0\), \(f'(0)=1\) and \(g\in K(1,2\alpha +1).\) This characterization gives the author a way of extending the definition of Bazilvich functions for \(\alpha\geq -\). Then he shows that such functions are univalent in U, and given \(g\in K(1,2\alpha +1)\) the differential equation above has a solution f which is analytic in U and is of Bazilevich type \(\gamma\). The author also proves several theorems about membership of functions in K(\(\alpha\),\(\beta)\) which are related through convolution of power series with \(\sum^{\infty}_{n=1}(\gamma /(n+\gamma))z^ n\). One corollary asserts that each Bazilevich function of type \(\gamma =-+i\beta\) is spiral-like. The author obtains facts about Bazilevich functions of real type, including which K(\(\alpha\),\(\beta)\) class f(z)/z belongs to. When \(\alpha\geq 1\) this result yields the following fact. If L is continuous linear functional on the space of functions analytic in U then the range of L(f) for f Bazilevich of type \(\alpha\), \(\alpha\geq 1\), is the same as the range of L over the functions \(z(1+xz)/(1-yz)^ 2\) where \(| x| =| y| =1\) or \(x=-y\in U\).
scientific article; zbMATH DE number 3902977