Growth results for a subclass of Bazilevich functions
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 subclass of S whose functions are Bazilevich of type \(\alpha\). Choosing the associated starlike function \(g(z)=z\) gives the proper subclass \(B_ 1(\alpha)\) of B(\(\alpha)\). For B(\(\alpha)\), correct growth estimates in terms of the area function A(r) are unknown. The authors give several results in this direction for \(B_ 1()\). For example, they prove that if \(f\in B_ 1()\) then \[ n| a_ n| \leq o(1)+O(1)\sqrt{A(1- \frac{1}{n})}\quad as\quad n\to \infty \quad and\quad L(r)=O(1)\{\sqrt{A(r)\log \frac{1}{1-r}}\}\quad as\quad r\to 1. \] For f regular in \(D_ 1\) define for \(\lambda\) real, \[ I_{\lambda}(r,f)=\int^{2\pi}_{0}| f(re^{i\theta})|^{\lambda} d\theta. \] Let \(M(r)=\max_{| z| =r}| f(re^{i\theta})|\). The authors prove that if \(f\in B_ 1()\) and \(\lambda >1\) then \[ I_{\lambda}(r^ 2,zf')\leq C(\lambda)\int^{r}_{0}\frac{M(p)^{\lambda /2}}{(1-p)^{\lambda}}dp \] where C(\(\lambda)\) is a constant depending only on \(\lambda\). Finally, the authors prove that if \(f\in B_ 1()\) then for \(0\leq r\leq 1\), \(I_ 1(r^ 2,f)\leq I_ 1(r^ 2,f_ 0)\) where \[ f_ 0(z^ 2)=(\int^{z}_{0}\frac{1+t^ 2}{1-t^ 2}dt)^ 2. \]
scientific article; zbMATH DE number 3944213