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Integral means and Dirichlet integral for analytic functions. (English) Zbl 1309.30019

Summary: For normalized analytic functions \(f\) in the unit disk, the estimate of the integral means
\[ L_1(r,f):=\frac{r^2}{2\pi}\int_{-\pi}^\pi\frac{d\theta}{| f(re^{i\theta})|^2} \]
is important in certain problems in fluid dynamics, especially when the functions \(f(z)\) are non-vanishing in the punctured unit disk \(0<| z|<1\). We consider the problem of finding the extremal function \(f\) which maximizes the integral means \(L_1(r,f)\) for \(f\) belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses \(\mathcal F\) of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional
\[ A(r)=\max_{f\in\mathcal F} \Delta\bigg(r,\frac{z}{f(z)}\bigg)\quad\text{for}\quad 0<r\leq 1, \] where \(\Delta\Big(r,\frac{z}{f(z)}\Big)\) denotes the area of the image of \(| z|<r\) under \(z/f(z)\). The first problem was originally discussed by L. Gromova and A. Vasil’ev [Proc. Indian Acad. Sci., Math. Sci. 112, No. 4, 563–570 (2002 Zbl 1032.30008)] while the second by S. Yamashita [Bull. Aust. Math. Soc. 41, No. 3, 435–439 (1990; Zbl 0698.30015).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30E99 Miscellaneous topics of analysis in the complex plane
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