Details

Autor(en) / Beteiligte

• Obradović, Milutin
• Ponnusamy, S.
• Wirths, Karl-Joachim

Titel

Integral means and Dirichlet integral for analytic functions

Ist Teil von

• Mathematische Nachrichten, 2015-02, Vol.288 (2-3), p.334-342

Ort / Verlag

Weinheim: Blackwell Publishing Ltd

Erscheinungsjahr

2015

Quelle

Wiley Online Library

Beschreibungen/Notizen

• For normalized analytic functions f in the unit disk, the estimate of the integral means L1(r,f):=r22π∫−ππdθ|f(reiθ)|2is 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 L1(r,f) for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses F of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional A(r)=maxf∈FΔr,zf(z)for0<r≤1,where Δr,zf(z) denotes the area of the image of |z|<r under z/f(z). The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.