Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral meansL 1 ( r , f ) : = r 2 2 π∫ − π πd θ| f ( r e i θ )| 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 meansL 1 ( 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 ) = max f ∈ F Δ r , z f ( z )for0 < r ≤ 1 , where Δ r , z f ( 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.