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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.