Let D_(r):={z=x+iy∈C:|z|<r},r≤1.For a normalized analytic function f in the unit disk D:=D1,estimating the Dirichlet integralΔ(r,f)=∫∫_(D_(r))|f'(z)|^(2) dxdy,z=x+iy,is an important classical problem in co...Let D_(r):={z=x+iy∈C:|z|<r},r≤1.For a normalized analytic function f in the unit disk D:=D1,estimating the Dirichlet integralΔ(r,f)=∫∫_(D_(r))|f'(z)|^(2) dxdy,z=x+iy,is an important classical problem in complex analysis.Geometrically,Δ(r,f)represents the area of the image of D_(r)under f counting multiplicities.In this paper,our main ob jective is to estimate areas of images of D_(r)under non-vanishing analytic functions of the form(z/f)^(μ),μ>0,in principal powers,when f ranges over certain classes of analytic and univalent functions in D.展开更多
文摘Let D_(r):={z=x+iy∈C:|z|<r},r≤1.For a normalized analytic function f in the unit disk D:=D1,estimating the Dirichlet integralΔ(r,f)=∫∫_(D_(r))|f'(z)|^(2) dxdy,z=x+iy,is an important classical problem in complex analysis.Geometrically,Δ(r,f)represents the area of the image of D_(r)under f counting multiplicities.In this paper,our main ob jective is to estimate areas of images of D_(r)under non-vanishing analytic functions of the form(z/f)^(μ),μ>0,in principal powers,when f ranges over certain classes of analytic and univalent functions in D.
