ON WEIGHTED DISTORTION IN CONFORMAL MAPPING II

Abstract

Some years ago, Blatter [1] gave a result of the form

formula here

for any function f regular and univalent in D[ratio ][mid ]z[mid ]<1, where ρ is the hyperbolic distance between z₁ and z₂. Kim and Minda [5] pointed out that the multiplier on the right is incorrect. They say that Blatter's proof gives the correct multiplier, but Blatter's formula for ρ in terms of z₁, z₂ is wrong. Kim and Minda proved the generalized formula

formula here

where D₁(f)=f′(z) (1−[mid ]z[mid ]²), valid for p[ges ]P with some P, 1<P[les ]3/2. In each case there was an appropriate equality statement. Kim and Minda made the important and easily verified remark that these problems are linearly invariant in the sense that if the result is proved for f, then it follows for f˜=UfT, where U is a linear transformation of the plane onto itself and T is a linear transformation of D onto itself. This means that we need to prove such results only in an appropriately normalized context.