On Weighted Distortion in Conformal Mapping II

Some years ago, Blatter [ 1 ] gave a result of the form| f ( z 1 ) = f ( z 2 ) | 2 ⩾ sinh   2 ρ 8   cosh   4 ρ∑ j = 1 , 2( 1 =| z j | 2 )2| f ′ ( z j ) | 2for any function f regular and univalent in D : | z | < 1, where ρ is the hyperbolic distance between z 1 and z 2 . 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 z 1 , z 2 is wrong. Kim and Minda proved the generalized formula| f ( z 1 ) = f ( z 2 ) | ⩾ sinh   2 ρ 2( 2   cosh   2 p ρ )1 / p(| D 1 f ( z 1 ) | p +| D 1 f ( z 2 ) | p )1 / p ,where D 1 ( f ) = f ′( z ) (1 − | z | 2 ), valid for p ⩾ P with some P , 1 < P ⩽ 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 forf ˜ = U f T , 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. 1991 Mathematics Subject Classification 30C75, 30F30.