An extremal problem for the invariant differential operators on a class of Cauchy-type integrals
Keywords:holomorphic function, pseudo-hyperbolic metric, Bloch class, integral of the Cauchy type, BMOA, extremal problem
The differential operators D1( f )(z) = (1- |z|2 )δf (z) / δz and D2( f ) = D21 ( f ) , defined on the space of holomor phic functions in a unit disk, are invariant with respect to compositions with fractionally linear functions. They arise naturally in studies of holomorphic functions from the Bloch class β , which plays an important role in the geometric theory of functions. It is known that the images of the operators Dj ( f ) , j =1,2 , are Lipschitz functions with respect to the pseudo-hyperbolic metric ρ(z,w) in a unit disk. Namely, we have that upfєβ || D1( f )(z)|-|D1( f )(w) ||/ ρ(z,w) = 3√ 3 / 2. In this paper, we solve the extremal problem about the exact value of the quantity supf |D1( f )(z)-D2( f )(w)| / ρ(z, w), where f runs a class of Cauchy-type integrals, which, as is well known, is a subclass of Bloch functions.
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