Let H(D) be the linear space of all analytic functions on the open
unit disc D. We define Sβ by the linear subspace
of all fβH(D) with bounded derivative fβ² on D. We
give the characterization of surjective, not necessarily linear, isometries on
Sβ with respect to the following two norms: β₯fβ₯ββ+β₯fβ²β₯ββ and β£f(a)β£+β₯fβ²β₯ββ for aβD, where
β₯β β₯ββ is the supremum norm on D.Comment: 46 page