Carleson measures induced by higher order Schwarzian derivatives and derivatives of analytic functions

Let G be a finitely generated Fuchsian group of the second kind without any parabolic element and f univalent analytic function in unit diskDcompatible withG. In this paper, westudy higher order Schwarzian derivatives: ?n+1( ) = ?? n( )?(n?1) f?? f???n( ), n ? 3, where ?3(f) stands for derivatives f, Sn(f) (f?) n?1/2 Dn( ?)? , 2. For p > 0, we show that if |?n(f)(z)|p(1?|z|2)p(n?1)?1dxdy (resp. |Sn( )(z)|p(1 |z|2)pn?1dxdy) satisfies Carleson condition on infinite boundary Dirichlet fundamental domainF ofG, then |?n( )(z)|p(1?|z|2)p(n?1)?1dxdy )(z)|p(1?|z|2)pn?1dxdy) is measure D. Similarly, 0 bounded disk D compatible with G, prove | ?(z)|p(1 |z|2)p?1dxdy domain F |f?(z)|p(1