Geodesic
Carleson measures induced by higher order Schwarzian derivatives and derivatives of analytic functions
Filomat, Tome 36 (2022) no. 15, p. 5297

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Let G be a finitely generated Fuchsian group of the second kind without any parabolic element and f be a univalent analytic function in the unit diskD compatible with G. In this paper, we study the higher order Schwarzian derivatives: σn+1( f ) = σ′n( f )−(n−1) f ′′ f ′ ·σn( f ), n ≥ 3,where σ3( f ) stands for the Schwarzian derivatives of f , and Sn( f ) = ( f ′) n−1 2 Dn( f ′)− n−12 , n ≥ 2. For p > 0, we show that if |σn( f )(z)|p(1−|z|2)p(n−1)−1dxdy (resp. |Sn( f )(z)|p(1 − |z|2)pn−1dxdy) satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domainF of G, then |σn( f )(z)|p(1−|z|2)p(n−1)−1dxdy (resp. |Sn( f )(z)|p(1−|z|2)pn−1dxdy) is a Carleson measure in D. Similarly, for p > 0 and a bounded analytic function f in the unit disk D compatible with G, we prove that if | f ′(z)|p(1 − |z|2)p−1dxdy satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G, then | f ′(z)|p(1 − |z|2)p−1dxdy is a Carleson measure in D.
DOI : 10.2298/FIL2215297H
Classification : 30C62, 30F35
Keywords: Fuchsian group, Carleson measure, Univalent analytic function, Higher order Schwarzian derivatives 
Zhenyong Hu; Jinhua Fan; Yutong Liu; Yu Sun. Carleson measures induced by higher order Schwarzian derivatives and derivatives of analytic functions. Filomat, Tome 36 (2022) no. 15, p. 5297 . doi: 10.2298/FIL2215297H
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     author = {Zhenyong Hu and Jinhua Fan and Yutong Liu and Yu Sun},
     title = {Carleson measures induced by higher order {Schwarzian} derivatives and derivatives of analytic functions},
     journal = {Filomat},
     pages = {5297 },
     year = {2022},
     volume = {36},
     number = {15},
     doi = {10.2298/FIL2215297H},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2215297H/}
}
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