Geodesic
On the Shwarz inequality on the boundary for functions regular in the disk
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 74-84

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The classical Schwarz inequality on the boundary of the disk for the functions regular in the disk is refined in various directions. Inequalities involving zeros of the function, an inequality for points mapped to symmetric points on the circle, and an inverse estimate for univalent functions are presented. Other inequalities are discussed, and the possibility of applying them to estimates of polynomials and rational functions is indicated. 
@article{ZNSL_2002_286_a5,
     author = {V. N. Dubinin},
     title = {On the {Shwarz} inequality on the boundary for functions regular in the disk},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {74--84},
     publisher = {mathdoc},
     volume = {286},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_286_a5/}
}
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V. N. Dubinin. On the Shwarz inequality on the boundary for functions regular in the disk. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 74-84. http://geodesic.mathdoc.fr/item/ZNSL_2002_286_a5/