Geodesic
The Carath\'{e}odory inequality on the boundary for holomorphic functions in the unit disc
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2016), pp. 287-301

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In this paper, a boundary version of the Carathéodory inequality is studied. For the function $f(z)$, defined in the unit disc with $f(0)=0$, $\Re f(z)\leq A$, we estimate a modulus of angular derivative at the boundary point $z_{0}$, $\Re f(z_{0})=A$, by taking into account the first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved. 
@article{JMAG_2016_12_a0,
     author = {B. N. \"Ornek},
     title = {The {Carath\'{e}odory} inequality on the boundary for holomorphic functions in the unit disc},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {287--301},
     publisher = {mathdoc},
     volume = {12},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2016_12_a0/}
}
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B. N. Örnek. The Carath\'{e}odory inequality on the boundary for holomorphic functions in the unit disc. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2016), pp. 287-301. http://geodesic.mathdoc.fr/item/JMAG_2016_12_a0/