Geodesic
Inequalities for moduli of the circumferentially mean $p$-valent functions
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 44-54

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Let $f$ be a circumferentially mean $p$-valent function in the disk $|z|1$ with Montel's normalization: $f(0)=0$, $f(\omega)=\omega$ $(0\omega1)$. Under an additional constraint on the covering of the concentric circles by $f$, precise lower and upper bounds of modulus $|f(z)|$ for some $z\in(-1,0)$ are established. The necessity of such constraint for the non-trivial estimates to be true is shown. 
@article{ZNSL_2014_429_a4,
     author = {V. N. Dubinin},
     title = {Inequalities for moduli of the circumferentially mean $p$-valent functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {44--54},
     publisher = {mathdoc},
     volume = {429},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_429_a4/}
}
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V. N. Dubinin. Inequalities for moduli of the circumferentially mean $p$-valent functions. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 29, Tome 429 (2014), pp. 44-54. http://geodesic.mathdoc.fr/item/ZNSL_2014_429_a4/