Geodesic
Estimates of the Distances to Direct Lines and Rays from the Poles of Simplest Fractions Bounded in the Norm of~$L_p$ on These Sets
Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 803-810

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For each $p>1$, we obtain a lower bound for the distances to the real axis from the poles of simplest fractions (i.e., logarithmic derivatives of polynomials) bounded by 1 in the norm of $L_p$ on this axis; this estimate improves the first estimate of such kind derived by Danchenko in 1994. For $p=2$, the estimate turns out to be sharp. Similar estimates are obtained for the distances from the poles of simplest fractions to the vertices of angles and rays.
Keywords: simplest fraction, logarithmic derivative, rational function, Euler beta function, Hölder's inequality, Hardy space.
Mots-clés : algebraic polynomial, $L_p$-norm 
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     author = {P. A. Borodin},
     title = {Estimates of the {Distances} to {Direct} {Lines} and {Rays} from the {Poles} of {Simplest} {Fractions} {Bounded} in the {Norm} of~$L_p$ on {These} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {803--810},
     publisher = {mathdoc},
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P. A. Borodin. Estimates of the Distances to Direct Lines and Rays from the Poles of Simplest Fractions Bounded in the Norm of~$L_p$ on These Sets. Matematičeskie zametki, Tome 82 (2007) no. 6, pp. 803-810. http://geodesic.mathdoc.fr/item/MZM_2007_82_6_a0/