# Gorin's problem for individual simple partial fractions

## TL;DR

This paper provides new lower bounds for the imaginary parts of poles of simple partial fractions, considering residues, under norm constraints, advancing understanding of Gorin's and Gelfond's problems.

## Contribution

It introduces residue-aware estimates for pole locations in simple partial fractions under specific norm conditions, extending previous results.

## Key findings

- Derived lower bounds for pole imaginary parts considering residues.
- Established new estimates for the moduli of poles when the derivative's norm is constrained.
- Enhanced the theoretical understanding of Gorin's and Gelfond's problems.

## Abstract

The main result of the paper is a lower estimate for the moduli of imaginary parts of the poles of a simple partial fraction (i.e. the logarithmic derivative of an algebraic polynomial) under the condition that the $L^\infty(\mathbb{R})$-norm of the fraction is unit (Gorin's problem). In contrast to the preceding results, the estimate takes into account the residues associated with the poles. Moreover, a new estimate for the moduli is obtained in the case when the $L^\infty(\mathbb{R})$-norm of the derivative of the simple partial fraction is unit (Gelfond's problem).