Gorin's problem for individual simple partial fractions

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).