Remarks on the paper: "Bounds for functions involving ratios of modified Bessel functions"

TL;DR

This paper critically examines recent claims about bounds for ratios of modified Bessel functions, clarifying errors and demonstrating that previous results already encompass the new bounds, thus correcting the scientific record.

Contribution

It provides a detailed critique of recent bounds claims, showing they do not extend or improve existing inequalities, and clarifies the correct relationships between these bounds.

Findings

The bounds in Kokologiannaki's paper are not improvements over previous results.

Errors in the comparison of bounds are identified and explained.

Existing literature already contains the bounds claimed as new.

Abstract

A recent paper by C.G. Kokologiannaki published in J. Math. Anal. Appl. \cite{Kolo:2012:BFI} gives some properties for ratios of modified Bessel functions and, in particular, some bounds. These bounds are said to improve the range of some inequalities in \cite{Segura:2011:BRM} or to be sharper. Unfortunately, Kokologiannaki made some mistakes in the comparison and no improvement or extension is made over the results in \cite{Segura:2011:BRM}. We explain the errors in these comments and show that the bounds given in \cite{Kolo:2012:BFI} are already contained in \cite{Segura:2011:BRM}.