ES Why sums of log 2 ( x ) ? What conjecture ? ( A comment on a recent preprint by Borisov & al . [ 1 ] , posted on RG )

This note corrects some intentionallymisleading statements about my work [4] (posted earlier on RG) in a recent preprint by Borisov,Neff, Sra & Thiel [1] (posted on the same platform). In the danger of a possible confusion of interested readers, and solely because of that, I decided, after much thought, to take the exception and reply in some detail. Both [4] and [1] deal with the inequality of the form f x1 +Ü+f xn 2 f y1 +Ü+f yn Ù for a real valued function f Ú 0Ù ð r RÙ conditioned by the inequalities S x 2 S y for k ̈ 1Ù Ü Ù n − 1Ù S x ̈ S y with elementary symmetric functions SÙ see Section 1 for a precise and full formulation. In Borisov, Neff, Sra & Thiel [1] only the particular case f x ̈ log x Ù is treated and is accordingly called the sum of squared logarithms inequality in that case by the authors. In contrast, in [4], Inequality is established for a large class C of functions f which includes as an easy particular case. That particular case is in no way preferred over the other members of C Ù and therefore any fuss with fancy concocted names (see above and below) seems inappropriate. To demonstrate the power of my approach, I give a short proof of in the Borisov, Neff, Sra & Thiel’s special case by repeating [4; Example 2.3]. Not only that the proof is much shorter and more straightforward than the later one given by Borisov, Neff, Sra & Thiel [1]; it also indicates the right track to a generalization. It is therefore conceptual, a virtue that [1] does not seem to possess. [4; Example 2.3] requires no knowledge beyond the additivity of the logarithmon the real positive semi–axis and the elementary integration of fractions over the same semi–axis.