ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 209–220. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © G. V. Kalachev, S. Yu. Sadov, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 210–222.
A Logarithmic Inequality
G. V. Kalachev
and S. Yu. Sadov
Lomonosov Moscow State University, Moscow, Russia
Received February 12, 2017; in ﬁnal form, April 23, 2017
ln ln(r − ln r)+1< min
(ln x + x
ln(r − x)) < ln ln(r − ln(r − 2
ln r)) + 1,
where r>2, is proved. A combinatorial optimization problem which involves the function to be
minimized is described.
Keywords: logarithmic inequality, two-sided estimate, extremal graph.
1. THE THEOREM AND DISCUSSION
The real function
k(x, r)=lnx +
ln(r − x)
is deﬁned for 0 <x<r. In the case r>2, the function k( · ,r) has a local minimum at some point
x = ξ(r) <r− 1 and a local maximum at the point x = r − 1,sothatξ(r) is a point of global minimum
of k( · ,r) on the interval (0,r − 1]. In this paper, we give the proof of the two-sided inequality providing
the approximate value of the minimum
Let us extend by continuity (see Sec. 2): ξ(2) = 1, K(2) = 0.
Theorem 1. For all r ≥ 2,
ln ln(r − ln r) <K(r) − 1 < ln ln
r − ln
It is easy to show that K(r) = ln ln r +1+o(1) as r →∞. The inequalities in Theorem 1 resulted
from the search of reﬁned and nonasymptotic versions of this simple estimate. As analogs, we
can indicate similar two-sided estimates of some useful mathematical functions, such as variants of
Stirling’s formula , , estimates for (1 + x
, etc. The graphs (Figs. 1, 2) illustrate the accuracy
of the inequalities of Theorem 1.
The paper consists of the main part (Secs. 1–4) and two appendices. In the main part, after the
preparotory Sec. 2, we prove the lower bound (Sec. 3) and, in greater detail, the upper bound (Sec. 4).
There are two appendices: Appendix 1 contains the derivation of asymptotic formulas (2), (3) (given
below) and Appendix 2 describes a combinatorial optimization problem in which the answer (for the
case of an integer r)isgivenbythefunction