# A Logarithmic Inequality

A Logarithmic Inequality The inequality $$\ln {\kern 1pt} \ln \left( {r - \ln r} \right) + 1 < \mathop {\min }\limits_{0 < x \leqslant r - 1} \left( {\ln x + {x^{ - 1}}\ln \left( {r - x} \right)} \right) < \ln {\kern 1pt} \ln \left( {r - \ln \left( {r - {2^{ - 1}}\ln r} \right)} \right) + 1,$$ ln ln ( r − ln r ) + 1 < min 0 < x ≤ r − 1 ( ln x + x − 1 ln ( r − x ) ) < ln ln ( r − ln ( r − 2 − 1 ln r ) ) + 1 , where r > 2, is proved. A combinatorial optimization problem which involves the function to be minimized is described. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Notes Springer Journals

# A Logarithmic Inequality

, Volume 103 (2) – Mar 14, 2018
12 pages

/lp/springer_journal/a-logarithmic-inequality-jAy0k50p44
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-4346
eISSN
1573-8876
D.O.I.
10.1134/S0001434618010224
Publisher site
See Article on Publisher Site

### Abstract

The inequality $$\ln {\kern 1pt} \ln \left( {r - \ln r} \right) + 1 < \mathop {\min }\limits_{0 < x \leqslant r - 1} \left( {\ln x + {x^{ - 1}}\ln \left( {r - x} \right)} \right) < \ln {\kern 1pt} \ln \left( {r - \ln \left( {r - {2^{ - 1}}\ln r} \right)} \right) + 1,$$ ln ln ( r − ln r ) + 1 < min 0 < x ≤ r − 1 ( ln x + x − 1 ln ( r − x ) ) < ln ln ( r − ln ( r − 2 − 1 ln r ) ) + 1 , where r > 2, is proved. A combinatorial optimization problem which involves the function to be minimized is described.

### Journal

Mathematical NotesSpringer Journals

Published: Mar 14, 2018

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