Topsøe, Flemming (2004) Some Bounds for the Logarithmic Function. Research report collection, 7 (2).

Abstract

Bounds for the logarithmic function are studied. In particular, we establish bounds with rational functions as approximants. The study leads into the fascinating areas of Padé approximations ([2], [6]), continued fractions ([7], [11]) and orthogonal polynomials ([14], [4]) as well as the somewhat frightening jungle of special functions and associated identities ([5], [9]). Originally, the results aimed at establishing certain inequalities for Shannon entropy but are here discussed in their own right (the applications to entropy inequalities will be published elsewhere). The exposition is informal, a kind of essay, with only occasional indications of proofs. The reader may take it as an invitation to further studies. Enough details are provided to enable the reader to verify all statements. To the expert in the fields pointed to there is little or nothing new.

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