A new lower bound in the second Kershaw's double inequality

Abstract

In the paper, a new and elegant lower bound in the second Kershaw's double inequality is established, some alternative simple and polished proofs are given, several deduced functions involving the gamma and psi functions are proved to be decreasingly monotonic and logarithmically completely monotonic, and some remarks and comparisons are stated.

Integral Representations for Bivariate Complex Geometric Mean and Applications

“…Lévy-Khintchine representation of bivariate logarithmic mean. In [22,Remark 3.7] and its preprint [23,Remark 3.7], the logarithmic mean L a,b (x) was proved to be increasing and concave in x ∈ (− min{a, b}, ∞).…”

A new upper bound in the second Kershaw's double inequality and its generalizations

“…In [13], the second Kershaw's double inequality (12) was generalized, extended and refined as follows: for s, t ∈ R with s = t, the function…”

“…Inequalities(12),(13),(15),(16),(19),(21),(22) and the increasingly monotonic property of the generalized logarithmic mean L p (a, b) suggest us a question: What are the best constants and such that e (L (s,t)) | (n)(L (s, t))| | (n−1) (s)| − | (n−1) (t)| s − t − | (n) (L (s, t))|(27)are valid for positive real numbers s and t?…”

Refinements, extensions and generalizations of the second Kershaw's double inequality

“…For more information, please refer to [5,7,9,10,11,12,14,19,20,21,22,23,30,32,34,35,36,38,39,40,41,49] and the references therein.…”

“…In [23], the following generalization, extension and refinement of the second Kershaw's double inequality (2) were obtained: For s, t ∈ R with s = t, the function…”