Abstract
Some complete monotonicity results that the functions±1/e±t-1are logarithmically completely monotonic, and that differences between consecutive derivatives of these two functions are completely monotonic, and that the ratios between consecutive derivatives of these two functions are decreasing on0, ∞are discovered. As applications of these newly discovered results, some complete monotonicity results concerning the polylogarithm are found. Finally a conjecture on the complete monotonicity of the above-mentioned ratios is posed.
Highlights
Introduction and Main ResultsThroughout this paper, we denote the set of all positive integers by N.Recently, the following problem was posed in [1, page 569]
), where t ∈ (0, ∞) and i ∈ {0} ∪ N, and we firstly discover in this paper the following results
For more information on the theory of completely monotonic functions, please refer to [4, Chapter XIII], [5, Chapter IV], and the newly published monograph [6]. This means that it is useful to confirm the complete monotonicity of functions
Summary
Throughout this paper, we denote the set of all positive integers by N. For t ≠ 0 and k ∈ N, determine the numbers ak,i−1 for 1 ≤ i ≤ k such that This problem, among other things, was answered in [1] by eight identities. The functions F0(t) and G0(t) are logarithmically completely monotonic on (0, ∞). Gi+1 (t) = Gi+2 (t) − Gi+1 (t) are completely monotonic functions on (0, ∞). We pose a conjecture on the complete monotonicity of the functions Fi(t) and Gi(t) defined in (10)
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