Abstract

The monotonic functions were first introduced by S. Bernstein as functions which are non-negative with non-negative derivatives of all orders. He proved that such functions are necessarily analytic and he showed later that if a function is absolutely monotonic on the negative real axis then it can be represented there by a Laplace- Stieitjes integral with non-decreasing determining function and converse. Somewhat earlier F. Hausdorff had proved a similar result for completely monotonic sequences which essentially contained the Bernstein result. Bernstein was evidently unaware of Hausdorff's result, and his proof followed entirely independent lines. Since then many studies have been written on monotonic functions. In this work, we mainly have proved that a certain function involving ratio of the Euler gamma functions and some parameters is completely and logarithmically completely monotonic. Also, we have given the sufficient conditions for this function to be respectively completely and logarithmically completely monotonic. As applications, some inequalities involving the volume of the unite ball in the Euclidian space Rn are obtained. The established results not only unify and improve certain known inequalities including, but also can generate some new inequalities and the given results could trigger a new research direction in the theory of inequalities and special functions.

Highlights

  • A function is said to be completely monotonic [1] on interval ⊆ R if it has derivatives of all orders on and satisfies for all > 0 and ∈ N

  • We prove that the function h%,/ is logarithmically completely monotonic function in some cases

  • The established results could trigger a new research direction in the theory of inequalities and special functions

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Summary

Introduction

A positive function is said to be logarithmically completely monotonic (see for example [2]) on an interval It is known that any logarithmically completely monotonic function must be completely monotonic, but not [4]. The following results were investigated in [13]: The function is decreasing with respect to ≥ 1 for fixed 4 ≥ 0. The function (1) was proved to be logarithmically completely monotonic with respect to ∈ 0, ∞ for 4 ≥ 0 and so is its reciprocal for −1 < 4 ≤ − ) [14].

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