www.sciencepubco.com/index.php/GJMA doi: 10.14419/gjma.v2i3.3062
Research Paper
Absolute monotonicity of a function involving the exponential function
Feng Qi
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China
E-mail: [email protected],[email protected], [email protected] URL:http: // qifeng618. wordpress. com
Copyright c°2014 Feng Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In the paper, the author verifies the absolute monotonicity of a function involving the exponential function.
Keywords: absolute monotonicity; absolutely monotonic function; completely monotonic function; completely monotonic degree; ex- ponential function
MSC : Primary 26A48; Secondary 33B10, 44A10
1. Preliminaries and main results
Recall from [5, 15, 16] that a function f is said to be completely monotonic on an interval I if it has derivatives of all orders on I such that (−1)
kf
(k)(x) ≥ 0 for x ∈ I and k ≥ 0. Recall also from [5, 15, 16] that a function f is said to be absolutely monotonic on an interval I if it has derivatives of all orders and f
(k−1)(t) ≥ 0 for t ∈ I and k ∈ N, where N denotes the set of all positive integers.
It is easy to see that a function f (x) is completely monotonic in (a, b) if and only if f (−x) is absolutely monotonic in (−b, −a). See [16, p. 145, Definition 2c].
Theorem 12a in [16, p. 160] reads that a necessary and sufficient condition that f (x) should be completely monotonic in 0 ≤ x < ∞ is that f (x) = R
∞0
e
−xtd α(t), where α(t) is bounded and non-decreasing and the integral converges for 0 ≤ x < ∞. Theorem 12c in [16, p. 162] states that a necessary and sufficient condition that f (x) should be absolutely monotonic in −∞ < x < 0 is that f (x) = R
∞0
e
xtd α(t), where α(t) is non-decreasing and the integral converges for −∞ < x < 0.
For more information on these kinds of functions, please refer to [5, Chapter XIII], [16, Chapter IV], [2, 12, 15]
and closely related references therein.
The classical Euler’s gamma function Γ(x) may be defined for <(z) > 0 by Γ(z) = R
∞0
