Some Properties of a Function Originating from Geometric Probability for Pairs of Hyperplanes Intersecting with a Convex Body

Feng Qi,Mansour Mahmoud

doi:10.3390/mca21030027

Abstract

In the paper, the authors derive an integral representation, present a double inequality, supply an asymptotic formula, find an inequality, and verify complete monotonicity of a function involving the gamma function and originating from geometric probability for pairs of hyperplanes intersecting with a convex body.

Highlights

The problem of studying the increasing property of the sequence pm m−1 2 π/2 sinm−1 t d t arises from geometric probability for pairs of hyperplanes intersecting with a convex body, see [1]
Remark 4.√Using the second conclusion in Theorem 4, for x ∈ N and r = 1, we can see that the sequence m αQm is increasing with respect to m ∈ N if and only if 0 < α ≤ 2
In the papers [13,14], the authors investigated by probabilistic methods and approaches the monotonicity of incomplete gamma functions and their ratios and applied their results to probability and actuarial area

Summary

Introduction

The problem of studying the increasing property of the sequence pm =m−1 2 π/2 sinm−1 t d t, m∈N arises from geometric probability for pairs of hyperplanes intersecting with a convex body, see [1]. In 2015, Qi et al [5] established an asymptotic formula for the function φ(x) = 2 ln Γ They posed two problems about the monotonicity of the sequence m αQm for 0 < α ≤ 2. We will derive an integral representation, present a double inequality, supply an asymptotic formula, find an inequality, and verify complete monotonicity of the function φ(x) or Q(x) = eφ(x).

Results

Conclusion