On the absolute monotonicity of generalized elliptic integral of the first kind
type: Item
instance of: scholarly article
title: On the absolute monotonicity of generalized elliptic integral of the first kind
zbMATH Open document ID: 1522.33011
author: Yajun Chen
author: Jiahui Wu
author: Tiehong Zhao
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2023-09-04T00:00:00Z
review text: The purpose of this paper is to prove that the second derivative of the generalized elliptic integral of the first kind \(K_a(r)\), is absolutely monotonic on (0, 1). The generalized elliptic integrals of the first kind, which can be expressed as a hypergeometric series, plays an important role in different branches of modern mathematics. Starting with the inequality for \(K_{\tfrac 12}(r)\), several functions are shown to be absolutely monotonic on (0, 1). Then several abbreviations are introduced for brevity. The proofs use generalized Wallis' ratio, long, long differentiations, partial differentiations, numerical experiment, induction. Finally, the main theorems are proved by using Maclaurin series with hypergeometric series. Several new functional inequalities are proved in the end of paper by using formulas for the derivative of hypergeometric series. Inequalities with the best possible constants are proved and open problems are given.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 7733899
zbMATH Keywords: Gaussian hypergeometric function
zbMATH Keywords: generalized elliptic integrals
zbMATH Keywords: absolutely monotonic property
MaRDI profile type: Publication
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DOI: 10.1007/S13398-023-01472-0
description: scientific article; zbMATH DE number 7733899