Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind

Let K r be the complete elliptic integral of the first kind defined on 0 , 1 . By virtue of the auxiliary function H f , g = f ′ / g ′ g - f , we prove that the function Q p x = ln p / 1 - x K x is strictly convex on 0 , 1 if and only if 0 < p ≤ 4 , thus answering a conjecture. Moreover, we compl...

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Bibliographic Details
Published in:Resultate der Mathematik Vol. 78; no. 1; p. 29
Main Authors: Tian, Jing-Feng, Yang, Zhen-Hang
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.02.2023
Springer Nature B.V
Subjects:
Convexity
Mathematics
Mathematics and Statistics
inequality
Primary 33C05
26A51
Secondary 39B62
hypergeometric function
convexity
Complete elliptic integral of the first kind
ISSN:1422-6383, 1420-9012
Online Access:Get full text
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Summary:Let K r be the complete elliptic integral of the first kind defined on 0 , 1 . By virtue of the auxiliary function H f , g = f ′ / g ′ g - f , we prove that the function Q p x = ln p / 1 - x K x is strictly convex on 0 , 1 if and only if 0 < p ≤ 4 , thus answering a conjecture. Moreover, we completely described the monotonicity of Q p x on 0 , 1 for different p ∈ 0 , ∞ .
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-022-01799-x