A Look at Generalized Trigonometric Functions as Functions of Their Two Parameters and Further New Properties

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this pape...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 12; no. 21; p. 3383
Main Authors: Karp, Dmitrii, Prilepkina, Elena
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.11.2024
Subjects:
(p,q)-Laplacian
Concavity
Convexity
Distributions, Theory of (Functional analysis)
generalized hyperbolic functions
generalized trigonometric functions
Hyperbolic functions
Hypergeometric functions
integral representation
log-concavity
log-convexity
Mathematical research
Methods
Parameter estimation
Parameters
Trigonometric functions
Trigonometrical functions
Israel
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math12213383