Next Article in Journal
Relationships of Associated Curves of Mixed-Type Curves and Their Singularities in Minkowski Plane
Previous Article in Journal
Minification Integer-Valued Split-BREAK Process with Power Series Innovations and Application in Fire Safety Dynamics
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Axioms
Volume 15
Issue 6
10.3390/axioms15060389
axioms-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Hyperbolic Extension of Parabolic Trigonometry: Wilker, Lazarević, Wu-Debnah, Cusa-Huygens and Shafer Type Inequalities

by
Jan Jekl
and
Josef Rebenda
*
Department of Mathematics and Physics, Faculty of Military Technology, University of Defence, 662 10 Brno, Czech Republic
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(6), 389; https://doi.org/10.3390/axioms15060389 (registering DOI)
Submission received: 10 April 2026 / Revised: 15 May 2026 / Accepted: 19 May 2026 / Published: 23 May 2026
(This article belongs to the Section Mathematical Analysis)
Versions Notes

Abstract

Recently, a new generalization of hyperbolic functions called para-hyperbolic functions has been introduced. However, properties of the para-hyperbolic functions have not been investigated yet. In this paper, we derive the correct explicit formulas of the para-hyperbolic sine and cosine, study elementary properties of these functions, and explore which of the inequalities that hold for trigonometric and hyperbolic functions find their counterparts for para-hyperbolic functions. Namely, we prove a Wilker type inequality, Cusa-Huygens and Lazarević type inequality, Wu-Debnah modification of Wilker type inequality and Shafer type inequality for para-hyperbolic functions. Our results provide the first thorough exploration of para-hyperbolic functions which establishes groundwork for further discoveries and applications.
Keywords: parabolic trigonometry; Wilker inequality; Lazarević inequality; Wu-Debnah inequality; Shafer inequality; Cusa-Huygens inequality; hyperbolic functions; para-hyperbolic functions parabolic trigonometry; Wilker inequality; Lazarević inequality; Wu-Debnah inequality; Shafer inequality; Cusa-Huygens inequality; hyperbolic functions; para-hyperbolic functions

Share and Cite

MDPI and ACS Style

Jekl, J.; Rebenda, J. Hyperbolic Extension of Parabolic Trigonometry: Wilker, Lazarević, Wu-Debnah, Cusa-Huygens and Shafer Type Inequalities. Axioms 2026, 15, 389. https://doi.org/10.3390/axioms15060389

AMA Style

Jekl J, Rebenda J. Hyperbolic Extension of Parabolic Trigonometry: Wilker, Lazarević, Wu-Debnah, Cusa-Huygens and Shafer Type Inequalities. Axioms. 2026; 15(6):389. https://doi.org/10.3390/axioms15060389

Chicago/Turabian Style

Jekl, Jan, and Josef Rebenda. 2026. "Hyperbolic Extension of Parabolic Trigonometry: Wilker, Lazarević, Wu-Debnah, Cusa-Huygens and Shafer Type Inequalities" Axioms 15, no. 6: 389. https://doi.org/10.3390/axioms15060389

APA Style

Jekl, J., & Rebenda, J. (2026). Hyperbolic Extension of Parabolic Trigonometry: Wilker, Lazarević, Wu-Debnah, Cusa-Huygens and Shafer Type Inequalities. Axioms, 15(6), 389. https://doi.org/10.3390/axioms15060389

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop