Optimal bounds for the tangent and hyperbolic sine means III

Monika Nowicka; Alfred Witkowski

doi:10.33205/cma.1667818

Optimal bounds for the tangent and hyperbolic sine means III

Abstract

In this note, the most accurate possible bounds for the tangent and hyperbolic sine means are established. These bounds are articulated using diverse weighted combinations of the arithmetic mean and the minimum mean.

Keywords

References

I. Lazarević: Certain inequalities with hyperbolic functions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 170 (1966), 41–48.
M. Nowicka, A. Witkowski: Optimal bounds for the tangent and hyperbolic sine means, Aequationes Math., 94(5) (2020), 817–827.
M. Nowicka, A. Witkowski: Optimal bounds for the tangent and hyperbolic sine means II, J. Math. Inequal., 14(1) (2020), 23–33.
M. Nowicka, A. Witkowski: Optimal bounds for the sine and hyperbolic tangent means III, Jaen J. Approx., 13 (2022), 29–41.
M. Nowicka, A. Witkowski: Optimal bounds for the arithmetic mean in terms of new Seiffert-like means, Math. Ineqal. Apps., 23(1) (2020), 383–392.
A. Witkowski: On Seiffert-like means, J. Math. Inequal., 9(4) (2015), 1071–1092.
T.-H. Zhao, W.-M. Qian and Y.-M. Chu: Sharp power mean bounds for the tangent and hyperbolic sine means, J. Math. Inequal., 15(4) (2021), 1459–1472.
T.-H. Zhao, W.-M. Qian and Y.-P. Lv: Refinements of bounds for the arithmetic mean by new Seiffert-like means, AIMS Mathematics, 6(8) (2021), 9036–9047.

Details

Primary Language

English

Subjects

Real and Complex Functions (Incl. Several Variables), Applied Mathematics (Other)

Journal Section

Research Article

Authors

Monika Nowicka ^*
0000-0001-8193-4080
Poland

Alfred Witkowski This is me
0000-0003-1901-013X
Poland

Publication Date

June 16, 2026

Submission Date

March 28, 2025

Acceptance Date

April 22, 2026

Published in Issue

Year 2026 Volume: 9 Number: 2

DOI

https://doi.org/10.33205/cma.1667818

IZ

https://izlik.org/JA63TM95WM

Cite

RIS / Bibtex

APA

Nowicka, M., & Witkowski, A. (2026). Optimal bounds for the tangent and hyperbolic sine means III. Constructive Mathematical Analysis, 9(2), 55-61. https://doi.org/10.33205/cma.1667818

AMA

1.Nowicka M, Witkowski A. Optimal bounds for the tangent and hyperbolic sine means III. CMA. 2026;9(2):55-61. doi:10.33205/cma.1667818

Chicago

Nowicka, Monika, and Alfred Witkowski. 2026. “Optimal Bounds for the Tangent and Hyperbolic Sine Means III”. Constructive Mathematical Analysis 9 (2): 55-61. https://doi.org/10.33205/cma.1667818.

EndNote

Nowicka M, Witkowski A (June 1, 2026) Optimal bounds for the tangent and hyperbolic sine means III. Constructive Mathematical Analysis 9 2 55–61.

IEEE

[1]M. Nowicka and A. Witkowski, “Optimal bounds for the tangent and hyperbolic sine means III”, CMA, vol. 9, no. 2, pp. 55–61, June 2026, doi: 10.33205/cma.1667818.

ISNAD

Nowicka, Monika - Witkowski, Alfred. “Optimal Bounds for the Tangent and Hyperbolic Sine Means III”. Constructive Mathematical Analysis 9/2 (June 1, 2026): 55-61. https://doi.org/10.33205/cma.1667818.

JAMA

1.Nowicka M, Witkowski A. Optimal bounds for the tangent and hyperbolic sine means III. CMA. 2026;9:55–61.

MLA

Nowicka, Monika, and Alfred Witkowski. “Optimal Bounds for the Tangent and Hyperbolic Sine Means III”. Constructive Mathematical Analysis, vol. 9, no. 2, June 2026, pp. 55-61, doi:10.33205/cma.1667818.

Vancouver

1.Monika Nowicka, Alfred Witkowski. Optimal bounds for the tangent and hyperbolic sine means III. CMA. 2026 Jun. 1;9(2):55-61. doi:10.33205/cma.1667818