Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function

Mahmoud, Mansour; Almuashi, Hanan; Mortici, Cristinel

doi:10.3390/math14111898

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Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function

by

Mansour Mahmoud

^1,*

,

Hanan Almuashi

²

and

Cristinel Mortici

^3,4,5

¹

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²

Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

³

Academy of Romanian Scientists, Strada IIfov nr. 3, 050044 Bucureşti, Romania

⁴

Department of Science and Advanced, Faculty of Sciences and Arts, Technologies, Valahia University of Târgovişte, Aleea Sinaia 13, 130004 Târgovişte, Romania

⁵

Doctoral School of Applied Sciences , National University of Science and Technology Politehnica Bucureşti, Splaiul Independenţei 313, 060042 Bucureşti, Romania

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(11), 1898; https://doi.org/10.3390/math14111898

Submission received: 17 April 2026 / Revised: 28 May 2026 / Accepted: 29 May 2026 / Published: 29 May 2026

(This article belongs to the Special Issue Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications)

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Abstract

This paper develops two novel arcsine-type analytic approximation formulas for

arcsl (x)

, the Gauss lemniscate sine function, each equipped with a monotonic and bounded remainder term of order

x^{29}

as

x \to 0

, which demonstrates the high accuracy of the obtained formulas near the origin. Based on these approximations, we establish several new sharp inequalities valid on the interval

(0, 1)

. Numerical evidence confirms that the resulting bounds provide improved accuracy compared with existing estimates in the literature, particularly in a neighborhood of the origin.

Keywords: Gauss lemniscate sine function; arcsine function; monotonicity; approximation formula; inequality

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MDPI and ACS Style

Mahmoud, M.; Almuashi, H.; Mortici, C. Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function. Mathematics 2026, 14, 1898. https://doi.org/10.3390/math14111898

AMA Style

Mahmoud M, Almuashi H, Mortici C. Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function. Mathematics. 2026; 14(11):1898. https://doi.org/10.3390/math14111898

Chicago/Turabian Style

Mahmoud, Mansour, Hanan Almuashi, and Cristinel Mortici. 2026. "Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function" Mathematics 14, no. 11: 1898. https://doi.org/10.3390/math14111898

APA Style

Mahmoud, M., Almuashi, H., & Mortici, C. (2026). Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function. Mathematics, 14(11), 1898. https://doi.org/10.3390/math14111898

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Sharp Arcsine-Type Bounds and Analytic Approximations for the Gauss Lemniscate Sine Function

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