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Keywords = L’Hôpital’s rule of monotonicity

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9 pages, 248 KiB  
Article
Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus
by Vuk Stojiljković, Slobodan Radojević, Eyüp Çetin, Vesna Šešum Čavić and Stojan Radenović
Symmetry 2022, 14(6), 1260; https://doi.org/10.3390/sym14061260 - 18 Jun 2022
Cited by 5 | Viewed by 2374
Abstract
Sharp bounds for cosh(x)x,sinh(x)x, and sin(x)x were obtained, as well as one new bound for ex+arctan(x)x. A new situation to [...] Read more.
Sharp bounds for cosh(x)x,sinh(x)x, and sin(x)x were obtained, as well as one new bound for ex+arctan(x)x. A new situation to note about the obtained boundaries is the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary. New consequences of the inequalities were obtained in terms of the Riemann–Liovuille fractional integral and in terms of the standard integral. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)
10 pages, 271 KiB  
Article
Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions
by Yogesh J. Bagul, Ramkrishna M. Dhaigude, Marko Kostić and Christophe Chesneau
Axioms 2021, 10(4), 308; https://doi.org/10.3390/axioms10040308 - 18 Nov 2021
Cited by 7 | Viewed by 2391
Abstract
Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form [...] Read more.
Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given. Full article
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