Generalized bounds for sine and cosine functions

Bagul, Yogesh J.; Chesnea, Christophe

doi:10.1142/s1793557122500127

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…The lower bound of the cosine function in (7) is sharper than that in (5) for the interval (λ 2 , π/2), where λ 2 ≈ 1.2221, and we also obtained the upper bound for the cosine function in (7).…”

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confidence: 51%

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Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Bagul¹,

Dhaigude

Kostić

et al. 2021

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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 (1−αx2)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.

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confidence: 51%

“…Now, we aim to present sharp polynomial-exponential bounds for the hyperbolic sinc and hyperbolic cosine functions. In particular, we establish hyperbolic counterparts of ( 6) and (7) in the following theorems. Theorem 3.…”

Section: Main Theoremsmentioning

confidence: 99%

Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Bagul¹,

Dhaigude

Kostić

et al. 2021

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“…In this paper, we will sharpen these bounds in a simple and efficient manner. More about such inequalities can be found in the following papers [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus

et al. 2022

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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.

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