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 (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.
Highlights
We know that the sinc and cosine functions, i.e., sin x/x and cos x, are less than 1 for 0 < x < π/2
In the first part of this work, we aim to provide new sharp bounds for polynomialexponential types of the form (1 − αx2)eβx2 for the sinc and cosine functions
We aim to present sharp polynomial-exponential bounds for the hyperbolic sinc and hyperbolic cosine functions
Summary
We know that the sinc and cosine functions, i.e., sin x/x and cos x, are less than 1 for 0 < x < π/2. 2. Main Theorems We begin with our new polynomial-exponential bounds of the form (1 − αx2)eβx for the sinc and cosine functions. Hold; β = −(ln 2)/π2 and β = 1/π2 − 1/6 are the best possible constants for lower and upper bounds for sin x/x of the form (1 − αx2)eβx with α = 1/π2, respectively. Hold; β = 4 ln(π/4)/π2 and β = 4/π2 − 1/2 are the best possible constants for lower and upper bounds for cos x of the form (1 − αx2)eβx with α = 4/π2, respectively. 4/π2 are the best possible constants for lower and upper bounds for cosh x of the form (1 − αx2)eβx with α = −4/π2, respectively The proofs of these new results, several applications, and a discussion of the significance of the findings and existing literature results are presented in the remainder of the work. The above preliminaries are the basis of the proofs of the main results, which are the subject of the section
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