Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions
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 <inline-formula><math xmlns="http://www....
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MDPI AG
2021-11-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/4/308 |
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| author | Yogesh J. Bagul Ramkrishna M. Dhaigude Marko Kostić Christophe Chesneau |
| author_facet | Yogesh J. Bagul Ramkrishna M. Dhaigude Marko Kostić Christophe Chesneau |
| author_sort | Yogesh J. Bagul |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi><mrow><mi>β</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></semantics></math></inline-formula> for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula>, while our main result for the cosine function is a double inequality holding on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>π</mi><mo>/</mo><mn>2</mn><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> 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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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T04:35:08Z |
| publishDate | 2021-11-01 |
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| spelling | doaj.art-d75d7db5f3ec4ac488ec70d081e2cbf52023-11-23T03:49:52ZengMDPI AGAxioms2075-16802021-11-0110430810.3390/axioms10040308Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic FunctionsYogesh J. Bagul0Ramkrishna M. Dhaigude1Marko Kostić2Christophe Chesneau3Department of Mathematics, K. K. M. College, Manwath 431505, IndiaDepartment of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati 444604, IndiaFaculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, SerbiaDepartment of Mathematics, University of Caen-Normandie, 14032 Caen, FranceRecent 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><msup><mi>e</mi><mrow><mi>β</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></mrow></semantics></math></inline-formula> for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula>, while our main result for the cosine function is a double inequality holding on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>π</mi><mo>/</mo><mn>2</mn><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> 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.https://www.mdpi.com/2075-1680/10/4/308polynomial-exponential boundsl’Hôpital’s rule of monotonicityBernoulli numbersJordan’s inequalityKober’s inequalitytrigonometric functions |
| spellingShingle | Yogesh J. Bagul Ramkrishna M. Dhaigude Marko Kostić Christophe Chesneau Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions Axioms polynomial-exponential bounds l’Hôpital’s rule of monotonicity Bernoulli numbers Jordan’s inequality Kober’s inequality trigonometric functions |
| title | Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions |
| title_full | Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions |
| title_fullStr | Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions |
| title_full_unstemmed | Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions |
| title_short | Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions |
| title_sort | polynomial exponential bounds for some trigonometric and hyperbolic functions |
| topic | polynomial-exponential bounds l’Hôpital’s rule of monotonicity Bernoulli numbers Jordan’s inequality Kober’s inequality trigonometric functions |
| url | https://www.mdpi.com/2075-1680/10/4/308 |
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