New Inequalities of Cusa–Huygens Type
Using the power series expansions of the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">cot</mo><mi>x</mi><mo>,</mo><mn>1<...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/17/2101 |
| _version_ | 1797521118480826368 |
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| author | Ling Zhu |
| author_facet | Ling Zhu |
| author_sort | Ling Zhu |
| collection | DOAJ |
| description | Using the power series expansions of the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">cot</mo><mi>x</mi><mo>,</mo><mn>1</mn><mo>/</mo><mo form="prefix">sin</mo><mi>x</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mo form="prefix">sin</mo><mn>2</mn></msup><mi>x</mi></mrow></semantics></math></inline-formula>, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mi>π</mi><mo>/</mo><mn>2</mn></mfenced></semantics></math></inline-formula>. Our results are much better than those in the existing literature. |
| first_indexed | 2024-03-10T08:07:07Z |
| format | Article |
| id | doaj.art-adaf10de8a3f42ddb31daf75f5f73bb8 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T08:07:07Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-adaf10de8a3f42ddb31daf75f5f73bb82023-11-22T10:57:55ZengMDPI AGMathematics2227-73902021-08-01917210110.3390/math9172101New Inequalities of Cusa–Huygens TypeLing Zhu0Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, ChinaUsing the power series expansions of the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">cot</mo><mi>x</mi><mo>,</mo><mn>1</mn><mo>/</mo><mo form="prefix">sin</mo><mi>x</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mo form="prefix">sin</mo><mn>2</mn></msup><mi>x</mi></mrow></semantics></math></inline-formula>, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mn>0</mn><mo>,</mo><mi>π</mi><mo>/</mo><mn>2</mn></mfenced></semantics></math></inline-formula>. Our results are much better than those in the existing literature.https://www.mdpi.com/2227-7390/9/17/2101sharp the double inequalities of Cusa–Huygens typecircular functionsBernoulli numbers |
| spellingShingle | Ling Zhu New Inequalities of Cusa–Huygens Type Mathematics sharp the double inequalities of Cusa–Huygens type circular functions Bernoulli numbers |
| title | New Inequalities of Cusa–Huygens Type |
| title_full | New Inequalities of Cusa–Huygens Type |
| title_fullStr | New Inequalities of Cusa–Huygens Type |
| title_full_unstemmed | New Inequalities of Cusa–Huygens Type |
| title_short | New Inequalities of Cusa–Huygens Type |
| title_sort | new inequalities of cusa huygens type |
| topic | sharp the double inequalities of Cusa–Huygens type circular functions Bernoulli numbers |
| url | https://www.mdpi.com/2227-7390/9/17/2101 |
| work_keys_str_mv | AT lingzhu newinequalitiesofcusahuygenstype |