Inequalities for Information Potentials and Entropies
We consider a probability distribution <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>p</mi><mn>0</mn></msub><mrow><mo>(</mo>&l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/11/2056 |
| _version_ | 1797547548972417024 |
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| author | Ana Maria Acu Alexandra Măduţa Diana Otrocol Ioan Raşa |
| author_facet | Ana Maria Acu Alexandra Măduţa Diana Otrocol Ioan Raşa |
| author_sort | Ana Maria Acu |
| collection | DOAJ |
| description | We consider a probability distribution <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>p</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>p</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo></mfenced></semantics></math></inline-formula> depending on a real parameter <i>x</i>. The associated information potential is <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msubsup><mi>p</mi><mrow><mi>k</mi></mrow><mn>2</mn></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The Rényi entropy and the Tsallis entropy of order 2 can be expressed as <inline-formula><math display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo form="prefix">log</mo><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We establish recurrence relations, inequalities and bounds for <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which lead immediately to similar relations, inequalities and bounds for the two entropies. We show that some sequences <inline-formula><math display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula>, associated with sequences of classical positive linear operators, are concave and increasing. Two conjectures are formulated involving the information potentials associated with the Durrmeyer density of probability, respectively the Bleimann–Butzer–Hahn probability distribution. |
| first_indexed | 2024-03-10T14:45:38Z |
| format | Article |
| id | doaj.art-37569925d437449fabfc593b6e7e97a2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T14:45:38Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-37569925d437449fabfc593b6e7e97a22023-11-20T21:21:37ZengMDPI AGMathematics2227-73902020-11-01811205610.3390/math8112056Inequalities for Information Potentials and EntropiesAna Maria Acu0Alexandra Măduţa1Diana Otrocol2Ioan Raşa3Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No. 5-7, RO-550012 Sibiu, RomaniaDepartment of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, RomaniaDepartment of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, RomaniaDepartment of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114 Cluj-Napoca, RomaniaWe consider a probability distribution <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><msub><mi>p</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>p</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mo>…</mo></mfenced></semantics></math></inline-formula> depending on a real parameter <i>x</i>. The associated information potential is <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msubsup><mi>p</mi><mrow><mi>k</mi></mrow><mn>2</mn></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The Rényi entropy and the Tsallis entropy of order 2 can be expressed as <inline-formula><math display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo form="prefix">log</mo><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We establish recurrence relations, inequalities and bounds for <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which lead immediately to similar relations, inequalities and bounds for the two entropies. We show that some sequences <inline-formula><math display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi>R</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi>T</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfenced><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula>, associated with sequences of classical positive linear operators, are concave and increasing. Two conjectures are formulated involving the information potentials associated with the Durrmeyer density of probability, respectively the Bleimann–Butzer–Hahn probability distribution.https://www.mdpi.com/2227-7390/8/11/2056probability distributionRényi entropyTsallis entropyinformation potentialfunctional equationsinequalities |
| spellingShingle | Ana Maria Acu Alexandra Măduţa Diana Otrocol Ioan Raşa Inequalities for Information Potentials and Entropies Mathematics probability distribution Rényi entropy Tsallis entropy information potential functional equations inequalities |
| title | Inequalities for Information Potentials and Entropies |
| title_full | Inequalities for Information Potentials and Entropies |
| title_fullStr | Inequalities for Information Potentials and Entropies |
| title_full_unstemmed | Inequalities for Information Potentials and Entropies |
| title_short | Inequalities for Information Potentials and Entropies |
| title_sort | inequalities for information potentials and entropies |
| topic | probability distribution Rényi entropy Tsallis entropy information potential functional equations inequalities |
| url | https://www.mdpi.com/2227-7390/8/11/2056 |
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