Monotonicity and positivity of several functions involving ratios and products of polygamma functions
Abstract Let ψ ( x ) $\psi (x)$ denote the digamma function, that is, the logarithmic derivative of the classical Euler gamma function Γ ( x ) $\Gamma (x)$ . In the paper, the authors discover the monotonic properties of the functions ψ ( n ) ( x ) x ψ ( n + 1 ) ( x ) and ψ ( n ) ( x ) ψ ( n ) ( 1 x...
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| Format: | Article |
| Language: | English |
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2025-01-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-024-03245-8 |
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| author | Feng Qi Dongkyu Lim Kwara Nantomah |
| author_facet | Feng Qi Dongkyu Lim Kwara Nantomah |
| author_sort | Feng Qi |
| collection | DOAJ |
| description | Abstract Let ψ ( x ) $\psi (x)$ denote the digamma function, that is, the logarithmic derivative of the classical Euler gamma function Γ ( x ) $\Gamma (x)$ . In the paper, the authors discover the monotonic properties of the functions ψ ( n ) ( x ) x ψ ( n + 1 ) ( x ) and ψ ( n ) ( x ) ψ ( n ) ( 1 x ) $$ \frac{\psi ^{(n)}(x)}{x\psi ^{(n+1)}(x)} \quad \text{and}\quad \psi ^{(n)}(x) \psi ^{(n)}\biggl(\frac{1}{x}\biggr) $$ for n ≥ 0 $n\ge 0$ on ( 0 , ∞ ) $(0,\infty )$ . With the aid of these monotonic properties, the authors confirm the positivity of the function ψ ( x ) + x ψ ′ ( x ) − ψ ( x ) ψ ( 1 x ) $$ \psi (x)+x\psi '(x)-\psi (x)\psi \biggl(\frac{1}{x}\biggr) $$ on ( 0 , ∞ ) $(0,\infty )$ . The authors also pose five open problems, generalizing the latter two of the three functions mentioned above and their related conclusions. |
| format | Article |
| id | doaj-art-a25f9252325943d386374929e617c2e0 |
| institution | Kabale University |
| issn | 1029-242X |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj-art-a25f9252325943d386374929e617c2e02025-01-19T12:42:59ZengSpringerOpenJournal of Inequalities and Applications1029-242X2025-01-012025111010.1186/s13660-024-03245-8Monotonicity and positivity of several functions involving ratios and products of polygamma functionsFeng Qi0Dongkyu Lim1Kwara Nantomah2School of Mathematics and Informatics, Henan Polytechnic UniversityDepartment of Mathematics Education, Andong National UniversitySchool of Mathematical Sciences, C. K. Tedam University of Technology and Applied SciencesAbstract Let ψ ( x ) $\psi (x)$ denote the digamma function, that is, the logarithmic derivative of the classical Euler gamma function Γ ( x ) $\Gamma (x)$ . In the paper, the authors discover the monotonic properties of the functions ψ ( n ) ( x ) x ψ ( n + 1 ) ( x ) and ψ ( n ) ( x ) ψ ( n ) ( 1 x ) $$ \frac{\psi ^{(n)}(x)}{x\psi ^{(n+1)}(x)} \quad \text{and}\quad \psi ^{(n)}(x) \psi ^{(n)}\biggl(\frac{1}{x}\biggr) $$ for n ≥ 0 $n\ge 0$ on ( 0 , ∞ ) $(0,\infty )$ . With the aid of these monotonic properties, the authors confirm the positivity of the function ψ ( x ) + x ψ ′ ( x ) − ψ ( x ) ψ ( 1 x ) $$ \psi (x)+x\psi '(x)-\psi (x)\psi \biggl(\frac{1}{x}\biggr) $$ on ( 0 , ∞ ) $(0,\infty )$ . The authors also pose five open problems, generalizing the latter two of the three functions mentioned above and their related conclusions.https://doi.org/10.1186/s13660-024-03245-8MonotonicityPositivityRatioProductGamma functionDigamma function |
| spellingShingle | Feng Qi Dongkyu Lim Kwara Nantomah Monotonicity and positivity of several functions involving ratios and products of polygamma functions Journal of Inequalities and Applications Monotonicity Positivity Ratio Product Gamma function Digamma function |
| title | Monotonicity and positivity of several functions involving ratios and products of polygamma functions |
| title_full | Monotonicity and positivity of several functions involving ratios and products of polygamma functions |
| title_fullStr | Monotonicity and positivity of several functions involving ratios and products of polygamma functions |
| title_full_unstemmed | Monotonicity and positivity of several functions involving ratios and products of polygamma functions |
| title_short | Monotonicity and positivity of several functions involving ratios and products of polygamma functions |
| title_sort | monotonicity and positivity of several functions involving ratios and products of polygamma functions |
| topic | Monotonicity Positivity Ratio Product Gamma function Digamma function |
| url | https://doi.org/10.1186/s13660-024-03245-8 |
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