Öz
Anahtar Kelimeler
Dirichlet beta function Euler number increasing property logarithmic convexity ratio integral representation
Kaynakça
- [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.
- [2] J. A. Adell and A. Lekuona, Dirichlet’s eta and beta functions: concavity and fast computation of their derivatives, J. Number Theory 157, 215–222, 2015.
- [3] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.
- [4] B.-N. Guo and F. Qi, Increasing property and logarithmic convexity of functions involving Riemann zeta function, https://arxiv.org/abs/2201.06970, 2022.
- [5] Kh. Hessami Pilehrood and T. Hessami Pilehrood, Series acceleration formulas for beta values, Discrete Math. Theor. Comput. Sci. 12 (2), 223–236, 2010.
- [6] D. Lim and F. Qi, Increasing property and logarithmic convexity of two functions involving Dirichlet eta function, J. Math. Inequal. 16 (2), 463–469, 2022.
- [7] F. Qi, Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish J. Anal. Number Theory 6 (5), 129–131, 2018.
- [8] F. Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math. 351, 1–5, 2019.
- [9] F. Qi, Decreasing properties of two ratios defined by three and four polygamma functions, C. R. Math. Acad. Sci. Paris 360, 89–101, 2022.
- [10] F. Qi, W.-H. Li, S.-B. Yu, X.-Y. Du, and B.-N. Guo, A ratio of finitely many gamma functions and its properties with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 115 (2), 2021.
- [11] Y. Shuang, B.-N. Guo, and F. Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (3), 2021.
- [12] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.
- [13] C.-F. Wei, Integral representations and inequalities of extended central binomial coefficients, Math. Methods Appl. Sci. 45 (9), 5412–5422, 2022.
- [14] Z.-H. Yang and J.-F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math. 364, 112359, 2020.
- [15] L. Zhu, New bounds for the ratio of two adjacent even-indexed Bernoulli numbers, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2), 2020.
Increasing property and logarithmic convexity concerning Dirichlet beta function, Euler numbers, and their ratios
Öz
In the paper, by virtue of an integral representation of the Dirichlet beta function, with the aid of a relation between the Dirichlet beta function and the Euler numbers, and by means of a monotonicity rule for the ratio of two definite integrals with a parameter, the author finds increasing property and logarithmic convexity of two functions and two sequences involving the Dirichlet beta function, the Euler numbers, and their ratios.
Anahtar Kelimeler
Dirichlet beta function Euler number increasing property logarithmic convexity ratio integral representation
Kaynakça
- [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.
- [2] J. A. Adell and A. Lekuona, Dirichlet’s eta and beta functions: concavity and fast computation of their derivatives, J. Number Theory 157, 215–222, 2015.
- [3] J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.
- [4] B.-N. Guo and F. Qi, Increasing property and logarithmic convexity of functions involving Riemann zeta function, https://arxiv.org/abs/2201.06970, 2022.
- [5] Kh. Hessami Pilehrood and T. Hessami Pilehrood, Series acceleration formulas for beta values, Discrete Math. Theor. Comput. Sci. 12 (2), 223–236, 2010.
- [6] D. Lim and F. Qi, Increasing property and logarithmic convexity of two functions involving Dirichlet eta function, J. Math. Inequal. 16 (2), 463–469, 2022.
- [7] F. Qi, Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish J. Anal. Number Theory 6 (5), 129–131, 2018.
- [8] F. Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, J. Comput. Appl. Math. 351, 1–5, 2019.
- [9] F. Qi, Decreasing properties of two ratios defined by three and four polygamma functions, C. R. Math. Acad. Sci. Paris 360, 89–101, 2022.
- [10] F. Qi, W.-H. Li, S.-B. Yu, X.-Y. Du, and B.-N. Guo, A ratio of finitely many gamma functions and its properties with applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 115 (2), 2021.
- [11] Y. Shuang, B.-N. Guo, and F. Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (3), 2021.
- [12] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.
- [13] C.-F. Wei, Integral representations and inequalities of extended central binomial coefficients, Math. Methods Appl. Sci. 45 (9), 5412–5422, 2022.
- [14] Z.-H. Yang and J.-F. Tian, Sharp bounds for the ratio of two zeta functions, J. Comput. Appl. Math. 364, 112359, 2020.
- [15] L. Zhu, New bounds for the ratio of two adjacent even-indexed Bernoulli numbers, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114 (2), 2020.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Yayımlanma Tarihi | 15 Şubat 2023 |
| Yayımlandığı Sayı | Yıl 2023 Cilt: 52 Sayı: 1 |