Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function

• Autorzy: Yin Hong-Ping , Han Ling-Xiong , Qi Feng

Identyfikatory

DOI

10.1515/dema-2024-0041

• Języki publikacji: EN

Abstrakty

EN

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

Słowa kluczowe

EN

decreasing monotonicity; complete monotonicity; completely monotonic function; trigamma function; derivative; ratio; convolution theorem; inequality; monotonicity rule; Laplace transform; Bernstein’s theorem; exponential function guess

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20240041
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Yin Hong-Ping

• College of Mathematical Science, Inner Mongolia Minzu University, Tongliao, Inner Mongolia, 028043, China

autor

Han Ling-Xiong

• College of Mathematical Science, Inner Mongolia Minzu University, Tongliao, Inner Mongolia, 028043, China

autor

Qi Feng

• School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, Henan, 454010, China
• School of Mathematics and Physics, Hulunbuir University, Hulunbuir, Inner Mongolia, 021008, China
• ndependent researcher, University Village, Dallas, TX 75252, USA

Bibliografia

• [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, vol. 55, 10th printing, Washington, 1972.
• [2] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993, DOI: https://doi.org/10.1007/978-94-017-1043-5.
• [3] R. L. Schilling, R. Song, and Z. Vondracccek, Bernstein Functions, 2nd ed., de Gruyter Studies in Mathematics, vol. 37, Walter de Gruyter, Berlin, Germany, 2012, DOI: https://doi.org/10.1515/9783110269338.
• [4] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.
• [5] F. Qi, Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function, Appl. Anal. Discrete Math. 15 (2021), no. 2, 378–392, DOI: https://doi.org/10.2298/AADM191111014Q.
• [6] F. Qi, Necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic, Turkish J. Inequal. 5 (2021), no. 1, 50–59.
• [7] F. Qi, Some properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold, São Paulo J. Math. Sci. 14 (2020), no. 2, 614–630, DOI: https://doi.org/10.1007/s40863-020-00193-1.
• [8] F. Qi and R. P. Agarwal, Several functions originating from Fisher-Rao geometry of Dirichlet distributions and involving polygamma functions, Mathematics 12 (2024), no. 1, 44, DOI: https://doi.org/10.3390/math12010044.
• [9] Z.-H. Yang and J.-F. Tian, Monotonicity and inequalities for the gamma function, J. Inequal. Appl. 2017 (2017), no. 1, 317, DOI: https://doi.org/10.1186/s13660-017-1591-9.
• [10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015, DOI: https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
• [11] P. S. Bullen, Handbook of Means and Their Inequalities, Mathematics and its Applications, vol. 560, Kluwer Academic Publishers Group, Dordrecht, 2003, DOI: https://doi.org/10.1007/978-94-017-0399-4.
• [12] F. Qi and D. Lim, Integral representations of bivariate complex geometric mean and their applications, J. Comput. Appl. Math. 330 (2018), 41–58, DOI: https://doi.org/10.1016/j.cam.2017.08.005.

Uwagi

1. Dedicated to Professor Dr. Mourad E. H. Ismail at University of Central Florida.

2. Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2026).