An inequality involving the gamma and digamma functions

• Autorzy: Qi, F. , Guo, B.-N.
• Języki publikacji: EN

Abstrakty

EN

In the paper, the authors establish an inequality involving the gamma and digamma functions and apply it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.

Słowa kluczowe

PL

funkcja gamma; funkcja psi; funkcja polygamma; nierówność; negatywność; monotoniczność

EN

gamma function; psi function; polygamma function; inequality; negativity; monotonicity; application

• Czasopismo: Journal of Applied Analysis
• Rocznik: 2016
• Tom: Vol. 22, nr 1
• Strony: 49--54
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Qi, F.

• Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, P. R. China
• College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, P. R. China
• Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, P. R. China

autor

Guo, B.-N.

• School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, P. R. China

Bibliografia

• [1] J. Dubourdieu, Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes, Compos. Math. 7 (1939), 96-111.
• [2] N. Elezovic, C. Giordano and J. Pecarîc, The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000), 239-252.
• [3] N. Elezović and J. Pecarić, Differential and integral f-means and applications to digamma function, Math, Inequal. Appl. 3 (2000), no. 2,189-196.
• [4] B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1,103-111.
• [5] B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208.
• [6] S. Guo, F. Qi and H. M. Srivastava, Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic, Integral Transforms Spec. Funct. 18 (2007), no. 11,819-826.
• [7] B.-N. Guo, F. Qi, J.-L. Zhao and Q.-M. Luo, Sharp inequalities for polygamma functions, Math. Slovaca 65 (2015), no. 1, 103-120.
• [8] J.-C. Kuang, Changyông Bùdëngshî (in Chinese), Appl. Inequal., 3rd ed., Shandong Science and Technology Press, Ji'nan City, 2004.
• [9] D. S. Mitrinovic, Analytic Inequalities, Springer, Berlin, 1970.
• [10] F. Qi, A new lower bound in the second Kershaw's double inequality, J. Comput Appl. Math. 214 (2008), no. 2, 610-616.
• [11] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058.
• [12] F. Qi, Bounds for the ratio of two gamma functions: From Gautschi's and Kershaw's inequalities to complete monotonicity, Turkish J. Anal. Number Theory 2 (2014), no. 5,152-164.
• [13] F. Qi, R.-Q. Cui, C.-P. Chen and B.-N. Guo, Some completely monotonic functions involving polygamma functions and an application, J. Math. Anal. Appl. 310 (2005), no. 1, 303-308.
• [14] F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving a ratio of gamma functions to be logarithmically completely monotonic, preprint (2009), http://arxiv.org/abs/0904.1101.
• [15] F. Qi and B.-N. Guo, An inequality involving the gamma and digamma functions, preprint (2011), http://arxiv.org/abs/1101.4698.
• [16] F. Qi, S. Guo and S.-X. Chen, A new upper bound in the second Kershaw's double inequality and its generalizations, J. Comput. Appl. Math. 220 (2008), no. 1-2,111-118.
• [17] F. QÎ, S. Guo and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149-2160.
• [18] F. Qiand W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput. 5 (2015), no. 4, 626-634.
• [19] F. Qi, X.-A. Li and S.-X. Chen, Refinements, extensions and generalizations of the second Kershaw's double inequality, Math. Inequal. Appl. 11 (2008), no. 3, 457-465.
• [20] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: From Wenders and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132-158.
• [21] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: From Wendel's asymptotic relation to Elezovic-Giordano-Pecaric's theorem, J. Inequal Appl. 2013 (2013), Article ID 542.
• [22] F. Qi, C.-F. Wei and B.-N. Guo, Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal. 6 (2012), no. 1, 35-44.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.

Identyfikatory

DOI

10.1515/jaa-2016-0005