Complete monotonicity of a function involving the p-psi function and alternative proofs
Abstract
In the paper, the authors prove that the function $x^\alpha\big[\ln\frac{px}{x+p+1}-\psi_p(x)\big]$ is completely monotonic on $(0,\infty)$ if and only if $\alpha \le 1$, where $p\in\mathbb{N}$ and $\psi_p(x)$ is the $p$-analogue of the classical psi function $\psi(x)$.
Keywords: completely monotonic function; necessary and sufficient condition; p-gamma function; p-psi function; Inequality
MSC: Primary 33D05; Secondary 26A48, 33B15, 33E50
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References
[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, 9th printing, Washington, 1970.
[2] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373--389; Available online at http://dx.doi.org/10.1090/S0025-5718-97-00807-7.
[3] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
[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; Available online at http://dx.doi.org/10.4134/bkms.2010.47.1.103.
[5] B.-N. Guo, A.-Q. Liu, and F. Qi, Monotonicity and logarithmic convexity of three functions involving exponential function, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 15 (2008), no. 4, 387--392.
View more references (13)
[6] B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms 52 (2009), no. 1, 89--92; Available online at http://dx.doi.org/10.1007/s11075-008-9259-7.
[7] S. Guo and F. Qi, A class of completely monotonic functions related to the remainder of Binet's formula with applications, Tamsui Oxf. J. Math. Sci. 25 (2009), no. 1, 9--14.
[8] V. Krasniqi, T. Mansour, and A. Sh. Shabani, Some monotonicity properties and inequalities for and - functions, Math. Commun. 15 (2010), no. 2, 365--376.
[9] V. Krasniqi and F. Qi, Complete monotonicity of a function involving the -psi function and alternative proofs, available online at http://arxiv.org/abs/1105.4928.
[10] V. Krasniqi and A. Sh. Shabani, Convexity properties and inequalities for a generalized gamma function, Appl. Math. E-Notes 10 (2010), 27--35.
[11] A.-Q. Liu, G.-F. Li, B.-N. Guo, and F. Qi, Monotonicity and logarithmic concavity of two functions involving exponential function, Internat. J. Math. Ed. Sci. Tech. 39 (2008), no. 5, 686--691; Available online at http://dx.doi.org/10.1080/00207390801986841.
[12] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.
[13] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058.
[14] F. Qi, P. Cerone, S. S. Dragomir, and H. M. Srivastava, Alternative proofs for monotonic and logarithmically convex properties of one-parameter mean values, Appl. Math. Comput. 208 (2009), no. 1, 129--133; Available online at http://dx.doi.org/10.1016/j.amc.2008.11.023.
[15] F. Qi and B.-N. Guo, Some properties of extended remainder of Binet's first formula for logarithm of gamma function, Math. Slovaca 60 (2010), no. 4, 461--470; Available online at http://dx.doi.org/10.2478/s12175-010-0025-7.
[16] R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions, de Gruyter Studies in Mathematics 37, De Gruyter, Berlin, Germany, 2010.
[17] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
[18] S.-Q. Zhang, B.-N. Guo, and F. Qi, A concise proof for properties of three functions involving the exponential function, Appl. Math. E-Notes 9 (2009), 177--183.