A completely monotonic function involving the gamma and trigamma functions

Feng Qi

Abstract


In the paper the author provides necessary and sufficient conditions on $a$ for the function\begin{equation*}\frac{1}{2}\ln(2\pi)-x+\biggl(x-\frac{1}{2}\biggr)\ln x-\ln\Gamma(x)+\frac1{12}{\psi'(x+a)}\end{equation*}and its negative to be completely monotonic on $(0,\infty)$, where $a\ge0$ is a real number, $\Gamma(x)$ is the classical gamma function, and $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ is the digamma function. As applications, some known results and new inequalities are derived.

Keywords


Completely monotonic function; gamma function; inequality; logarithmically completely monotonic function; trigamma function.

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References


Abramowitz, M. & Stegun, I.A. (Eds) (1972). Handbook of

Mathematical Functions with Formulas, Graphs, and Mathematical

Tables. National Bureau of Standards, Applied Mathematics Series 55,

th printing, with corrections, Washington.

Alzer, H. (1993). Some gamma function inequalities. Mathematics of

Computation, 60:337–346, doi:10.2307/2153171.

Atanassov, R.D. & Tsoukrovski, U.V. (1988). Some properties of

a class of logarithmically completely monotonic functions. Comptes

rendus de l’Académie bulgare des Sciences, 41:21–23.

Berg, C. (2004). Integral representation of some functions related to

the gamma function. Mediterranean Journal of Mathematics, 1:433–

, doi:10.1007/s00009-004-0022-6.

Dubourdieu, J. (1939). Sur un théorème de M. S. Bernstein relatif

la transformation de Laplace-Stieltjes. Compositio Mathematica, 7:96–

(French)

Guo, B.N., Chen, R.J. & Qi, F. (2006). A class of completely monotonic

functions involving the polygamma functions. Journal of Mathematical

Analysis and Approximation Theory, 1:124–134.

Guo, B.N. & Qi, F. (2010a). A property of logarithmically absolutely

monotonic functions and the logarithmically complete monotonicity

of a power-exponential function. University Politehnica of Bucharest

Scientific Bulletin Series A—Applied Mathematics and Physics,

:21–30.

Guo, B.N. & Qi, F. (2010b). Some properties of the psi and polygamma

functions. Hacettepe Journal of Mathematics and Statistics, 39:219–

Guo, B.N. & Qi, F. (2010c). Two new proofs of the complete

monotonicity of a function involving the psi function. Bulletin

of the Korean Mathematical Society, 47:103–111, doi:10.4134/

bkms.2010.47.1.103.

Guo, B.N. & Qi, F. (2011a). A class of completely monotonic functions

involving divided differences of the psi and tri-gamma functions

and some applications. Journal of the Korean Mathematical Society,

:655–667, doi:10.4134/JKMS.2011.48.3.655.

Guo, B.N. & Qi, F. (2011b). An alternative proof of Elezović-Giordano-

Pečarić’s theorem. Mathematical Inequalities & Applications, 14:73–

, doi:10.7153/mia-14-06.

Guo, B.N. & Qi, F. (2012a). A completely monotonic function involving

the tri-gamma function and with degree one. Applied Mathematics and

Computation, 218:9890–9897, doi:10.1016/j.amc.2012.03.075.

Guo, B.N. & Qi, F. (2012b). Monotonicity of functions connected with

the gamma function and the volume of the unit ball. Integral Transforms

and Special Functions, 23:701–708, doi:10.1080/10652469.2011.6275

Guo, B.N. & Qi, F. (2013a). Monotonicity and logarithmic convexity

relating to the volume of the unit ball. Optimization Letters, 7:1139–

, doi:10.1007/s11590-012-0488-2.

Guo, B.N. & Qi, F. (2013b). Refinements of lower bounds for

polygamma functions. Proceedings of the American Mathematical

Society, 141:1007–1015, doi:10.1090/S0002-9939-2012-11387-5.

Guo, B.N., Qi, F. & Srivastava, H.M. (2010). Some uniqueness results

for the non-trivially complete monotonicity of a class of functions

involving the polygamma and related functions. Integral Transforms and

Special Functions, 21:849–858, doi:10.1080/10652461003748112.

Guo, B.N., Zhang, Y.J. & Qi, F. (2008). Refinements and sharpenings

of some double inequalities for bounding the gamma function. Journal

of Inequalities in Pure and Applied Mathematics, 9:(1)1, Art. 17; http://

www.emis.de/journals/JIPAM/article953.html.

Guo, S., Qi, F. & Srivastava, H.M. (2012). A class of logarithmically

completely monotonic functions related to the gamma function with

applications. Integral Transforms and Special Functions 23:557–566,

doi:10.1080/10652469.2011.611331.

Li, W.H., Qi, F. & Guo, B.N. (2013). On proofs for monotonicity of

a function involving the psi and exponential functions. Analysis—

International mathematical Journal of Analysis and its Applications,

:45–50, doi:10.1524/anly.2013.1175.

Li, A.J., Zhao, W.Z., & Chen, C.P. (2006). Logarithmically complete

monotonicity properties for the ratio of gamma function. Advanced

Studies in Contemporary Mathematics (Kyungshang), 13:183–191.

Lü, Y.P., Sun, T.C. & Y.M. Chu (2011). Necessary and sufficient

conditions for a class of functions and their reciprocals to be

logarithmically completely monotonic. Journal of Inequalities and

Applications, 2011:8 pages, doi:10.1186/1029-242X-2011-36.

Magnus, W., Oberhettinger, F. & Soni R.P. (1966). Formulas and

Theorems for the Special Functions of Mathematical Physics. Springer,

Berlin, doi:10.1137/1009129.

Merkle, M. (1998). Convexity, Schur-convexity and bounds

for the gamma function involving the digamma function. Rocky

Mountain Journal of Mathematics, 28:1053–1066, doi:10.1216/

rmjm/1181071755.

Mitrinović, D.S., Pečarić, J.E. & Fink, A.M. (1993). Classical

and New Inequalities in Analysis. Kluwer Academic Publishers,

doi:10.1007/978-94-017-1043-5.

Qi, F. (2007). A completely monotonic function involving the divided

difference of the psi function and an equivalent inequality involving

sums. Australian & New Zealand Industrial and Applied Mathematics

Journal, 48:523–532, doi:10.1017/S1446181100003199.

Qi, F. (2013). A completely monotonic function involving the

gamma and tri-gamma functions. arXiv preprint, http://arxiv.org/

abs/1307.5407.

Qi, F. (2010). Bounds for the ratio of two gamma functions. Journal

of Inequalities and Applications, 2010:Article ID 493058, 84 pages,

doi:10.1155/2010/493058.

Qi, F. (2014). Bounds for the ratio of two gamma functions: from

Gautschi’s and Kershaw’s inequalities to complete monotonicity.

Turkish Journal of Analysis and Number Theory, 2:152–164,

doi:10.12691/tjant-2-5-1.

Qi, F. & Berg, C. (2013). Complete monotonicity of a difference

between the exponential and trigamma functions and properties related

to a modified Bessel function. Mediterranean Journal of Mathematics,

:1685–1696, doi:10.1007/s00009-013-0272-2.

Qi, F., Cerone, P. & Dragomir, S.S. (2013a). Complete monotonicity

of a function involving the divided difference of psi functions. Bulletin

of the Australian Mathematical Society 88:309–319, doi:10.1017/

S0004972712001025.

Qi, F., Luo, Q.M. & Guo, B.N. (2013b). Complete monotonicity of

a function involving the divided difference of digamma functions.

Science China Mathematics, 56:2315–2325, doi:10.1007/s11425-012-

-0.

Qi, F. & Chen, C.P. (2004). A complete monotonicity property of the

gamma function. Journal of Mathematical Analysis and Applications,

:603–607, doi:10.1016/j.jmaa.2004.04.026.

Qi, F., Cui, R.Q., Chen, C.P. & Guo, B.N. (2005). Some completely

monotonic functions involving polygamma functions and an application.

Journal of Mathematical Analysis and Applications, 310:303–308,

doi:10.1016/j.jmaa.2005.02.016.

Qi, F. & Guo, B.N. (2004). Complete monotonicities of functions

involving the gamma and digamma functions. RGMIA Research Report

Collection, 7:Art. 8:63–72, http://rgmia.org/v7n1.php.

Qi, F. & Guo, B.N. (2009). Completely monotonic functions involving

divided differences of the di- and tri-gamma functions and some

applications. Communications on Pure and Applied Analysis, 8:1975–

, doi:10.3934/cpaa.2009.8.1975.

Qi, F. & Guo, B.N. (2010a). Necessary and sufficient conditions for

functions involving the tri- and tetra-gamma functions to be completely

monotonic. Advances in Applied Mathematics, 44:71–83, doi:10.1016/j.

aam.2009.03.003.

Qi, F. & Guo, B.N. (2010b). Some logarithmically completely

monotonic functions related to the gamma function. Journal of

the Korean Mathematical Society, 47:1283–1297, doi:10.4134/

JKMS.2010.47.6.1283.

Qi, F. & Guo, B.N. (2010c). Some properties of extended remainder

of Binet’s first formula for logarithm of gamma function. Mathematica

Slovaca, 60:461–470, doi:10.2478/s12175-010-0025-7.

Qi, F., Guo, B.N. & Chen, C.P. (2006). Some completely monotonic

functions involving the gamma and polygamma functions. Journal

of the Australian Mathematical Society, 80:81–88, doi:10.1017/

S1446788700011393.

Qi, F., Guo, B.N. & Chen, C.P. (2004). Some completely monotonic

functions involving the gamma and polygamma functions. RGMIA

Research Report Collection, 7:Art. 5:31–36, http://rgmia.org/v7n1.

php.

Qi, F., Guo, S. & Guo, B.N. (2010). Complete monotonicity of some

functions involving polygamma functions. Journal of Computational

and Applied Mathematics, 233:2149–2160, doi:10.1016/j.

cam.2009.09.044.

Qi, F., Li, W. & Guo, B.N. (2006). Generalizations of a theorem of I.

Schur. Applied Mathematics E-Notes, 6:244–250.

Qi, F. & Luo, Q.M. (2012). Bounds for the ratio of two gamma

functions—From Wendel’s and related inequalities to logarithmically

completely monotonic functions. Banach Journal of Mathematical

Analysis, 6:132–158, doi:10.15352/bjma/1342210165.

Qi, F. & Luo, Q.M. (2013). Bounds for the ratio of two gamma

functions: from Wendel’s asymptotic relation to Elezović-Giordano-

Pečarić’s theorem. Journal of Inequalities and Applications, 542, 20

pages, doi:10.1186/1029-242X-2013-542.

Qi, F., Wei, C.F. & Guo, B.N. (2012). Complete monotonicity of a

function involving the ratio of gamma functions and applications.

Banach Journal of Mathematical Analysis, 6:35–44, doi:10.15352/

bjma/1337014663.

Qi, F., Xu, S.L. & Debnath, L. (1999). A new proof of

monotonicity for extended mean values. International Journal of Mathematics and Mathematical Sciences, 22:417–421, doi:10.1155/

S0161171299224179.

Qi, F. & Zhang, S.Q. (1999). Note on monotonicity of generalized

weighted mean values. Proceedings of the Royal Society of London

Series A—Mathematical, Physical and Engineering Sciences,

:3259–3260, doi:10.1098/rspa.1999.0449.

Srivastava, H.M., Guo, S. & Qi, F. (2012). Some properties of a class

of functions related to completely monotonic functions. Computers

& Mathematics with Applications, 64:1649–1654, doi:10.1016/j.

camwa.2012.01.016.

Widder, D.V. (1946). The Laplace Transform. Princeton University

Press, Princeton.

Zhao, T.H., Chu, Y.M. & Wang, H. (2011). Logarithmically

complete monotonicity properties relating to the gamma function.

Abstract and Applied Analysis, 2011:Article ID 896483, 13 pages,

doi:10.1155/2011/896483.

Zhao, J.L., Guo, B.N. & Qi, F. (2012a). A refinement of a double

inequality for the gamma function. Publ. Math. Debrecen, 80:333–342,

doi:10.5486/PMD.2012.5010.

Zhao, J.L., Guo, B.N. & Qi, F. (2012b). Complete monotonicity of

two functions involving the tri- and tetra-gamma functions. Period.

Math. Hungar., 65:147–155, doi:10.1007/s10998-012-9562-x.


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