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On complete monotonicity for several classes of functions related to ratios of gamma functions. (English) Zbl 1499.33013

Summary: Let \(\varGamma (x)\) denote the classical Euler gamma function. The logarithmic derivative \(\psi (x)=[\ln \varGamma (x)]'=\frac{\varGamma'(x)}{ \varGamma (x)}\), \(\psi'(x)\), and \(\psi''(x)\) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function \([\psi'(x)]^2+ \psi''(x)\), its \(q\)-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.

MSC:

33B15 Gamma, beta and polygamma functions
26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
44A10 Laplace transform
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[1] Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th printing edn. National Bureau of Standards, Applied Mathematics Series, vol. 55. Dover, New York (1972) · Zbl 0543.33001
[2] Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66(217), 373-389 (1997). Available online at https://doi.org/10.1090/S0025-5718-97-00807-7 · Zbl 0854.33001 · doi:10.1090/S0025-5718-97-00807-7
[3] Alzer, H.: Sharp inequalities for the digamma and polygamma functions. Forum Math. 16(2), 181-221 (2004). Available online at https://doi.org/10.1515/form.2004.009 · Zbl 1048.33001 · doi:10.1515/form.2004.009
[4] Alzer, H.: Sharp inequalities for the harmonic numbers. Expo. Math. 24(4), 385-388 (2006). Available online at https://doi.org/10.1016/j.exmath.2006.02.001 · Zbl 1105.11003 · doi:10.1016/j.exmath.2006.02.001
[5] Alzer, H.: Sub- and superadditive properties of Euler’s gamma function. Proc. Am. Math. Soc. 135(11), 3641-3648 (2007). Available online at https://doi.org/10.1090/S0002-9939-07-09057-0 · Zbl 1126.33001 · doi:10.1090/S0002-9939-07-09057-0
[6] Alzer, H.: Complete monotonicity of a function related to the binomial probability. J. Math. Anal. Appl. 459(1), 10-15 (2018). Available online at https://doi.org/10.1016/j.jmaa.2017.10.077 · Zbl 1377.33001 · doi:10.1016/j.jmaa.2017.10.077
[7] Alzer, H., Grinshpan, A.Z.: Inequalities for the gamma and q-gamma functions. J. Approx. Theory 144(1), 67-83 (2007). Available online at https://doi.org/10.1016/j.jat.2006.04.008 · Zbl 1108.33001 · doi:10.1016/j.jat.2006.04.008
[8] Alzer, H., Wells, J.: Inequalities for the polygamma functions. SIAM J. Math. Anal. 29(6), 1459-1466 (1998). Available online at https://doi.org/10.1137/S0036141097325071 · Zbl 0907.33003 · doi:10.1137/S0036141097325071
[9] Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra. CBMS Regional Conference Series in Mathematics, vol. 66. Am. Math. Soc., Providence (1986) Published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0594.33001
[10] Andrews, G.E., Askey, R.A., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999). Available online at https://doi.org/10.1017/CBO9781107325937 · Zbl 0920.33001 · doi:10.1017/CBO9781107325937
[11] Atanassov, R.D., Tsoukrovski, U.V.: Some properties of a class of logarithmically completely monotonic functions. C. R. Acad. Bulgare Sci. 41(2), 21-23 (1988) · Zbl 0658.26010
[12] Batır, N.: An interesting double inequality for Euler’s gamma function. J. Inequal. Pure Appl. Math. 5(4), Article ID 97 (2004). Available online at http://www.emis.de/journals/JIPAM/article452.html · Zbl 1078.33001
[13] Batır, N.: Some new inequalities for gamma and polygamma functions. J. Inequal. Pure Appl. Math. 6(4), Article ID 103 (2005). Available online at http://www.emis.de/journals/JIPAM/article577.html · Zbl 1089.33001
[14] Batır, N.: On some properties of digamma and polygamma functions. J. Math. Anal. Appl. 328(1), 452-465 (2007). Available online at https://doi.org/10.1016/j.jmaa.2006.05.065 · Zbl 1105.33002 · doi:10.1016/j.jmaa.2006.05.065
[15] Berg, C.: Integral representation of some functions related to the gamma function. Mediterr. J. Math. 1(4), 433-439 (2004). Available online at https://doi.org/10.1007/s00009-004-0022-6 · Zbl 1162.33300 · doi:10.1007/s00009-004-0022-6
[16] Bochner, S.: Harmonic Analysis and the Theory of Probability. California Monographs in Mathematical Sciences. University of California Press, Berkeley (1955) · Zbl 0068.11702
[17] Boyadzhiev, K.N.: Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of exp(1/x)\( \exp (1/x)\). Acta Univ. Sapientiae Math. 8(1), 22-31 (2016). Available online at https://doi.org/10.1515/ausm-2016-0002 · Zbl 1398.11108 · doi:10.1515/ausm-2016-0002
[18] Bullen, P.S.: Handbook of Means and Their Inequalities. Mathematics and Its Applications, vol. 560. Kluwer Academic, Dordrecht (2003) · Zbl 1035.26024 · doi:10.1007/978-94-017-0399-4
[19] Chen, C.-P.: Monotonicity and convexity for the gamma function. J. Inequal. Pure Appl. Math. 6(4), Article ID 100 (2005). Available online at http://www.emis.de/journals/JIPAM/article574.html · Zbl 1081.33004
[20] Chen, C.-P., Qi, F.: Logarithmically completely monotonic functions relating to the gamma function. J. Math. Anal. Appl. 321(1), 405-411 (2006). Available online at https://doi.org/10.1016/j.jmaa.2005.08.056 · Zbl 1099.33002 · doi:10.1016/j.jmaa.2005.08.056
[21] Chen, C.-P., Qi, F., Srivastava, H.M.: Some properties of functions related to the gamma and psi functions. Integral Transforms Spec. Funct. 21(2), 153-164 (2010). Available online at https://doi.org/10.1080/10652460903064216 · Zbl 1188.33003 · doi:10.1080/10652460903064216
[22] Daboul, S., Mangaldan, J., Spivey, M.Z., Taylor, P.J.: The Lah numbers and the nth derivative of e1/x \(e^{1/x}\). Math. Mag. 86(1), 39-47 (2013). Available online at https://doi.org/10.4169/math.mag.86.1.039 · Zbl 1274.05014 · doi:10.4169/math.mag.86.1.039
[23] Elezović, N., Giordano, C., Pečarić, J.: The best bounds in Gautschi’s inequality. Math. Inequal. Appl. 3(2), 239-252 (2000). Available online at https://doi.org/10.7153/mia-03-26 · Zbl 0947.33001 · doi:10.7153/mia-03-26
[24] Gao, P.: Some monotonicity properties of gamma and q-gamma functions. ISRN Math. Anal. 2011, Article ID 375715 (2011) · Zbl 1216.33006
[25] Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 96. Cambridge University Press, Cambridge (2004). Available online at https://doi.org/10.1017/CBO9780511526251 · Zbl 1129.33005 · doi:10.1017/CBO9780511526251
[26] Guo, B.-N., Qi, F.: Properties and applications of a function involving exponential functions. Commun. Pure Appl. Anal. 8(4), 1231-1249 (2009). Available online at https://doi.org/10.3934/cpaa.2009.8.1231 · Zbl 1179.33001 · doi:10.3934/cpaa.2009.8.1231
[27] Guo, B.-N., Qi, F.: A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 72(2), 21-30 (2010) · Zbl 1299.26022
[28] Guo, B.-N., Qi, F.: Some properties of the psi and polygamma functions. Hacet. J. Math. Stat. 39(2), 219-231 (2010) · Zbl 1203.33002
[29] Guo, B.-N., Qi, F.: Two new proofs of the complete monotonicity of a function involving the psi function. Bull. Korean Math. Soc. 47(1), 103-111 (2010). Available online at https://doi.org/10.4134/bkms.2010.47.1.103 · Zbl 1183.33002 · doi:10.4134/bkms.2010.47.1.103
[30] Guo, B.-N., Qi, F.: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J. Korean Math. Soc. 48(3), 655-667 (2011). Available online at https://doi.org/10.4134/JKMS.2011.48.3.655 · Zbl 1270.26010 · doi:10.4134/JKMS.2011.48.3.655
[31] Guo, B.-N., Qi, F.: An alternative proof of Elezović-Giordano-Pečarić’s theorem. Math. Inequal. Appl. 14(1), 73-78 (2011). Available online at https://doi.org/10.7153/mia-14-06 · Zbl 1207.26019 · doi:10.7153/mia-14-06
[32] Guo, B.-N., Qi, F.: A completely monotonic function involving the tri-gamma function and with degree one. Appl. Math. Comput. 218(19), 9890-9897 (2012). Available online at https://doi.org/10.1016/j.amc.2012.03.075 · Zbl 1245.26007 · doi:10.1016/j.amc.2012.03.075
[33] Guo, B.-N., Qi, F.: Refinements of lower bounds for polygamma functions. Proc. Am. Math. Soc. 141(3), 1007-1015 (2013). Available online at https://doi.org/10.1090/S0002-9939-2012-11387-5 · Zbl 1267.33003 · doi:10.1090/S0002-9939-2012-11387-5
[34] Guo, B.-N., Qi, F.: Sharp inequalities for the psi function and harmonic numbers. Analysis (Berlin) 34(2), 201-208 (2014). Available online at https://doi.org/10.1515/anly-2014-0001 · Zbl 1294.33003 · doi:10.1515/anly-2014-0001
[35] Guo, B.-N., Qi, F.: Some integral representations and properties of Lah numbers. J. Algebra Number Theory Acad. 4(3), 77-87 (2014)
[36] Guo, B.-N., Qi, F.: On the degree of the weighted geometric mean as a complete Bernstein function. Afr. Math. 26(7), 1253-1262 (2015). Available online at https://doi.org/10.1007/s13370-014-0279-2 · Zbl 1329.26054 · doi:10.1007/s13370-014-0279-2
[37] Guo, B.-N., Qi, F.: Six proofs for an identity of the Lah numbers. Online J. Anal. Comb. 10, 5 pages (2015) · Zbl 1306.05014
[38] Guo, B.-N., Qi, F., Srivastava, H.M.: Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions. Integral Transforms Spec. Funct. 21(11), 849-858 (2010). Available online at https://doi.org/10.1080/10652461003748112 · Zbl 1207.26020 · doi:10.1080/10652461003748112
[39] Guo, B.-N., Qi, F., Zhao, J.-L., Luo, Q.-M.: Sharp inequalities for polygamma functions. Math. Slovaca 65(1), 103-120 (2015). Available online at https://doi.org/10.1515/ms-2015-0010 · Zbl 1349.33001 · doi:10.1515/ms-2015-0010
[40] Guo, B.-N., Zhao, J.-L., Qi, F.: A completely monotonic function involving the tri- and tetra-gamma functions. Math. Slovaca 63(3), 469-478 (2013). Available online at https://doi.org/10.2478/s12175-013-0109-2 · Zbl 1349.26024 · doi:10.2478/s12175-013-0109-2
[41] Ismail, M.E.H., Lorch, L., Muldoon, M.E.: Completely monotonic functions associated with the gamma function and its q-analogues. J. Math. Anal. Appl. 116, 1-9 (1986). Available online at https://doi.org/10.1016/0022-247X(86)90042-9 · Zbl 0589.33001 · doi:10.1016/0022-247X(86)90042-9
[42] Ismail, M. E.H.; Muldoon, M. E.; Zahar, R. V.M. (ed.), Inequalities and monotonicity properties for gamma and q-gamma functions, No. 119, 309-323 (1994), Basel · Zbl 0819.33001 · doi:10.1007/978-1-4684-7415-2_19
[43] Ismail, M.E.H., Muldoon, M.E.: Inequalities and monotonicity properties for gamma and q-gamma functions. arXiv preprint (2013). Available online at http://arxiv.org/abs/1301.1749 · Zbl 0819.33001
[44] Katriel, J.: The q-Lah numbers and the n-th q-derivative of expq(1/x)\( \exp_q(1/x)\). Notes Number Theory Discrete Math. 23(2), 45-47 (2017) · Zbl 1386.11035
[45] Kazarinoff, D.K.: On Wallis’ formula. Edinb. Math. Notes 1956(40), 19-21 (1956) · Zbl 0072.28401 · doi:10.1017/S095018430000029X
[46] Koornwinder, T.H.: Jacobi functions as limit cases of q-ultraspherical polynomials. J. Math. Anal. Appl. 148(1), 44-54 (1990). Available online at https://doi.org/10.1016/0022-247X(90)90026-C · Zbl 0713.33010 · doi:10.1016/0022-247X(90)90026-C
[47] Koumandos, S.: Monotonicity of some functions involving the gamma and psi functions. Math. Comput. 77(264), 2261-2275 (2008). Available online at https://doi.org/10.1090/S0025-5718-08-02140-6 · Zbl 1210.33002 · doi:10.1090/S0025-5718-08-02140-6
[48] Koumandos, S., Lamprecht, M.: Some completely monotonic functions of positive order. Math. Comput. 79(271), 1697-1707 (2010). Available online at https://doi.org/10.1090/S0025-5718-09-02313-8 · Zbl 1194.33003 · doi:10.1090/S0025-5718-09-02313-8
[49] Koumandos, S., Lamprecht, M.: Complete monotonicity and related properties of some special functions. Math. Comput. 82(282), 1097-1120 (2013). Available online at https://doi.org/10.1090/S0025-5718-2012-02629-9 · Zbl 1272.33003 · doi:10.1090/S0025-5718-2012-02629-9
[50] Koumandos, S., Pedersen, H.L.: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 355(1), 33-40 (2009). Available online at https://doi.org/10.1016/j.jmaa.2009.01.042 · Zbl 1169.33001 · doi:10.1016/j.jmaa.2009.01.042
[51] Koumandos, S., Pedersen, H.L.: Absolutely monotonic functions related to Euler’s gamma function and Barnes’ double and triple gamma function. Monatshefte Math. 163(1), 51-69 (2011). Available online at https://doi.org/10.1007/s00605-010-0197-9 · Zbl 1241.33004 · doi:10.1007/s00605-010-0197-9
[52] Leblanc, A., Johnson, B.C.: On a uniformly integrable family of polynomials defined on the unit interval. J. Inequal. Pure Appl. Math. 8(3), Article ID 67 (2007). https://www.emis.de/journals/JIPAM/article878.html · Zbl 1130.05006
[53] Li, W.-H., Qi, F., Guo, B.-N.: On proofs for monotonicity of a function involving the psi and exponential functions. Analysis (Munich) 33(1), 45-50 (2013). Available online at https://doi.org/10.1524/anly.2013.1175 · Zbl 1273.33004 · doi:10.1524/anly.2013.1175
[54] Liu, F.-F., Shi, X.-T., Qi, F.: A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function. Glob. J. Math. Anal. 3(4), 140-144 (2015). Available online at https://doi.org/10.14419/gjma.v3i4.5187 · doi:10.14419/gjma.v3i4.5187
[55] Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht (1993). Available online athttps://doi.org/10.1007/978-94-017-1043-5 · Zbl 0771.26009 · doi:10.1007/978-94-017-1043-5
[56] Muldoon, M.E.: Some monotonicity properties and characterizations of the gamma function. Aequ. Math. 18, 54-63 (1978). Available online at https://doi.org/10.1007/BF01844067 · Zbl 0386.33001 · doi:10.1007/BF01844067
[57] Ouimet, F.: Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex. arXiv preprint (2018). Available online at https://arxiv.org/abs/1804.02108 · Zbl 1394.62038
[58] Ouimet, F.: Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex. J. Math. Anal. Appl. 466(2), 1609-1617 (2018). Available online at https://doi.org/10.1016/j.jmaa.2018.06.049 · Zbl 1394.62038 · doi:10.1016/j.jmaa.2018.06.049
[59] Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings, and Statistical Applications. Mathematics in Science and Engineering, vol. 187. Academic Press, San Diego (1992) · Zbl 0749.26004
[60] Qi, F.: Three-log-convexity for a class of elementary functions involving exponential function. J. Math. Anal. Approx. Theory 1(2), 100-103 (2006) · Zbl 1204.26015
[61] Qi, F.: A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw’s double inequality. J. Comput. Appl. Math. 206(2), 1007-1014 (2007). Available online at https://doi.org/10.1016/j.cam.2006.09.005 · Zbl 1113.33004 · doi:10.1016/j.cam.2006.09.005
[62] Qi, F.: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums. ANZIAM J. 48(4), 523-532 (2007). Available online at https://doi.org/10.1017/S1446181100003199 · Zbl 1131.33002 · doi:10.1017/S1446181100003199
[63] Qi, F.: Three classes of logarithmically completely monotonic functions involving gamma and psi functions. Integral Transforms Spec. Funct. 18, 503-509 (2007). Available online at https://doi.org/10.1080/10652460701358976 · Zbl 1144.26013 · doi:10.1080/10652460701358976
[64] Qi, F.: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, Article ID 493058 (2010). https://doi.org/10.1155/2010/493058 · Zbl 1194.33005 · doi:10.1155/2010/493058
[65] Qi, F.: Completely monotonic degree of a function involving the tri- and tetra-gamma functions. arXiv preprint (2013). Available online at http://arxiv.org/abs/1301.0154 · Zbl 1349.26024
[66] Qi, F.: Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities. Filomat 27(4), 601-604 (2013). Available online at https://doi.org/10.2298/FIL1304601Q · Zbl 1324.33004 · doi:10.2298/FIL1304601Q
[67] Qi, F.: A completely monotonic function related to the q-trigamma function. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 76(1), 107-114 (2014) · Zbl 1374.26022
[68] Qi, F.: Absolute monotonicity of a function involving the exponential function. Glob. J. Math. Anal. 2(3), 184-203 (2014). Available online at https://doi.org/10.14419/gjma.v2i3.3062 · doi:10.14419/gjma.v2i3.3062
[69] Qi, F.: Bounds for the ratio of two gamma functions: from Gautschi’s and Kershaw’s inequalities to complete monotonicity. Turk. J. Anal. Number Theory 2(5), 152-164 (2014). Available online at https://doi.org/10.12691/tjant-2-5-1 · doi:10.12691/tjant-2-5-1
[70] Qi, F.: Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind. Filomat 28(2), 319-327 (2014). Available online at https://doi.org/10.2298/FIL1402319O · Zbl 1385.11011 · doi:10.2298/FIL1402319O
[71] Qi, F.: Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function. J. Comput. Appl. Math. 268, 155-167 (2014). Available online at https://doi.org/10.1016/j.cam.2014.03.004 · Zbl 1293.33002 · doi:10.1016/j.cam.2014.03.004
[72] Qi, F.: Complete monotonicity of a function involving the tri- and tetra-gamma functions. Proc. Jangjeon Math. Soc. 18(2), 253-264 (2015). Available online at https://doi.org/10.17777/pjms.2015.18.2.253 · Zbl 1321.33002 · doi:10.17777/pjms.2015.18.2.253
[73] Qi, F.: Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 109(2), 419-429 (2015). Available online at https://doi.org/10.1007/s13398-014-0193-3 · Zbl 1321.33020 · doi:10.1007/s13398-014-0193-3
[74] Qi, F.: Derivatives of tangent function and tangent numbers. Appl. Math. Comput. 268, 844-858 (2015). Available online at https://doi.org/10.1016/j.amc.2015.06.123 · Zbl 1410.11018 · doi:10.1016/j.amc.2015.06.123
[75] Qi, F.: Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. Math. Inequal. Appl. 18(2), 493-518 (2015). Available online at https://doi.org/10.7153/mia-18-37 · Zbl 1326.33001 · doi:10.7153/mia-18-37
[76] Qi, F.: A completely monotonic function involving the gamma and trigamma functions. Kuwait J. Sci. Eng. 43(3), 32-40 (2016) · Zbl 1474.33010
[77] Qi, F.: An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers. Mediterr. J. Math. 13(5), 2795-2800 (2016). Available online at https://doi.org/10.1007/s00009-015-0655-7 · Zbl 1419.11045 · doi:10.1007/s00009-015-0655-7
[78] Qi, F.: Diagonal recurrence relations for the Stirling numbers of the first kind. Contrib. Discrete Math. 11, 22-30 (2016). Available online at http://hdl.handle.net/10515/sy5wh2dx6 and https://doi.org/10515/sy5wh2dx6 · Zbl 1360.11051
[79] Qi, F.: A logarithmically completely monotonic function involving the q-gamma function. HAL archives (2018). Available online at https://hal.archives-ouvertes.fr/hal-01803352
[80] Qi, F.: Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers. Turk. J. Anal. Number Theory 6(5), 129-131 (2018). Available online at https://doi.org/10.12691/tjant-6-5-1 · doi:10.12691/tjant-6-5-1
[81] Qi, F.: A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers. J. Comput. Appl. Math. 351, 1-5 (2019). Available online at https://doi.org/10.1016/j.cam.2018.10.049 · Zbl 1425.11043 · doi:10.1016/j.cam.2018.10.049
[82] Qi, F., Berg, C.: Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function. Mediterr. J. Math. 10(4), 1685-1696 (2013). Available online at https://doi.org/10.1007/s00009-013-0272-2 · Zbl 1280.26017 · doi:10.1007/s00009-013-0272-2
[83] Qi, F., Cerone, P., Dragomir, S.S.: Complete monotonicity of a function involving the divided difference of psi functions. Bull. Aust. Math. Soc. 88(2), 309-319 (2013). Available online at https://doi.org/10.1017/S0004972712001025 · Zbl 1280.26018 · doi:10.1017/S0004972712001025
[84] Qi, F., Chapman, R.J.: Two closed forms for the Bernoulli polynomials. J. Number Theory 159, 89-100 (2016). Available online at https://doi.org/10.1016/j.jnt.2015.07.021 · Zbl 1400.11070 · doi:10.1016/j.jnt.2015.07.021
[85] Qi, F., Chen, C.-P.: A complete monotonicity property of the gamma function. J. Math. Anal. Appl. 296(2), 603-607 (2004). Available online at https://doi.org/10.1016/j.jmaa.2004.04.026 · Zbl 1046.33001 · doi:10.1016/j.jmaa.2004.04.026
[86] Qi, F., Cui, R.-Q., Chen, C.-P., Guo, B.-N.: Some completely monotonic functions involving polygamma functions and an application. J. Math. Anal. Appl. 310(1), 303-308 (2005). Available online at https://doi.org/10.1016/j.jmaa.2005.02.016 · Zbl 1074.33005 · doi:10.1016/j.jmaa.2005.02.016
[87] Qi, F., Guo, B.-N.: Wendel’s and Gautschi’s inequalities: refinements, extensions, and a class of logarithmically completely monotonic functions. Appl. Math. Comput. 205(1), 281-290 (2008). Available online at https://doi.org/10.1016/j.amc.2008.07.005 · Zbl 1173.26315 · doi:10.1016/j.amc.2008.07.005
[88] Qi, F., Guo, B.-N.: Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Commun. Pure Appl. Anal. 8(6), 1975-1989 (2009). Available online at https://doi.org/10.3934/cpaa.2009.8.1975 · Zbl 1181.26024 · doi:10.3934/cpaa.2009.8.1975
[89] Qi, F., Guo, B.-N.: Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic. Adv. Appl. Math. 44(1), 71-83 (2010) Availale online at. https://doi.org/10.1016/j.aam.2009.03.003 · Zbl 1179.26036 · doi:10.1016/j.aam.2009.03.003
[90] Qi, F., Guo, B.-N.: Some properties of extended remainder of Binet’s first formula for logarithm of gamma function. Math. Slovaca 60(4), 461-470 (2010). Available online at https://doi.org/10.2478/s12175-010-0025-7 · Zbl 1240.26019 · doi:10.2478/s12175-010-0025-7
[91] Qi, F., Guo, B.-N.: Complete monotonicity of divided differences of the di- and tri-gamma functions with applications. Georgian Math. J. 23(2), 279-291 (2016). Available online at https://doi.org/10.1515/gmj-2016-0004 · Zbl 1339.33007 · doi:10.1515/gmj-2016-0004
[92] Qi, F., Guo, B.-N.: Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 111(2), 425-434 (2017). Available online at https://doi.org/10.1007/s13398-016-0302-6 · Zbl 1360.33004 · doi:10.1007/s13398-016-0302-6
[93] Qi, F., Guo, B.-N.: Integral representations of the Catalan numbers and their applications. Mathematics 5(3), Article ID 40 (2017). https://doi.org/10.3390/math5030040 · Zbl 1402.11032 · doi:10.3390/math5030040
[94] Qi, F., Guo, B.-N.: Lévy-Khintchine representation of Toader-Qi mean. Math. Inequal. Appl. 21(2), 421-431 (2018). Available online at https://doi.org/10.7153/mia-2018-21-29 · Zbl 1384.44002 · doi:10.7153/mia-2018-21-29
[95] Qi, F., Guo, B.-N.: On the sum of the Lah numbers and zeros of the Kummer confluent hypergeometric function. Acta Univ. Sapientiae Math. 10(1), 125-133 (2018). Available online at https://doi.org/10.2478/ausm-2018-0011 · Zbl 1439.11087 · doi:10.2478/ausm-2018-0011
[96] Qi, F.; Guo, B.-N.; Ruzhansky, M. (ed.); Dutta, H. (ed.); Agarwal, R. P. (ed.), Some properties and generalizations of the Catalan, Fuss, and Fuss-Catalan numbers, 101-133 (2018), New York · doi:10.1002/9781119414421.ch5
[97] Qi, F., Guo, B.-N.: The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function. Quaest. Math. 41(5), 653-664 (2018). Available online at https://doi.org/10.2989/16073606.2017.1396508 · Zbl 1400.30046 · doi:10.2989/16073606.2017.1396508
[98] Qi, F., Guo, B.-N., Chen, C.-P.: Some completely monotonic functions involving the gamma and polygamma functions. J. Aust. Math. Soc. 80(1), 81-88 (2006). Available online at https://doi.org/10.1017/S1446788700011393 · Zbl 1094.33002 · doi:10.1017/S1446788700011393
[99] Qi, F., Guo, B.-N., Chen, C.-P.: The best bounds in Gautschi-Kershaw inequalities. Math. Inequal. Appl. 9(3), 427-436 (2006). Available online at https://doi.org/10.7153/mia-09-41 · Zbl 1101.33001 · doi:10.7153/mia-09-41
[100] Qi, F., Guo, S., Guo, B.-N.: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 233(9), 2149-2160 (2010). Available online at https://doi.org/10.1016/j.cam.2009.09.044 · Zbl 1188.26007 · doi:10.1016/j.cam.2009.09.044
[101] Qi, F., Li, W.-H.: A logarithmically completely monotonic function involving the ratio of gamma functions. J. Appl. Anal. Comput. 5(4), 626-634 (2015). Available online at https://doi.org/10.11948/2015049 · Zbl 1447.33002 · doi:10.11948/2015049
[102] Qi, F., Li, W.-H.: Integral representations and properties of some functions involving the logarithmic function. Filomat 30(7), 1659-1674 (2016). Available online at https://doi.org/10.2298/FIL1607659Q · Zbl 1474.30251 · doi:10.2298/FIL1607659Q
[103] Qi, F., Lim, D.: Integral representations of bivariate complex geometric mean and their applications. J. Comput. Appl. Math. 330, 41-58 (2018). Available online at https://doi.org/10.1016/j.cam.2017.08.005 · Zbl 1375.26045 · doi:10.1016/j.cam.2017.08.005
[104] Qi, F., Lim, D., Guo, B.-N.: Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(1), 1-9 (2019). Available online at https://doi.org/10.1007/s13398-017-0427-2 · Zbl 1439.11088 · doi:10.1007/s13398-017-0427-2
[105] Qi, F., Liu, F.-F., Shi, X.-T.: Comments on two completely monotonic functions involving the q-trigamma function. J. Inequal. Spec. Funct. 7(4), 211-217 (2016)
[106] Qi, F., Luo, Q.-M.: Bounds for the ratio of two gamma functions—from Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 6(2), 132-158 (2012). Available online at https://doi.org/10.15352/bjma/1342210165 · Zbl 1245.33004 · doi:10.15352/bjma/1342210165
[107] Qi, F., Luo, Q.-M.: Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem. J. Inequal. Appl. 2013, 542 (2013). 20 pages, Available online at https://doi.org/10.1186/1029-242X-2013-542 · Zbl 1294.33004 · doi:10.1186/1029-242X-2013-542
[108] Qi, F., Luo, Q.-M., Guo, B.-N.: Complete monotonicity of a function involving the divided difference of digamma functions. Sci. China Math. 56(11), 2315-2325 (2013). Available online at https://doi.org/10.1007/s11425-012-4562-0 · Zbl 1286.26007 · doi:10.1007/s11425-012-4562-0
[109] Qi, F.; Luo, Q.-M.; Guo, B.-N.; Milovanović, G. V. (ed.); Rassias, M. Th. (ed.), The function (bx−ax)/x \((b^x-a^x)/x\): ratio’s properties, 485-494 (2014), Berlin · Zbl 1329.26006 · doi:10.1007/978-1-4939-0258-3_16
[110] Qi, F., Mahmoud, M., Shi, X.-T., Liu, F.-F.: Some Properties of the Catalan-Qi Function Related to the Catalan Numbers, vol. 5. Springer, Berlin (2016). 20 pages, Available online at https://doi.org/10.1186/s40064-016-2793-1 · Zbl 1420.11056 · doi:10.1186/s40064-016-2793-1
[111] Qi, F., Mortici, C.: Some inequalities for the trigamma function in terms of the digamma function. Appl. Math. Comput. 271, 502-511 (2015). Available online at https://doi.org/10.1016/j.amc.2015.09.039 · Zbl 1410.33009 · doi:10.1016/j.amc.2015.09.039
[112] Qi, F., Mortici, C., Guo, B.-N.: Some properties of a sequence arising from geometric probability for pairs of hyperplanes intersecting with a convex body. Comput. Appl. Math. 37(2), 2190-2200 (2018). Available online at https://doi.org/10.1007/s40314-017-0448-7 · Zbl 1393.33002 · doi:10.1007/s40314-017-0448-7
[113] Qi, F., Niu, D.-W., Lim, D., Guo, B.-N.: Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions. HAL archives (2018). Available online at https://hal.archives-ouvertes.fr/hal-01769288
[114] Qi, F., Rahman, G., Hussain, S.M., Du, W.-S., Nisar, K.S.: Some inequalities of Čebyšev type for conformable k-fractional integral operators. Symmetry 10(11), Article ID 614 (2018). https://doi.org/10.3390/sym10110614 · Zbl 1423.26013 · doi:10.3390/sym10110614
[115] Qi, F., Shi, X.-T., Liu, F.-F.: An integral representation, complete monotonicity, and inequalities of the Catalan numbers. Filomat 32(2), 575-587 (2018). Available online at https://doi.org/10.2298/FIL1802575Q · Zbl 1513.11088 · doi:10.2298/FIL1802575Q
[116] Qi, F., Wang, S.-H.: Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. Glob. J. Math. Anal. 2(3), 91-97 (2014). Available online at https://doi.org/10.14419/gjma.v2i3.2919 · doi:10.14419/gjma.v2i3.2919
[117] Qi, F., Yao, S.-W., Guo, B.-N.: Arithmetic means for a class of functions and the modified Bessel functions of the first kind. Mathematics 7(1), Article ID 60 (2019). Available online at https://doi.org/10.3390/math7010060 · doi:10.3390/math7010060
[118] Qi, F., Zhang, X.-J.: Complete monotonicity of a difference between the exponential and trigamma functions. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 21(2), 141-145 (2014). Available online at https://doi.org/10.7468/jksmeb.2014.21.2.141 · Zbl 1305.33002 · doi:10.7468/jksmeb.2014.21.2.141
[119] Qi, F., Zhang, X.-J., Li, W.-H.: Lévy-Khintchine representations of the weighted geometric mean and the logarithmic mean. Mediterr. J. Math. 11(2), 315-327 (2014). Available online at https://doi.org/10.1007/s00009-013-0311-z · Zbl 1290.30043 · doi:10.1007/s00009-013-0311-z
[120] Qi, F., Zhang, X.-J., Li, W.-H.: The harmonic and geometric means are Bernstein functions. Bol. Soc. Mat. Mex. (3) 23(2), 713-736 (2017). Available online at https://doi.org/10.1007/s40590-016-0085-y · Zbl 1381.26031 · doi:10.1007/s40590-016-0085-y
[121] Qi, F., Zheng, M.-M.: Explicit expressions for a family of the Bell polynomials and applications. Appl. Math. Comput. 258, 597-607 (2015). Available online at https://doi.org/10.1016/j.amc.2015.02.027 · Zbl 1338.33002 · doi:10.1016/j.amc.2015.02.027
[122] Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions—Theory and Applications, 2nd edn. de Gruyter Studies in Mathematics, vol. 37. de Gruyter, Berlin (2012). Available online athttps://doi.org/10.1515/9783110269338 · Zbl 1257.33001 · doi:10.1515/9783110269338
[123] Sun, B.-C., Liu, Z.-M., Li, Q., Zheng, S.-Z.: The monotonicity and convexity of a function involving psi function with applications. J. Inequal. Appl. 2016, Article ID 151 (2016). 17 pp., available online at https://doi.org/10.1186/s13660-016-1084-2 · Zbl 1338.33012 · doi:10.1186/s13660-016-1084-2
[124] Temme, N.M.: Special Functions: An Introduction to Classical Functions of Mathematical Physics. Wiley-Interscience, New York (1996). Available online at https://doi.org/10.1002/9781118032572 · Zbl 0856.33001 · doi:10.1002/9781118032572
[125] Trimble, S.Y., Wells, J., Wright, F.T.: Superadditive functions and a statistical application. SIAM J. Math. Anal. 20(5), 1255-1259 (1989). Available online at https://doi.org/10.1137/0520082 · Zbl 0688.44002 · doi:10.1137/0520082
[126] Wang, S.-H., Qi, F.: Hermite-Hadamard type inequalities for s-convex functions via Riemann-Liouville fractional integrals. J. Comput. Anal. Appl. 22(6), 1124-1134 (2017)
[127] Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946) · JFM 67.0384.01
[128] Yang, Z.-H.: Some properties of the divided difference of psi and polygamma functions. J. Math. Anal. Appl. 455(1), 761-777 (2017). Available online at https://doi.org/10.1016/j.jmaa.2017.05.081 · Zbl 1369.33007 · doi:10.1016/j.jmaa.2017.05.081
[129] Yang, Z.-H., Chu, Y.-M.: Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications. J. Inequal. Appl. 2016, Article ID 200 (2016). 10 pp., available online at https://doi.org/10.1186/s13660-016-1143-8 · Zbl 1346.33001 · doi:10.1186/s13660-016-1143-8
[130] Yang, Z.-H., Chu, Y.-M., Tao, X.-J.: A double inequality for the trigamma function and its applications. Abstr. Appl. Anal. 2014, Article ID 702718 (2014). https://doi.org/10.1155/2014/702718 · Zbl 1474.33012 · doi:10.1155/2014/702718
[131] Yang, Z.-H., Tian, J.-F.: A comparison theorem for two divided differences and applications to special functions. J. Math. Anal. Appl. 464, 580-595 (2018). Available online at https://doi.org/10.1016/j.jmaa.2018.04.024 · Zbl 1388.33002 · doi:10.1016/j.jmaa.2018.04.024
[132] Yang, Z.-H., Tian, J.-F.: A class of completely mixed monotonic functions involving the gamma function with applications. Proc. Am. Math. Soc. 146(11), 4707-4721 (2018). Available online at https://doi.org/10.1090/proc/14199 · Zbl 1408.33006 · doi:10.1090/proc/14199
[133] Yang, Z.-H., Zheng, S.-Z.: Complete monotonicity involving some ratios of gamma functions. J. Inequal. Appl. 2017, Article ID 255 (2017). 17 pp., Available online at https://doi.org/10.1186/s13660-017-1527-4 · Zbl 1372.33002 · doi:10.1186/s13660-017-1527-4
[134] Zhang, X.-J., Qi, F., Li, W.-H.: Properties of three functions relating to the exponential function and the existence of partitions of unity. Int. J. Open Probl. Comput. Sci. Math. 5(3), 122-127 (2012). Available online at https://doi.org/10.12816/0006128 · doi:10.12816/0006128
[135] Zhao, J.-L.: A completely monotonic function relating to the q-trigamma function. J. Math. Inequal. 9(1), 53-60 (2015). Available online at https://doi.org/10.7153/jmi-09-05 · Zbl 1314.33015 · doi:10.7153/jmi-09-05
[136] Zhao, J.-L., Guo, B.-N., Qi, F.: Complete monotonicity of two functions involving the tri- and tetra-gamma functions. Period. Math. Hung. 65(1), 147-155 (2012). Available online at https://doi.org/10.1007/s10998-012-9562-x · Zbl 1289.26024 · doi:10.1007/s10998-012-9562-x
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