Turan's inequalities for the Kummer function in a simultaneous shift of the two parameters

D. B. Karp

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 25, Tome 383 (2010), pp. 110-125

Résumé

Direct and reverse Turan's inequalities are proved for the confluent hypergeometric function (the Kummer function) viewed as a function of a simultaneous shift in the upper and lower parameters. The reverse Turan inequality is derived from a stronger result on the log-convexity of a function of a sufficiently general form, whose particular case is the Kummer function. Two conjectures above the log-concavity of the Kummer function are formulated. The paper continues the research of a number of authors who studied the log-convexity and log-concavity of hypergeometric functions in parameters. Bibl. 18 titles.

Bibliographie

[1] D. S. Mitrinovć, J. E. Pecarić, A. M. Fink, Classical and new Inequalities in Analysis, Kluwer Acad. Publ., 1993 | MR

[2] C. Berg, R. Szwarc, “Bounds on Turán determinants”, J. Approx. Theory, 161:1 (2009), 127–141 | DOI | MR | Zbl

[3] R. Szwarc, “Positivity of Turán determinants for orthogonal polynomials”, Harmonic analysis and hypergroups (Delhi, 1995), Trends Math., Birkhauser Boston, Boston, MA, 1998, 165–182 | MR | Zbl

[4] F. Brenti, “Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update”, Jerusalem combinatorics' 93, Contemp. Math., 178, Amer. Math. Soc., Providence, RI, 1994, 71–89 | DOI | MR | Zbl

[5] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, v. 1, McGraw-Hill Book Company, Inc., New York, 1953 | MR | Zbl

[6] W. Gautschi, “A note on succesive remainders of the exponential series”, Elem. Math., 37 (1982), 46–49 | MR | Zbl

[7] S. M. Sitnik, Neravenstva dlya ostatochnogo chlena ryada Teilora eksponentsialnoi funktsii, Preprint, In-t avtomatiki i protsessov upravleniya DVO RAN, 1993

[8] H. Alzer, “An inequality for the exponential function”, Arch. Math., 55 (1990), 462–464 | DOI | MR | Zbl

[9] A. Baricz, “Functional inequalities involving Bessel and modified Bessel functions of the first kind”, Expo. Math., 26:3 (2008), 279–293 | DOI | MR | Zbl

[10] A. Baricz, “Turán type inequalities for generalized complete elliptic integrals”, Math. Z., 256 (2007), 895–911 | DOI | MR | Zbl

[11] A. Baricz, “Turán type inequalities for hypergeometric functions”, Proc. Amer. Math. Soc., 136:9 (2008), 3223–3229 | DOI | MR | Zbl

[12] M. Carey, M. B. Gordy, The Bank as Grim Reaper: Debt Composition and Recoveries on Defaulted Debt, Preprint, 2007

[13] R. W. Barnard, M. Gordy, K. C. Richards, “A note on Turán type and mean inequalities for the Kummer function”, J. Math. Anal. Appl., 349:1 (2009), 259–263 | DOI | MR | Zbl

[14] D. Karp, S. M. Sitnik, “Log-convexity and log-concavity of hypergeometric-like functions”, J. Math. Anal. Appl., 364 (2010), 384–394 | DOI | MR | Zbl

[15] D. B. Karp, “Ob odnoi zadache iz mnogomernoi statistiki dlya funktsii Kummera”, Tr. Vserossiiskoi konf. “XXXV Dalnevostochnaya matematicheskaya shkola–seminar im. akad. E. V. Zolotova” (Vladivostok, 31 avgusta – 5 sentyabrya 2010 g.) (to appear)

[16] A. Cuyt, V. B. Petersen, B. Verdonk, H. Waadeland, W. B. Jones, Handbook of Continued Fractions for Special Functions, Springer, 2008 | MR

[17] K. B. Stolarsky, “From Wythoff's Nim to Chebyshev's inequality”, Amer. Math. Monthly, 98 (1991), 889–900 | DOI | MR | Zbl

[18] H. Alzer, “On some inequalities for the gamma and psi functions”, Math. Comput., 66:217 (1997), 373–389 | DOI | MR | Zbl