Geodesic
On monotonicity of ratios of some hypergeometric functions
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 260-268.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the preprint [1] one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and were not proved from 1993. In this paper we prove some conjectures from [1] for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turán–type inequalities for special functions.
Keywords: hypergeometric functions, exponential, series sections, Turán–type inequalities. 
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S. M. Sitnik; Kh. Mehrez. On monotonicity of ratios of some hypergeometric functions. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 260-268. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a102/

[1] S. M. Sitnik, Inequalities for the exponential remainder, preprint, Institute of Automation and Control Process, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok, 1993, 31 pp. (in Russian)

[2] S. M. Sitnik, “Conjectures on Monotonicity of Ratios of Kummer and Gauss Hypergeometric Functions”, RGMIA Research Report Collection, 17 (2014), 107, 4 pp.

[3] A. Edrei, E. B. Saff, R. S. Varga, Zeros of Sections of Power Series, Springer, 1983 | MR | Zbl

[4] G. H. Hardy, P. V. Seshu Aiyan, B. M. Wilson (eds.), Collected Papers of Srinivasa Ramanujan, Cambridge, 1927

[5] P. Kesava Menon, “Some integral inequalities”, Math. Student, 11 (1943), 36–38 | MR

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

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

[8] D. B. Karp, S. M. Sitnik, “Log-convexity and log-concavity of hypergeometric-like functions”, Journal of Mathematical Analysis and Applications, 364:2 (2010), 384–394 | DOI | MR | Zbl

[9] D. B. Karp, S. M. Sitnik, “Inequalities and monotonicity of ratios for generalized hypergeometric function”, Journal of Approximation Theory, 161 (2009), 337–352 | DOI | MR | Zbl

[10] P. Turán, “On the zeros of the polynomials of Legendre”, Casopis Pest. Mat. Fys., 75 (1950), 113–122 | MR | Zbl

[11] M. Biernacki, J. Krzyz, “On the monotonicity of certain functionals in the theory of analytic functions”, Ann. Univ. M. Curie–Skłodowska, 2 (1995), 134–145

[12] S. Ponnusamy, M. Vuorinen, “Asymptotic expansions and inequalities for hypergeometric functions”, Mathematika, 44 (1997), 278–301 | DOI | MR | Zbl

[13] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010 | MR | Zbl

[14] Á. Baricz, “Functional inequalities involving Bessel and modified Bessel functions of the first kind”, Expositiones Mathematicae, 26 (2008), 279–293 | DOI | MR | Zbl

[15] S. M. Sitnik, “Factorization and Estimates of Norms in Weighted Lebesgue Spaces of Buschman–Erdelyi Operators”, Soviet Math. Dokl., 44:2 (1991), 641–646 | MR | Zbl

[16] V. V. Katrakhov, S. M. Sitnik, “Boundary value problem for the stationary Schrödinger equation with a singular potential”, Akademiia Nauk SSSR, Doklady, 30:2 (1984), 468–470 | MR | Zbl

[17] M. V. Zhuravlev, E. A. Kiselev, L. A. Minin, S. M. Sitnik, “Jacobi theta-functions and systems of integral shifts of Gaussian functions”, Journal of Mathematical Sciences, 173:2 (2011), 231–241 | DOI | MR | Zbl

[18] E. A. Kiselev, L. A. Minin, I. Ya. Novikov, S. M. Sitnik, “On the Riesz Constants for Systems of Integer Translates”, Mathematical Notes, 96:2 (2014), 228–238 | DOI | MR | Zbl

[19] A. B. Pevnyi, S. M. Sitnik, “On Gasparyan's inequality”, Journal Of Mathematical Sciences, 205:2 (2015), 304–307 | DOI | MR | Zbl

[20] K. Mehrez, S. M. Sitnik, “On monotonicity of ratios of $q$-Kummer confluent hypergeometric and $q$-hypergeometric functions and associated Turán types inequalities”, RGMIA Research Report Collection, 17 (2014), 150, 9 pp.

[21] G. Gasper, M. Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, 1990 | MR | Zbl