Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions

References

[1] Alpár L., In memory of Paul Turán, J. Number Theory 13 (1981), 271–278. 10.1016/0022-314X(81)90012-3Suche in Google Scholar

[2] Alzer H., An inequality for the exponential function, Arch. Math. 55 (1990), 462–464. 10.1007/BF01190267Suche in Google Scholar

[3] Anderson G. D. and Vuorinen M., Reflections on Ramanujan’s mathematical gems, preprint 2010, http://arxiv.org/abs/1006.5092. Suche in Google Scholar

[4] Baricz Á., Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (2008), 279–293. 10.1016/j.exmath.2008.01.001Suche in Google Scholar

[5] Baricz A., Raghavendarb K. and Swaminathan A., Turán type inequalities for q-hypergeometric functions, J. Approx. Theory 168 (2013), 69–79. 10.1090/S0002-9939-08-09353-2Suche in Google Scholar

[6] Barnard R. W., Gordy M. and Richards K. C., A note on Turán type and mean inequalities for the Kummer function, J. Math. Anal. Appl. 349 (2009), no. 1, 259–263. 10.1016/j.jmaa.2008.08.024Suche in Google Scholar

[7] Biernacki M. and Krzyz J., On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. M. Curie-Skłodowska 2 (1995), 134–145. Suche in Google Scholar

[8] Edrei A., Saff E. B. and Varga R. S., Zeros of Sections of Power Series, Lecture Notes in Math. 1002, Springer, Berlin, 1983. 10.1007/BFb0070472Suche in Google Scholar

[9] Gautschi W., A note on the successive remainders of the exponential series, Elem. Math. 37 (1982), 46–49. Suche in Google Scholar

[10] Hardy G. H., Seshu Aiyan P. V. and Wilson B. M. (eds.), Collected Papers of Srinivasa Ramanujan, Cambridge University Press, Cambridge, 1927. Suche in Google Scholar

[11] Kalmykov S. I. and Karp D. B., Log-concavity for series in reciprocal gamma functions, Integral Transforms Spec. Funct. 24 (2013), no. 11, 859–872. 10.1080/10652469.2013.764874Suche in Google Scholar

[12] Kalmykov S. I. and Karp D. B., Log-convexity and log-concavity for series in gamma ratios and applications, J. Math. Anal. Appl. 406 (2013), 400–418. 10.1016/j.jmaa.2013.04.061Suche in Google Scholar

[13] Karp D. B. and Sitnik S. M., Inequalities and monotonicity of ratios for generalized hypergeometric function, J. Approx. Theory 161 (2009), 337–352. 10.1016/j.jat.2008.10.002Suche in Google Scholar

[14] Karp D. B. and Sitnik S. M., Log-convexity and log-concavity of hypergeometric-like functions, J. Math. Anal. Appl. 364 (2010), no. 2, 384–394. 10.1016/j.jmaa.2009.10.057Suche in Google Scholar

[15] Karp D. B., Savenkova A. and Sitnik S. M., Series expansions for the third incomplete elliptic integral via partial fraction decompositions, J. Comput. Appl. Math. 207 (2007), no. 2, 331–337. 10.1016/j.cam.2006.10.019Suche in Google Scholar

[16] Katrakhov V. V. and Sitnik S. M., Boundary value problem for the stationary Schrödinger equation with a singular potential, Dokl. Acad. Nauk 278 (1984), 797–799. Suche in Google Scholar

[17] Kesava Menon P., Some integral inequalities, Math. Student 11 (1943), 36–38. Suche in Google Scholar

[18] Kiselev E. A., Minin L. A., Novikov I. Y. and Sitnik S. M., On the Riesz constants for systems of integer translates., Math. Notes 96 (2014), no. 2, 228–238. 10.1134/S0001434614070244Suche in Google Scholar

[19] Lindén H., Convexity and inequalities for power series, Helsinki Analysis Seminar, 2006. Suche in Google Scholar

[20] Mehrez K. and Sitnik S. M., Inequalities for sections of exponential function series and proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions, RGMIA Res. Rep. Collect. 17 (2014), Article ID 132. Suche in Google Scholar

[21] Mehrez K. and Sitnik S. M., On monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities., RGMIA Res. Rep. Collect. 17 (2014), Article ID 150. Suche in Google Scholar

[22] Mehrez K. and Sitnik S. M., Proofs of some conjectures on monotonicity of ratios of Kummer and Gauss hypergeometric functions and related Turán-type inequalities, preprint. Suche in Google Scholar

[23] Ponnusamy S. and Vuorinen M., Asymptotic expansions and inequalities for hypergeometric functions, Mathematika 44 (1997), 278–301. 10.1112/S0025579300012602Suche in Google Scholar

[24] Sitnik S. M., Factorization and estimates of norms in weighted Lebesgue spaces of Buschman–Erdelyi operators, Soviet Math. Dokl 44 (1991), 641–646. Suche in Google Scholar

[25] Sitnik S. M., Inequalities for the exponential remainder (in Russian), preprint 1993. Suche in Google Scholar

[26] Sitnik S. M., A conjecture on monotonicity of a ratio of Kummer hypergeometric functions, preprint 2012, http://arxiv.org/abs/1207.0936. Suche in Google Scholar

[27] Sitnik S. M., Conjectures on monotonicity of ratios of Kummer and Gauss hypergeometric functions, RGMIA Res. Rep. Collect. 17 (2014), Article ID 107. Suche in Google Scholar

[28] Sitnik S. M. and Zelezko O., unpublished, preprint 1994. Suche in Google Scholar

[29] Turán P., On the zeros of the polynomials of Legendre, Casopis Pest. Mat. Fys. 75 (1950), 113–122. 10.21136/CPMF.1950.123879Suche in Google Scholar

[30] Zhuravlev M. V., Kiselev E. A., Minin L. A. and Sitnik S. M., Jacobi theta-functions and systems of integral shifts of Gaussian functions, J. Math. Sci. 173 (2011), no. 2, 231–241. 10.1007/s10958-011-0246-5Suche in Google Scholar