Geodesic
Inequalities for some basic hypergeometric functions
Problemy analiza, Tome 8 (2019) no. 1, pp. 47-64.

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We establish conditions for the discrete versions of logarithmic concavity and convexity of the higher order regularized basic hypergeometric functions with respect to the simultaneous shift of all its parameters. For a particular case of Heine's basic hypergeometric function, we prove logarithmic concavity and convexity with respect to the bottom parameter. We, further, establish a linearization identity for the generalized Turánian formed by a particular case of Heine's basic hypergeometric function. Its $q=1$ case also appears to be new.
Keywords: basic hypergeometric function, log-convexity, log-concavity, multiplicative concavity, generalized Turánian, $q$-hypergeometric identity. 
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S. I. Kalmykov; D. B. Karp. Inequalities for some basic hypergeometric functions. Problemy analiza, Tome 8 (2019) no. 1, pp. 47-64. http://geodesic.mathdoc.fr/item/PA_2019_8_1_a3/

[1] A. Baricz, K. Raghavendar, A. Swaminathan, “Turan type inequalities for q-hypergeometric functions”, J. Approx. Theory, 168 (2013), 69–79 | DOI | MR | Zbl

[2] G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Second Edition, Cambridge University Press, 2004 | MR | Zbl

[3] M. E. H. Ismail, “The basic Bessel functions and polynomials”, SIAM J. Math. Anal., 12:3 (1981), 454–468 | DOI | MR | Zbl

[4] V. Kac, P. Cheung, Quantum Calculus, Springer, 2002 | MR | Zbl

[5] S. I. Kalmykov, D. B. Karp, “Log-convexity and log-concavity for seriesin gamma ratios and applications”, Journal of Mathematical Analysis and Applications, 406:2 (2013), 400–418 | DOI | MR | Zbl

[6] S. I. Kalmykov, D. B. Karp, “Log-concavity and Turan type inequalities for the generalized hypergeometric function”, Analysis Mathematica, 43:4 (2017), 567–580 | DOI | MR | Zbl

[7] S. I. Kalmykov, D. B. Karp, “Inequalities for series in q-shifted factorials and q-gamma functions”, Journal of Mathematical Analysis and Applications, 460:1 (2018), 332–351 | DOI | MR | Zbl

[8] S. I. Kalmykov, D. B. Karp, “New identities for a sum of products of the Kummer functions”, Siberian Electronic Mathematical Reports, 15 (2018), 267–276 | DOI | MR | Zbl

[9] S. Koumandos, H. L. Pedersen, “On the Laplace transform of absolutely monotonic functions”, Results in Mathematics, 72:3 (2017), 1041–1053 | DOI | MR | Zbl

[10] K. Mehrez, “Turan type inequalities for the q-exponential functions”, Arab.J. Math., 6:4 (2017), 309–314 | DOI | MR | Zbl

[11] K. Mehrez, S. M. Sitnik, “On monotonicity of ratios of some q-hypergeometric functions”, Matematicki Vesnik, 68:3 (2016), 225–231 | MR | Zbl

[12] A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of Majorization and Its applications, second edition, Springer, 2011 | MR | Zbl

[13] M. A. Olshanetsky, V.-B. K. Rogov, “Modified q-Bessel functions and q-Macdonald functions”, Mat. Sb., 187:10 (1996), 1525–1544 | DOI | MR

[14] M. Rahman, “An addition theorem and some product formulas for q-Bessel functions”, Can. J. Math., XL:5 (1988), 1203–1221 | DOI | MR | Zbl

[15] R. L. Schilling, R. Song, Z. Vondracek, Bernstein functions. Theory and applications, Studies in Mathematics, 37, Walter de Gruyter, Berlin | MR | Zbl

[16] R. Zhang, An Inequality for Basic Confluent Hypergeometric Series, 2006, arXiv: math/0611743