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Turán-type inequalities for Gauss and confluent hypergeometric functions via Cauchy-Bunyakovsky-Schwarz inequality. (English) Zbl 1416.26028

Summary: This paper is devoted to the study of Turán-type inequalities for some well-known special functions such as Gauss hypergeometric functions, generalized complete elliptic integrals and confluent hypergeometric functions which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality. We also apply these inequalities for some sample of interest such as incomplete beta function, incomplete gamma function, elliptic integrals and modified Bessel functions to obtain their corresponding Turán-type inequalities.

MSC:

26D07 Inequalities involving other types of functions
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
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[1] R. M. Ali, S. R. Mondal, and K. S. Nisar, Monotonicity properties of the generalized Struve functions, J. Korean Math. Soc. 54 (2017), no. 2, 575-598. · Zbl 1359.33004
[2] H. Alzer, On the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 234 (1999), no. 1, 6-14. · Zbl 0933.26010
[3] A. Baricz, Tur´an type inequalities for regular Coulomb wave functions, J. Math. Anal. Appl. 430 (2015), no. 1, 166-180. · Zbl 1318.33039
[4] P. K. Bhandari and S. K. Bissu, On some inequalities involving Tur´an-type inequalities, Cogent Math. 3 (2016), Art. ID 1130678, 7 pp. · Zbl 1426.26040
[5] , Inequalities for some classical integral transforms, Tamkang J. Math. 47 (2016), no. 3, 351-356. · Zbl 1368.44001
[6] , Tur´an-type inequalities for Struve, modified Struve and Anger-Weber functions, Integral Transforms Spec. Funct. 27 (2016), no. 12, 956-964. · Zbl 1357.26025
[7] D. K. Callebaut, Generalization of the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 12 (1965), 491-494. · Zbl 0136.03401
[8] C. M. Joshi and S. K. Bissu, Some inequalities of Bessel and modified Bessel functions, J. Austral. Math. Soc. Ser. A 50 (1991), no. 2, 333-342. · Zbl 0732.33002
[9] S. Karlin and G. Szeg¨o, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/1961), 1-157.
[10] A. Laforgia and P. Natalini, Tur´an-type inequalities for some special functions, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 22, 3 pp. · Zbl 1126.26017
[11] Z. Liu, Remark on a refinement of the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 218 (1998), no. 1, 13-21. · Zbl 0891.26011
[12] M. Masjed-Jamei, A functional generalization of the Cauchy-Schwarz inequality and some subclasses, Appl. Math. Lett. 22 (2009), no. 9, 1335-1339. · Zbl 1173.26323
[13] K. Mehrez, Functional inequalities for the Wright functions, Integral Transforms Spec. Funct. 28 (2017), no. 2, 130-144. · Zbl 1364.33010
[14] D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink,, Classical and New Inequalities in Analysis, Mathematics and its Applications (East European Series), 61, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0771.26009
[15] A. F. Nikiforov and V. B. Uvarov, Special functions of mathematical physics, translated from the Russian and with a preface by Ralph P. Boas, Birkh¨auser Verlag, Basel, 1988. · Zbl 0624.33001
[16] K. S. Nisar, S. R. Mondal, and J. Choi, Certain inequalities involving the k-Struve function, J. Inequal. Appl. 2017 (2017), Paper No. 71, 8 pp. · Zbl 1362.33003
[17] W. L. Steiger, Classroom Notes: On a Generalization of the Cauchy-Schwarz Inequality, Amer. Math. Monthly 76 (1969), no. 7, 815-816. · Zbl 0195.06401
[18] P. Tur´an, On the zeros of the polynomials of Legendre, ˇCasopis Pˇest. Mat. Fys. 75 (1950), 113-122.
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