Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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TURÁN-TYPE INEQUALITIES FOR GAUSS AND CONFLUENT HYPERGEOMETRIC FUNCTIONS VIA CAUCHY-BUNYAKOVSKY-SCHWARZ INEQUALITY

  • Received : 2017.11.21
  • Accepted : 2018.02.13
  • Published : 2018.10.31
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Abstract

This paper is devoted to the study of $Tur{\acute{a}}n$-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 $Tur{\acute{a}}n$-type inequalities.

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References

  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. https://doi.org/10.4134/JKMS.j160137
  2. H. Alzer, On the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 234 (1999), no. 1, 6-14. https://doi.org/10.1006/jmaa.1998.6252
  3. A. Baricz, Turan type inequalities for regular Coulomb wave functions, J. Math. Anal. Appl. 430 (2015), no. 1, 166-180. https://doi.org/10.1016/j.jmaa.2015.04.082
  4. P. K. Bhandari and S. K. Bissu, On some inequalities involving Turan-type inequalities, Cogent Math. 3 (2016), Art. ID 1130678, 7 pp.
  5. P. K. Bhandari and S. K. Bissu, Inequalities for some classical integral transforms, Tamkang J. Math. 47 (2016), no. 3, 351-356. https://doi.org/10.5556/j.tkjm.47.2016.1981
  6. P. K. Bhandari and S. K. Bissu, Turan-type inequalities for Struve, modified Struve and Anger-Weber functions, Integral Transforms Spec. Funct. 27 (2016), no. 12, 956-964. https://doi.org/10.1080/10652469.2016.1233404
  7. D. K. Callebaut, Generalization of the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 12 (1965), 491-494. https://doi.org/10.1016/0022-247X(65)90016-8
  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.
  9. S. Karlin and G. Szego, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/1961), 1-157.
  10. A. Laforgia and P. Natalini, Turan-type inequalities for some special functions, JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 22, 3 pp.
  11. Z. Liu, Remark on a refinement of the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 218 (1998), no. 1, 13-21. https://doi.org/10.1006/jmaa.1997.5720
  12. M. Masjed-Jamei, A functional generalization of the Cauchy-Schwarz inequality and some subclasses, Appl. Math. Lett. 22 (2009), no. 9, 1335-1339. https://doi.org/10.1016/j.aml.2009.03.001
  13. K. Mehrez, Functional inequalities for the Wright functions, Integral Transforms Spec. Funct. 28 (2017), no. 2, 130-144. https://doi.org/10.1080/10652469.2016.1254628
  14. D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink,, Classical and New Inequalities in Anal- ysis, Mathematics and its Applications (East European Series), 61, Kluwer Academic Publishers Group, Dordrecht, 1993.
  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, Birkhauser Verlag, Basel, 1988.
  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. https://doi.org/10.1186/s13660-016-1282-y
  17. W. L. Steiger, Classroom Notes: On a Generalization of the Cauchy-Schwarz Inequality, Amer. Math. Monthly 76 (1969), no. 7, 815-816. https://doi.org/10.1080/00029890.1969.12000339
  18. P. Turan, On the zeros of the polynomials of Legendre, Casopis Pest. Mat. Fys. 75 (1950), 113-122.