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Turán Type Inequalities for j Generalized \(p-k\) Mittag-Leffler Function

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Abstract

In the present paper, Turán type inequalities of j generalized \(p-k\) Mittag-Leffler function have been proved. The obtained results are the extension of Turán type inequalities for classical and generalized Mittag-Leffler functions, which provide fresh insights into the j generalized \(p-k\) Mittag-Leffler function and thus provides new properties and proofs. Even several corollaries have been derived as the particular cases.

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References

  1. Agarwal, P., Rogosin, S.V., Trujillo, J.J.: Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions. Proc. Math. Sci. 125(3), 291–306 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, P., Nieto, J.J.: Some fractional integral formulas for the Mittag-Leffler type function with four parameters. Open Math. 13(1), 1 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agarwal, P., Chand, M., Baleanu, D., O’Regan, D., Jain, S.: On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function. Adv. Differ. Equ. 2018(1), 1–13 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alzer, H., Gerhold, S., Kauers, M.: On Turán’s inequality for Legendre polynomials. Expo. Math. 172, 289–312 (2004)

    MATH  Google Scholar 

  5. Baricz, Á.: Turán type inequality for generalized complete elliptic integrals. Math. Z. 256, 895–911 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baricz, Á.: Turán type inequalities for hypergeometric functions. Proc. Am. Math. Soc. 136(9), 3223–3229 (2008)

    Article  MATH  Google Scholar 

  7. Baricz, Á.: Turán type inequalities for some probability density functions. Stud. Sci. Math. Hung. 47(2), 175–189 (2010)

    MATH  Google Scholar 

  8. Baricz, Á.: Turán type inequalities for modified Bessel functions. Bull. Aust. Math. Soc. 82(2), 254–264 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Baricz, Á., Jankov, D., Pogány, T.K.: Turán type inequalities for Krätzel functions. J. Math. Anal. Appl. 388(2), 716–724 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Baricz, Á., Ponnusamy, S.: On Turán type inequalities for modified Bessel functions. Proc. Am. Math. Soc. 141(2), 523–532 (2013)

    Article  MATH  Google Scholar 

  11. Baricz, Á., Ponnusamy, S., Singh, S.: Turán type inequalities for confluent hypergeometric functions of the second kind. Stud. Sci. Math. Hung. 53(1), 74–92 (2016)

    MATH  Google Scholar 

  12. Barnard, R.W., Gordy, M.B., Richards, K.C.: A note on Turán type and mean inequalities for the Kummer function. J. Math. Anal. Appl. 349, 259–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bhandari P. K., Bissu, S. K.: On some inequalities involving Turán-type inequalities. Cogent Math. 3(1), Article: 1130678 (2016)

  14. Bhandari, P.K., Bissu, S.K.: Turán type inequalities for Gauss and confluent hypergeometric functions via Cauchy–Bunyakovsky–Schwarz inequality. Commun. Korean Math. Soc. 33(4), 1285–1301 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Choi, J., Agarwal, P.: A note on fractional integral operator associated with multiindex Mittag-Leffler functions. Filomat 30(7), 1931–1939 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dharmadhikari, S., Joag-Dev, K.: Unimodality, Convexity, and Applications. Academic Press, Boston (1988)

    MATH  Google Scholar 

  17. Dou, X. K., Yin, L., Lin, X. L.: Several Turán-Type Inequalities for the Generalized Mittag-Leffler Function. J. Funct. Spaces. Article ID 9509728, 1–6 (2020)

  18. Gehlot, K.S.: Differential equation of generalized \(k\)-Mittag-Leffler function. Int. J. Contemp. Math. Sci. 8(14), 693–698 (2013)

    Article  MathSciNet  Google Scholar 

  19. Gehlot, K. S.: Two parameter gamma function and its properties. arXiv:1701.01052 (2017)

  20. Gehlot, K.S.: The \(p-k\) Mittag-Leffler function. PJM 7(2), 628–632 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Gehlot, K.S., Bhandari, A.: The \(j\)-Generalized \(p-k\) Mittag-Leffler Function Fract. Calc. Appl. Anal. 13(1), 122–129 (2022)

    MATH  Google Scholar 

  22. Gorenflo, R., Kilbas, A. A., Mainardi, F., Rogosin, S. V.: Mittag-Leffler Functions, Related Topics and Spplications. Springer, Berlin (2020)

  23. Jain, S., Agarwal, P., Kilicman, A.: Pathway fractional integral operator associated with 3m-parametric Mittag-Leffler functions. Int. J. Appl. Math. Comput. 4(5), 1–7 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jain, S., Agarwal, R.P., Agarwal, P., Singh, P.: Certain Unified Integrals Involving a Multivariate Mittag-Leffler Function. Axioms 10(2), 81 (2021)

    Article  Google Scholar 

  25. Luque, L.L.: On a Generalized Mittag-Leffler function. Int. J. Math. Anal. 13(5), 223–234 (2019)

    Article  Google Scholar 

  26. Mehrez, K., Sitnik, S.M.: Turán type inequalities for classical and generalized Mittag-Leffler functions. Anal. Math. 44(4), 521–541 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)

    MathSciNet  MATH  Google Scholar 

  28. Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  29. Shukla, A.K., Prajapati, J.C.: On the generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336, 797–811 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Turán, P.: On the zeros of the polynomials of Legendre, C\(\breve{a}\)sopis P\(\breve{e}\)st. Mat. Fys. 75, 113–122 (1950)

    MathSciNet  Google Scholar 

  31. Yin, L., Huang, L.: Turán type inequalities for generalized Mittag-Leffler functions. J. Math. Inequal. 13(3), 667–672 (2017)

    Article  MATH  Google Scholar 

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Acknowledgements

The first Author is thankful to Council of Scientific and Industrial Research(CSIR), India for financial assistance in the form of Junior Research Fellowship (File no: 09/1007(0010)/2020-EMR-I).

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Correspondence to R. K. Jana.

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In our paper, we proved Turán type inequalities for j Generalized \(p-k\) Mittag-Leffler function introduced by Gehlot and Bhandari [21].

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Gajera, J.B., Jana, R.K. Turán Type Inequalities for j Generalized \(p-k\) Mittag-Leffler Function. Int. J. Appl. Comput. Math 9, 28 (2023). https://doi.org/10.1007/s40819-023-01513-7

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