Abstract
In this paper, our aim is to establish some mean value inequalities for the Fox–Wright functions, such as Turán-type inequalities, Lazarević and Wilker-type inequalities. As applications we derive some new type inequalities for hypergeometric functions and the four-parametric Mittag–Leffler functions. Furthermore, we prove the monotonicity of ratios for sections of series of Fox–Wright functions. The results are also closely connected with Turán-type inequalities. Moreover, some other type inequalities are also presented. At the end of the paper, some problems are stated which may be of interest for further research.
Similar content being viewed by others
References
Mehrez, K., Sitnik, S.M.: Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions. Analysis 36(4), 263–268 (2016)
Mehrez, K., Sitnik, S.M.: Functional inequalities for the Mittag-Leffler functions. Results Math. 72(1), 703–714 (2017)
Mehrez, K., Sitnik, S.M.: Turán type inequalities for classical and generalized Mittag–Leffler functions. arXiv:1603.08504v2
Mehrez, K.: Functional inequalities for the Wright functions. Integral Transforms Spec. Funct. 28(2), 130–144 (2017)
Mathai, A.M., Saxena, R.K.: The H-Function with Applications in Statistics and Other Disciplines. Wiley, New York (1978)
Srivastava, H.M., Tomovski, Z.: Some problems and solutions involving Mathieus series and its generalization. J. Inequal. Pure Appl. Math. 5(2) (2004), 1–13 (Article 45, electronic)
Ponnusamy, S., Vuorinen, M.: Asymptotic expansions and inequalities for hypergeometric functions. Mathematika 44, 278–301 (1997)
Paris, R.B.: A Kummer-type transformation for a \(_2F_2\) hypergeometric function. J. Comput. Appl. Math. 173, 379–382 (2005)
Baricz, Á.: Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math. 26, 279–293 (2008)
Mitrinović, D.S.: Analytic Inequalities. Springer, Berlin (1970)
Wilker, J.B.: Problem E3306. Am. Math. Mon. 96, 1–55 (1989)
Zhu, L.: On Wilker-type inequalities. Math. Inequal. Appl. 10(4), 727–731 (2007)
Acknowledgements
S. M. Sitnik was supported by a State contract of the Russian Ministry of Education and Science (contract No 1.7311.2017/8.9).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mehrez, K., Sitnik, S.M. Functional inequalities for the Fox–Wright functions. Ramanujan J 50, 263–287 (2019). https://doi.org/10.1007/s11139-018-0071-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-018-0071-2
Keywords
- Fox–Wright functions
- Hypergeometric functions
- Four-parametric Mittag–Leffler functions
- Turán-type inequalities
- Lazarević and Wilker-type inequalities