Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

MONOTONICITY CRITERION AND FUNCTIONAL INEQUALITIES FOR SOME q-SPECIAL FUNCTIONS

  • Mehrez, Khaled (Department of Mathematics Faculty of Sciences of Tunis University of Tunis El Manar)
  • Received : 2019.12.30
  • Accepted : 2020.05.14
  • Published : 2021.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

Our aim in this paper is to derive several new monotonicity properties and functional inequalities of some functions involving the q-gamma, q-digamma and q-polygamma functions. More precisely, some classes of functions involving the q-gamma function are proved to be logarithmically completely monotonic and a class of functions involving the q-digamma function is showed to be completely monotonic. As applications of these, we offer upper and lower bounds for this special functions and new sharp upper and lower bounds for the q-analogue harmonic number harmonic are derived. Moreover, a number of two-sided exponential bounding inequalities are given for the q-digamma function and two-sided exponential bounding inequalities are then obtained for the q-tetragamma function.

Keywords

References

  1. H. Alzer, Sharp bounds for the ratio of q-gamma functions, Math. Nachr. 222 (2001), 5-14. https://doi.org/10.1002/1522-2616(200102)222:1<5::aid-mana5>3.0.co;2-q
  2. G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J. Math. 160 (1993), no. 1, 1-18. http://projecteuclid.org/euclid.pjm/1102624560 https://doi.org/10.2140/pjm.1993.160.1
  3. N. Batir, q-extensions of some estimates associated with the digamma function, J. Approx. Theory 174 (2013), 54-64. https://doi.org/10.1016/j.jat.2013.06.002
  4. N. Batir, Monotonicity properties of q-digamma and q-trigamma functions, J. Approx. Theory 192 (2015), 336-346. https://doi.org/10.1016/j.jat.2014.12.013
  5. J. El Kamel and K. Mehrez, A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions, Positivity 22 (2018), no. 5, 1403-1417. https://doi.org/10.1007/s11117-018-0584-3
  6. B.-N. Guo and F. Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 2, 21-30.
  7. M. E. H. Ismail, L. Lorch, and M. E. Muldoon, Completely monotonic functions associated with the gamma function and its q-analogues, J. Math. Anal. Appl. 116 (1986), no. 1, 1-9. https://doi.org/10.1016/0022-247X(86)90042-9
  8. C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Math. 10 (1974), 152-164. https://doi.org/10.1007/BF01832852
  9. T. H. Koornwinder, q-Special functions, a tutorial, arXiv:math/9403216v2
  10. K. Mehrez, A class of logarithmically completely monotonic functions related to the q-gamma function and applications, Positivity 21 (2017), no. 1, 495-507. https://doi.org/10.1007/s11117-016-0431-3
  11. K. Mehrez, Some geometric properties of a class of functions related to the Fox-Wright functions, Banach J. Math. Anal. (2020), https://doi.org/10.1007/s43037-020-00059-w
  12. A. Salem, A q-analogue of the exponential integral, Afr. Mat. 24 (2013), no. 2, 117-125. https://doi.org/10.1007/s13370-011-0046-6
  13. A. Salem, A certain class of approximations for the q-digamma function, Rocky Mountain J. Math. 46 (2016), no. 5, 1665-1677. https://doi.org/10.1216/RMJ-2016-46-5-1665
  14. A. Salem, Generalized the q-digamma and the q-polygamma functions via neutrices, Filomat 31 (2017), no. 5, 1475-1481. https://doi.org/10.2298/FIL1705475S
  15. C. Wei and Q. Gu, q-generalizations of a family of harmonic number identities, Adv. in Appl. Math. 45 (2010), no. 1, 24-27. https://doi.org/10.1016/j.aam.2009.11.007