n-Convexity via Delta-Integral Representation of Divided Difference on Time Scales

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-20-2022-8
Published: Jan 26, 2022
Cite as:
Hira Ashraf Baig, Naveed Ahmad, n-Convexity via Delta-Integral Representation of Divided Difference on Time Scales, Int. J. Anal. Appl., 20 (2022), 8.

Main Article Content

Hira Ashraf Baig, Naveed Ahmad

Abstract

We introduce the delta-integral representation of divided difference on arbitrary time scales and utilize it to set criteria for n-convex functions involving delta-derivative on time scales. Consequences of the theory appear in terms of estimates which generalize and extend some important facts in mathematical analysis.

Article Details

References

  1. M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Springer Science & Business Media, 2001.
  2. M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Springer Science & Business Media, 2002.
  3. M. Bohner, S.G. Georgiev, Multivariable dynamic calculus on time scales, Springer, 2016.
  4. L. Ciurdariu, A note concerning several hermite-hadamard inequalities for different types of convex functions, Int. J. Math. Anal. 6 (2012), 33–36.
  5. J. E. Pecˇaric´, Y. L. Tong, Convex functions, partial orderings, and statistical applications, Academic Press, 1992.
  6. S. Zaheer Ullah, M. Adil Khan, Y.M. Chu, A note on generalized convex functions, J Inequal Appl. 2019 (2019), 291. https://doi.org/10.1186/s13660-019-2242-0.
  7. D.E. Varberg, A. W. Roberts, Convex functions, Academic Press, New York-London, 1973.
  8. C. Dinu, Convex functions on time scales, Ann. Univ. Craiova, Math. Comp. Sci. Ser. 35 (2008), 87–96.
  9. R.P. Agarwal, D. O’Regan, S.H. Saker, Hardy type inequalities on time scales, Springer, 2016.
  10. R.P. Agarwal, D. O’Regan, S. Saker, Dynamic inequalities on time scales, Springer, 2014.
  11. E. Hopf, Über die zusammenhänge zwischen gewissen höheren differenzen-quotienten reeller funktionen einer reellen variablen und deren differenzierbarkeitseigenschaften, Ph.D. thesis, Norddeutsche Buchdr. u. Verlagsanst. (1926).
  12. T. Popoviciu, On some properties of functions of one or two variables réthem, Ph.D. thesis, Institutul de Arte Grafice "Ardealul (1933).
  13. T. Popoviciu, Les fonctions convexes, Actualites Sci. Ind. 992 (1945).
  14. P. Bullen, A criterion for n-convexity, Pac. J. Math. 36 (1971), 81–98. https://doi.org/10.2140/pjm.1971.36.81.
  15. R. Mikic, J. Pecˇaric´, Jensen-type inequalities on time scales for n-convex functions, Commun. Math. Anal. 21 (2018), 46–67.
  16. H.A. Baig, N. Ahmad, The weighted discrete dynamic inequalities for 4-convex functions, and its generalization on time scales with constant graininess function, J. Inequal. Appl. 2020 (2020), 168. https://doi.org/10.1186/s13660-020-02435-4.
  17. R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results. Math. 35 (1999), 3–22. https://doi.org/10.1007/BF03322019.
  18. D. Zwick, A divided difference inequality for n-convex functions, J. Math. Anal. Appl. 104 (1984), 435–436. https://doi.org/10.1016/0022-247X(84)90008-8.
  19. R. Farwig, D. Zwick, Some divided difference inequalities for n-convex functions, J. Math. Anal. Appl. 108 (1985), 430–437. https://doi.org/10.1016/0022-247X(85)90036-8.
  20. I. Brnetic´, Inequalities for n-convex functions, J. Math. Inequal. 5 (2011), 193–197.
  21. I. Brnetić, Inequalities for convex and 3-log convex functions, Rad HAZU (515) (2013), 189–194.
  22. E. Isaacson, H.B. Keller, Analysis of numerical methods, Courier Corporation, 2012.