Several Jensen–Grüss inequalities with applications in information theory

Keywords: Jensen difference, Gr\


UDC 517.5

Several integral Jensen–Grüss  inequalities are proved together with their refinements.  Some new bounds for integral Jensen–Chebyshev  inequality are obtained. The multidimensional integral variants are also presented.  In addition, some integral Jensen–Grüss  inequalities for monotone  and completely monotone functions are established.  Finally, as an application, we present the refinements  for Shannon's entropy.


M. Adil Khan, M. Anwar, J. Jakčetić, J. Pečarić, On some improvements of the Jensen inequality with some applications, J. Inequal. and Appl., Article ID 323615 (2009). DOI:

M. Adil Khan, Ð. Pečarić, J. Pečarić, New refinement of the Jensen inequality associated to certain functions with applications, J. Inequal. and Appl., Article~76 (2020). DOI:

K. M. Awan, J. Pečarić, A. Ur. Rehman, Steffensen's generalization of Čebyčev inequality, J. Math. Inequal., 9, №~1, 155–163 (2015). DOI:

M. K. Bakula, K. Nikodem, Converse Jensen inequality for strongly convex set-valued maps, J. Math. Inequal., 12, №~2, 545–550 (2018). DOI:

S. Bernstein, Sur la defnition et les properties des functions analytiques d'une variable reelle, Math. Ann., 75, 449–468 (1914). DOI:

I. Budimir, J. Pečarić, The Jensen–Grüss inequality, Math. Inequal. Appl., 5, №~2, 205–214 (2002). DOI:

S. I. Butt, T. Rasheed, Ð. Pečarić, J. Pečarić, Measure theoretic generalizations of Jensen's inequality by Fink's identity, Miskolc Math. Notes, 23, №~1, 131–154 (2022); DOI: 10.18514/MMN.2022.3656. DOI:

G. Grüss, Über das Maximum des absoluten Betrages von $dfrac{1}{b-a}displaystyleintnolimits_a^bf(x)g(x)dxduplicate-

dfrac{1}{b-a}displaystyleintnolimits_a^bf(x)dxdfrac{1}{b-a}displaystyleintnolimits_a^bg(x)dx$, Math. Z., 39, 215–226 (1934). DOI:

S. Khan, M. Adil Khan, Yu. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Methods Appl. Sci., 43, №~5, 2577–2587 (2020). DOI:

S. Khan, M. A. Khan, S. I. Butt, Y. Chu, A new bound for the Jensen gap pertaining twice differentiable functions with applications, Adv. Difference Equations (2020); DOI: 10.1186/s13662-020-02794-8. DOI:

N. Mehmood, S. I. Butt, Ð. Pečarić, J. Pečarić, Generalizations of cyclic refinements of Jensen's inequality by Lidstone's polynomial with applications in information theory, J. Math. Inequal., 14, №~1, 249–271 (2020). DOI:

D. S. Mitrinović, J. Pečarić, A. M. Fink, Classical and new inequalities in analysis, Kluwer Acad. Publ., Boston, London (1993). DOI:

G. V. Milovanović, I. Ž. Milovanović, On generalization of certain results of A.~Ostrwoski and A.~Lupas, Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. Appl., №~634–677, 62–69 (1979).

E. Landau, Über einige Ungleichungen von Herrn G. Grüss, Math. Z., 39, 742–744 (1935). DOI:

E. Landau, Über mehrfach monotone Folgen, Pr. Mat. Fiz., 44, 337–351 (1936).

G. H. Hardy, A note on two inequalities, J., London Math. Soc., 11, 167–170 (1936). DOI:

N. Latif, Ð. Pečarić, J. Pečarić, Majorization, "useful" Csiszar divergence and "useful" Zipf–Mandelbrot law, Open Math., 16, 1357–1373 (2018). DOI:

J. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical applications, Acad. Press, New York (1992).

J. Pečarić, J. Perić, New improvement of the converse Jensen inequality, Math. Inequal. Appl., 21, №~1, 217–234 (2018). DOI:

J. Pečarić, I. Perić, A multidimensional generalization of the Lupas–Ostrowski inequality, Acta Sci. Math. (Szeged), 72, 65–72 (2006).

M. Sababheh, Improved Jensen's inequality, Math. Inequal. Appl., 20, №~2, 389–403 (2017). DOI:

C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27, №~3, 379–423 (1948). DOI:

How to Cite
Butt, S. I., Ð. Pečarić, and J. Pečarić. “Several Jensen–Grüss Inequalities With Applications in Information Theory”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 12, Jan. 2023, pp. 1654 -72, doi:10.37863/umzh.v74i12.6554.
Research articles