Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications
-
László Horváth
and
Josip Pečarić
Abstract
In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using this result, we obtain new refinements of the classical Jensen’s inequality. We investigate m-exponential convexity of some functionals coming from the new refinements. To apply our results we define some new mixed symmetric means, generalized means, and Cauchy means, and study their properties.
Funding statement: The research of the third author was partially supported by the Croatian Science Foundation under project 5435.
References
[1] Brnetić I., Khan K. A. and Pečarić J., Refinement of Jensen’s inequality with applications to cyclic mixed symmetric means and Cauchy means, J. Math. Inequal. 9 (2015), no. 4, 1309–1321. 10.7153/jmi-09-100Search in Google Scholar
[2] Horváth L., Inequalities corresponding to the classical Jensen’s inequality, J. Math. Inequal. 3 (2009), no. 2, 189–200. 10.7153/jmi-03-19Search in Google Scholar
[3] Horváth L., Khan K. A. and Pečarić J., Combinatorial Improvements of Jensen’s Inequality, Monogr. Inequal. 8, Element, Zagreb, 2014. Search in Google Scholar
[4] Pečarić J. and Perić J., Improvement of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform. 39 (2012), no. 1, 65–75. Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Investigations about the Euler–Mascheroni constant and harmonic numbers
- A class of subsums of Euler’s sum
- Lyapunov type inequalities for second-order differential equations with mixed nonlinearities
- Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications
- Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions
- The time-dependent Stokes problem with Navier slip boundary conditions on Lp-spaces
- Mountain pass solutions for perturbed Hardy–Sobolev equations on compact manifolds
Articles in the same Issue
- Frontmatter
- Investigations about the Euler–Mascheroni constant and harmonic numbers
- A class of subsums of Euler’s sum
- Lyapunov type inequalities for second-order differential equations with mixed nonlinearities
- Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications
- Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions
- The time-dependent Stokes problem with Navier slip boundary conditions on Lp-spaces
- Mountain pass solutions for perturbed Hardy–Sobolev equations on compact manifolds
