A minimization problem connected with a generalized Jensen inequality

On the Jensen type inequality for generalized Sugeno integral

We prove necessary and sufficient conditions for the validity of Jensen type inequalities for generalized Sugeno integral. Our proofs make no appeal to the continuity of neither the fuzzy measure nor the operators. For several choices of operators, we characterize the classes of functions for which the corresponding inequalities are satisfied.

On the Jensen-Steffensen inequality for generalized convex functions

Jensen–Steffensen type inequalities for P -convex functions and functions with nondecreasing increments are presented. The obtained results are used to prove a generalization of Čebyšev’s inequality and several variants of Hölder’s inequality with weights satisfying the conditions as in the Jensen–Steffensen inequality. A few well-known inequalities for quasi-arithmetic means are generalized.

Cauchy-Rassias Stability of linear Mappings in Banach Modules Associated with a Generalized Jensen Type Mapping

Jensen Inequality with Subdifferential for Sugeno Integral

The classical Jensen inequality for concave function φ is adapted for the Sugeno integral using the notion of the subdifferential. Some examples in the framework of the Lebesgue measure to illustrate the results are presented.

A generalized Lyapunov's inequality for a fractional boundary value problem

We prove existence of positive solutions to a nonlinear fractional boundary value problem. Then, under some mild assumptions on the nonlinear term, we obtain a smart generalization of Lyapunov’s inequality. The new results are illustrated through examples.