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Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for a superquadratic functions. For superquadratic functions which are not convex we get inequalities analog to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function ϕ is y≥ϕ(x)+(1/(λ(β)- λ(α)))∫_{; ; ; ; ; ; α}; ; ; ; ; ; ^{; ; ; ; ; ; β}; ; ; ; ; ; ϕ(|f(t)-x|)dλ(t), where x= (1/(λ(β)-λ(α)))∫_{; ; ; ; ; ; α}; ; ; ; ; ; ^{; ; ; ; ; ; β}; ; ; ; ; ; f(t)dλ(t) and y= (1/(λ(β)-λ(α)))∫_{; ; ; ; ; ; α}; ; ; ; ; ; ^{; ; ; ; ; ; β}; ; ; ; ; ; ϕ(f(t))dλ(t), which under suitable conditions like those satisfied by functions of power equal or more than 2, is a refinement of the Jensen- Steffensen-Boas inequality. We also prove related results of Mercer's type.

jensen's inequality; jensen-steffensen inequality; jensen-mercer inequality; jensen-boas inequality; convex functions; superquadratic functions

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