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We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous 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 $$\overline{y}⩾\phi \left(\overline{x}\right)+\frac{1}{\lambda \left(\beta \right)-\lambda \left(\alpha \right)}{\int }_{\alpha }^{\beta }\phi \left(\left|f\left(t\right)-\overline{x}\right|\right)d\lambda \left(t\right)$$ , where $$\overline{x}=\frac{1}{\lambda \left(\beta \right)-\lambda \left(\alpha \right)}{\int }_{\alpha }^{\beta }f\left(t\right)d\lambda \left(t\right)$$ and $$\overline{y}=\frac{1}{\lambda \left(\beta \right)-\lambda \left(\alpha \right)}{\int }_{\alpha }^{\beta }\phi \left(f\left(t\right)\right)d\lambda \left(t\right)$$ which under...

This paper gives extensions and improvements of Sherman’s inequality for n-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.