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...

We present a couple of general inequalities related to the Jensen-Steffensen inequality in its discrete and integral form. The Jensen-Steffensen inequality, Slater’s inequality and a generalization of the counterpart to the Jensen-Steffensen inequality are deduced as special cases from these general inequalities.

Let $\Omega \subset {ℝ}^n$, $n\ge 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in W^1{L}^{\Phi }\left(\Omega \right),-div\left({\Phi }^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\text{in}\Omega ,\frac{\partial u}{\partial 𝐧}=0\text{on}\partial \Omega ,$$ where $\Phi$ is a Young function such that the space $W^1{L}^{\Phi }\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...