Exponentially convex functions generated by Wulbert’s inequality and Stolarsky‐type means

Let $-\infty <a<b<\infty$ . If $f$ is concave on $[a,b]$ and ${\psi }^{\prime }$ is convex on the interval of integration, then Wulbert proved that $\frac{1}{{\delta }_{+}-{\delta }_{-}}{\int }_{{\delta }_{-}}^{{\delta }_{+}}\psi \left(u\right)du\ge \frac{1}{b-a}{\int }_a^b\psi \left(f\left(x\right)\right)dx\text{,}$ where ${\delta }_{-}=\overline{f}-\sqrt{3}{({\Vert f\Vert }_2^2-{\left(\overline{f}\right)}^2)}^{1/2}$ , ${\delta }_{+}=\overline{f}+\sqrt{3}{({\Vert f\Vert }_2^2-{\left(\overline{f}\right)}^2)}^{1/2}$ , $\overline{f}=\frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx$ and ${\Vert f\Vert }_p={\left(\frac{1}{b-a}{\int }_a^b{\left|f\left(x\right)\right|}^{p}dx\right)}^{1/p}$ . We define new Cauchy type means using a functional defined via above inequality and give some related results as applications.