Generalized inequalities of the Mercer type for strongly convex functions

A generalization of the Mercer type inequality, for strongly convex functions with modulus $c>0$, is hereby established. Let $\mathfrak{h}:[\delta,\zeta] \rightarrow \mathbb{R}$ be a strongly convex function on the interval $[\delta,\zeta] \subset \mathbb{R}$. Let ${\bf a}=(a_1,….,a_s)$, ${\bf b}=(b_1,….,b_s)$ and ${\bf p}=(p_1,….,p_s)$, where $a_k, b_k \in [\delta,\zeta], p_k >0$ for each $k=\overline{1,s}$. If ${\bf n}\in{\mathbb{R}}^s$, $\langle {\bf a}-{\bf b}, {\bf n}\rangle=0$ and under some separability assumptions, then we prove that $$\sum_{l=1}^s p_l\mathfrak{h}(b_{l}) \leq \sum_{l=1}^s p_l\mathfrak{h}(a_l)-c\sum_{l=1}^sp_l(a_l-b_{l})^2.$$ Using the above result, we derive loads of inequalities for similarly separable vectors. We further applied our results to different types of tuples. Our results extend, complement and generalize known results in the literature.