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.