Sharp discrete inequalities and applications to discrete variational problems (Q843120)

The starting point is the well-known inequality \[ n\sum_{i=1}^na_i\sum_{i=1}^nb_i\geq\sum_{i=1}^na_ib_i, \] which holds for an increasing sequence \((a_n)\) and for a~decreasing sequence \((b_n)\) (cf.\ the Chebyshev functional and its discrete version). Many related inequalities involving monotonic and convex functions are proved. These results (interesting in their own right) are applied to solve some discrete variational problems.