Strongly Convex Sequences

Abstract

Let ω≥0 be a given number and I a subinterval of ℤ. We say that a sequence (f _k)_k∈I is ω-midconvex if

$$f_k \leq \frac{f_{k-1}+f_{k+1}}{2}-\omega \quad \mbox{for }k-1, k, k+1 \in I. $$

We give various characterizations of ω-midconvex sequences.

We also show that in a natural way one can derive from the above definition classical notions of convexity and strong convexity for functions defined on subintervals of ℝ.