Extension of an Inequality on Three Intervals and Applications to Csiszár ϕ-Divergence and Landau–Kolmogorov Inequality

In this paper, we generalize an inequality for a convex function in one dimension R1 on three intervals to a function with nondecreasing increments in k dimensions Rk on (2n+1) intervals. We prove all the situations when n=1,2 and prove a very special case for a general n as well as the discrete version. The proofs are based on a general conclusion for convex functions, and analogues of this conclusion are established. We apply the discrete case of the inequality to Csiszár ϕ-divergence Iϕ(p,q) in information theory, and the continuous case Iϕ(p1,q1)≤Iϕ(p2,q2) on a measurable set is also established. The same inequality for an ϵ-approximately convex function on a discrete set is also established and can be used to prove a similar Landau–Kolmogorov-type inequality.