Gruber P. Convex and Discrete Geometry

1 Convex Functions of One Variable 13
...it should be emphasized that the theory of convexity..., taken together with a few
elementary devices, can be used to derive a large number of the most familiar and
important inequalities of analysis.
Perhaps the most important elementary inequality dealing with convex functions is
the inequality of Jensen [544], proved earlier by Hölder [519] for differentiable convex
functions.
In this section it is proved by means of a simple convexity argument. As corollaries
we obtain a series of classical inequalities. Let I be an interval in R.
For a thorough treatment of Jensen’s inequality and its consequences, see Kuczma
[620] and Castillo and Ruiz Cobo [196]. See also Roberts and Varberg [841] and
Roberts [840] for an overview of inequalities in the context of convex functions. A
beautiful book on inequalities is Steele [953].
Jensen’s Inequality
Jensen’s inequality is as follows:
Theorem 1.9. Let f : I → R be convex, x1,...,xn ∈ I , and λ1,...,λn ≥ 0 such
that λ1 +···+λn = 1. Then λ1x1 +···+λnxn ∈ I and
We give two proofs.
f (λ1x1 +···+λnxn) ≤ λ1 f (x1) +···+λn f (xn).
Proof (by Induction). For n = 1 the assertion is trivial. Assume now that n > 1 and
that the assertion holds for n − 1. We have to prove it for n. Ifλn = 0 the assertion
reduces to the case n − 1 and thus holds by the induction assumption. If λn = 1, then
λ1 = ··· = λn−1 = 0 and the assertion is true trivially. It remains to consider the
case 0