Advanced High-School Mathematics

156 CHAPTER 3 Inequalities

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Next, recall from elementary differential calculus that a twice-differentiable
function f is concave down on an interval [a,b] if f′′(x) ≤ 0 for all
x∈[a,b]. Geometrically, this means that ifa ≤ c ≤d ≤ bthen the
convex combination of the pointsP =P(c,f(c)) andQ= Q(d,f(d))
lies on or below the graph ofy=f(x). Put more explicitly, this says
that whena≤c≤d≤b, and when 0≤t≤1,

f((1−t)c+td)≥(1−t)f(c) +tf(d).

Lemma 1. (Jensen’s Inequality) Assume that the twice-differentiable
functionf is concave down on the interval[a,b]and assume that
x 1 , x 2 , ..., xn ∈[a,b]. If t 1 , t 2 , ...,tn are non-negative real numbers
witht 1 +t 2 +···+tn= 1, then

f(t 1 x 1 +t 2 x 2 +···+tnxn)≥t 1 f(x 1 ) +t 2 f(x 2 ) +···+tnf(xn).

Proof. We shall argue by induction onnwith the result already being