144 3 Real Analysis
Solution.A bounded convex function on( 0 ,∞)has a horizontal asymptote, so its deriva-
tive tends to zero at infinity. Our problem is the discrete version of this result. The first
derivative of the sequence isbn=an+ 1 −an,n≥1. The convexity condition can be
written asan+ 1 −an≥an−an− 1 , which shows that(bn)nis increasing. Since(an)nis
bounded,(bn)nis bounded too, and being monotonic, by the Weierstrass theorem it con-
verges at a finite limitL.IfL>0, thenbneventually becomes positive, soanbecomes
increasing because it has a positive derivative. Again by the Weierstrass theorem,an
converges to some limitl, and thenL=l−l=0, a contradiction. A similar argument
rules out the caseL<0. We are left with the only possibilityL=0.
And now some problems.425.Letx 1 ,x 2 ,...,xnbe real numbers. Find the real numbersathat minimize the
expression
|a−x 1 |+|a−x 2 |+···+|a−xn|.
426.Leta, b >0 andx, c >1. Prove that
xa
c
+xb
c
≥ 2 x(ab)
c/ 2
.427.A triangle has side lengthsa ≥ b ≥cand vertices of measuresA, B,andC,
respectively. Prove that
Ab+Bc+Ca≥Ac+Ba+Cb.
428.Show that if a functionf:[a, b]→Ris convex, then it is continuous on(a, b).
429.Prove that a continuous function defined on a convex domain (for example, on an
interval of the real axis) is convex if and only if
f(
x+y
2)
≤
f(x)+f(y)
2, for allx, y∈D.430.Call a real-valued functionvery convexif
f(x)+f(y)
2≥f(
x+y
2)
+|x−y|holds for all real numbersxandy. Prove that no very convex function exists.431.Letf:[a, b]→Rbe a convex function. Prove that
f(x)+f(y)+f(z)+ 3 f(
x+y+z
3)
≥ 2
[
f(
x+y
2)
+f(
y+z
2)
+f(
z+x
2)]
,
for allx, y, z∈[a, b].