Homework 4 Math 8052, Spring 2009
1. Let I,Jbe intervals in R, let f:I→Jand g:J→Rbe convex functions with
gincreasing. Show that g◦fis convex.
2. Let Ibe an open interval in R,f:I→Rcontinuous. Show that fis convex iff
ZI
f(x)g00(x)dx ≥0 for all g∈C2
c(I) such that g≥0.
Here C2
c(I) is the set of C2functions on Isuch that the closure of {x:g(x)6= 0}
is a compact subset of I.
3. Supply the details of the proof of Hadamard’s Three Circles Theorem.
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