Page 56 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Convex analysis 43

1.5.1 Exercises
Exercise 1.5.1 Prove Theorem 1.50 when n =1.

1
Exercise 1.5.2 Prove Jensen inequality, when f ∈ C .
√
2
Exercise 1.5.3 Let f (x)= 1+ x .Compute f .
∗
Exercise 1.5.4 Establish (i), (ii) and (iv) of Example 1.53.

Exercise 1.5.5 Prove (i), (iii) and (iv) of Theorem 1.54. For proofs of (ii) and
(v) see the bibliography in the corrections of the present exercise and the exercise
below.

Exercise 1.5.6 Show (v) of Theorem 1.54 under the further restrictions that
2
n =1, f ∈ C (R) and
f (x) > 0, ∀x ∈ R .
00
2
Prove in addition that f ∈ C (R).
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1
Exercise 1.5.7 Let f ∈ C (R) be convex, p ≥ 1, α 1 > 0 and
p
|f (x)| ≤ α 1 (1 + |x| ) , ∀x ∈ R . (1.14)
Show that there exist α 2 ,α 3 > 0,sothat
³ ´
p−1
0
|f (x)| ≤ α 2 1+ |x| , ∀x ∈ R (1.15)
³ ´
p−1 p−1
|f (x) − f (y)| ≤ α 3 1+ |x| + |y| |x − y| , ∀x, y ∈ R . (1.16)
Note that (1.15) always implies (1.14) independently of the convexity of f.