Liapounov’s inequality follows from Holder’s inequality (2.22) by replacing Y with 1:
E(|X|) < (E(|X |p))1/p, where p > 1.
2.6.3. Minkowski’s Inequality
If for some p > 1, E[|X|p] < to and E [| Y|p] < to, then
E(|X + Y|) < (E(|X |p))1/p + (E(| Y |p))1/p. (2.23)
This inequality is due to Minkowski. For p = 1 the result is trivial. Therefore, let p > 1. First note that E[|X + Y|p] < E[(2 ■ max(|X|, |Y|))p] = 2pE[max(|X|p, |Y|p)] < 2pE[|X|p + |Y|p] < to; hence, we may apply Liapounov’s inequality:
E(|X + Y|) < (E(|X + Y|p))1/p. (2.24)
Next, observe that
+ E (| X + Y | p-1|Y |). (2.25)
Let q = p/(p — 1). Because 1/q + 1/p = 1 it follows from Holder’s inequality that
< (E(|X + Y|p))1—1/p(E(|X|p))1/p, (2.26)
If we combine (2.24)-(2.26), Minkowski’s inequality (2.23) follows.
2.6.4. Jensen’s Inequality
A real function y(x) on К is called convex if, for all a, b є К and 0 < X < 1,
y(Xa + (1 — X)b) < Xp(a) + (1 — X)y(b).
It follows by induction that, then also,
53 Xj aj) — 53 Xj v(aj),
j=i / j=i
where Xj > 0 for j = 1,…,n, and ^Xj = 1.
Consequently, it follows from (2.28) that, for a simple random variable X,
p(E(X)) — E(p(X)) for all convex real functions p on R. (2.29)
This is Jensen’s inequality. Because (2.29) holds for simple random variables, it holds for all random variables. Similarly, we have
p(E(X)) > E(p(X)) for all concave real functions p on R.