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§1.7 Inequalities

Contents

§1.7(i) Finite Sums

In this subsection A and B are positive constants.

Cauchy–Schwarz Inequality

1.7.1 (j=1najbj)2(j=1naj2)(j=1nbj2).

Equality holds iff aj=cbj, j; c= constant.

Conversely, if (j=1najbj)2AB for all bj such that j=1nbj2B, then j=1naj2A.

Hölder’s Inequality

For p>1, 1p+1q=1, aj0, bj0,

1.7.2 j=1najbj(j=1najp)1/p(j=1nbjq)1/q.

Equality holds iff ajp=cbjq, j; c= constant.

Conversely, if j=1najbjA1/pB1/q for all bj such that j=1nbjqB, then j=1najpA.

Minkowski’s Inequality

For p>1, aj0, bj0,

1.7.3 (j=1n(aj+bj)p)1/p(j=1najp)1/p+(j=1nbjp)1/p.

The direction of the inequality is reversed, that is, , when 0<p<1. Equality holds iff aj=cbj, j; c= constant.

§1.7(ii) Integrals

In this subsection a and b (>a) are real constants that can be , provided that the corresponding integrals converge. Also A and B are constants that are not simultaneously zero.

Cauchy–Schwarz Inequality

1.7.4 (abf(x)g(x)x)2ab(f(x))2xab(g(x))2x.

Equality holds iff Af(x)=Bg(x) for all x.

Hölder’s Inequality

For p>1, 1p+1q=1, f(x)0, g(x)0,

1.7.5 abf(x)g(x)x(ab(f(x))px)1/p(ab(g(x))qx)1/q.

Equality holds iff A(f(x))p=B(g(x))q for all x.

Minkowski’s Inequality

For p>1, f(x)0, g(x)0,

1.7.6 (ab(f(x)+g(x))px)1/p(ab(f(x))px)1/p+(ab(g(x))px)1/p.

The direction of the inequality is reversed, that is, , when 0<p<1. Equality holds iff Af(x)=Bg(x) for all x.

§1.7(iii) Means

For the notation, see §1.2(iv).

1.7.7 HGA,

with equality iff a1=a2==an.

1.7.8 min(a1,a2,,an)M(r)max(a1,a2,,an),

with equality iff a1=a2==an, or r<0 and some aj=0.

1.7.9 M(r)M(s),
r<s,

with equality iff a1=a2==an, or s0 and some aj=0.

§1.7(iv) Jensen’s Inequality

For f integrable on [0,1], a<f(x)<b, and ϕ convex on (a,b)1.4(viii)),

1.7.10 ϕ(01f(x)x)01ϕ(f(x))x,
1.7.11 exp(01ln(f(x))x)<01f(x)x.

For exp and ln see §4.2.