Inequalities | Research & Encyclopedia Articles

Inequalities

Inequalities are among the most important technical tools in mathematics. The concept of an inequality is at least as old as the concept of number, and yet it is only in relatively modern times that inequalities have been studied in a systematic way. Inequalities often occur implicitly in the statement that a function is convex or that a function of two variables defines a metric.

To begin with we will consider inequalities that involve finite sets of real numbers. Let x_1, x_2, ... , x_N and y_1, y_2, ... , y_N be two sets of real numbers. Then Cauchy's inequality asserts that

and Minkowski's inequality is

Both of these inequalities have important geometrical interpretations. Suppose that x is a vector in ℜ^N with coordinates x₁, x₂, ... , x_N and y is a vector in ℜ^N is coordinates y₁, y₂, ... , y_N. Then the Euclidean norm of x is defined by

and the norm of y is defined in a similar manner. The inner product of x and y is written ⟨x, y⟩ and defined by

We recall that two vectors x and y in ℜ^N are orthogonal if ⟨x, y⟩ = 0. Using these concepts Cauchy's inequality can be expressed as

|⟨x, y⟩| ∥x∥₂ ∥y∥₂,

and Minkowski's inequality becomes the triangle inequality

∥x + y∥₂ ∥x∥₂ + ∥y∥₂

for the Euclidean norm. There are also inequalities in which the inner product ⟨x, y⟩ is replaced by some other bilinear form in x and y. One of the most important of these is Hilbert's inequality. One form of Hilbert's inequality states that

The constant which occurs in this inequality is the smallest possible constant, independent of N, for which the inequality holds for all vectors x and y.

Both Cauchy's inequality and Minkowski's inequality can be generalized by introducing the p-norms, where 1 p . If 1 p < and x is a vector in ℜ^N we define

and in case p = we set

∥x∥ = max{|x₁|, |x₂|, ... , |x_N|}.

Of course the Euclidean norm is the special case p = 2. If 1 p we define q in the range 1 q by p^-1 + q^-1 = 1. Here we use the convention that if p = 1 then q = , and if p = then q = 1. Then Hölder's inequality states that

|⟨x, y⟩| ∥x∥_p ∥y∥_q,

and the general form of Minkowski's inequality for the p norm is

∥x + y∥_p ∥x∥_p + ∥y∥_p.

There is also an important inequality comparing these norms. In order to state this inequality we assume that 0 < r < s < , but we make no further assumptions about r and s. Then if x is a vector in ℜ^N we have

Usually an inequality involving a finite sum has an analogous, but more general, statement in which the finite sum is replaced by a suitable infinite series or by an integral. Suppose, for example, that 1 < p < , 1 < q < , p^-1 + q^-1 = 1, and both

Then it follows from Hölder's inequality that the infinite series converges absolutely and satisfies

Let (u,v) ⊆ ℜ be an open interval, and here we include the possibility of an infinite interval. A function ϕ:(u,v) ℜ is said to be convex on the interval (u,v) if it satisfies the inequality

ϕ(x₁ + (1 - )x₂) ϕ(x₁) + (1 - )ϕ(x₂)

for all real numbers x₁ and x₂ in the interval (u,v), and all real numbers with 0 < < 1. The graph of a convex function has a simple geometrical property: if ϕ is convex then the straight line segment connecting two distinct points on the graph of y = ϕ(x) always lies above the graph of y = ϕ(x). If ϕ: (u,v) has a second derivative at each point of the interval (u,v), then ϕ is convex on the interval if and only if ϕ'(x) 0 at each point x in (u,v). Using this criterion it is easy to see that the exponential function exp(x) (also written e^x) is convex on ℜ. Therefore we get the inequality

exp(x₁ + (1 - )x₂) exp(x₁) + (1 - ) exp(x₂)

for all real numbers x₁ and x₂, and all with 0 < < 1. Alternatively, if we write y₁ = exp(x₁) and y₂ = exp(x₂) we obtain the inequality

for all positive real numbers y₁ and y₂, and all with 0 < < 1. This is one form of the arithmetic mean-geometric mean inequality. A more common form of the inequality, which also follows from the fact that the exponential function is convex, is

which holds for all finite sets of nonnegative numbers y₁, y₂, ... ,y_N. The left hand side of the inequality is often called the geometric mean of the nonnegative numbers y₁, y₂, ... ,y_N, while the right hand side is the arithmetic mean of the numbers y₁, y_2, ... ,y_N.

Let A = (a_mn) be an N x N matrix with real entries. As usual we suppose that m = 1, 2, ... , N indexes rows and n = 1, 2, ... , N indexes columns. Hadamard's inequality provides an upper bound for the determinant of the matrix A in terms of the Euclidean norms of the columns (or rows) of the matrix. The precise statement of Hadamard's inequality is

An alternative formulation of the inequality can be given by writing a₁, a₂, ... , a_N for the N columns of the matrix A. Then the Euclidean norm of the vector a_n in ℜ^N is given by

In terms of these norms Hadamard's inequality can be written as

Now consider the parallelepiped in ℜ^N having the vectors a₁, a₂, ... , a_N as its edges. The volume of this parallelepiped is exactly |det A|. And then ∥a_n∥₂ is the length of the edge determined by the vector a_n. Thus we see that Hadamard's inequality has an important geometrical interpretation: the volume of a parallelepiped is less than or equal to the product of the lengths of its edges. If the vectors a₁, a₂, ... , a_N are orthogonal then the parallelepiped is rectangular and its volume is equal to the product of the lengths of its edges. This shows that there is equality in Hadamard's inequality in case the vectors a₁, a₂, ... , a_N are orthogonal.

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