Inequality (mathematics)

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).

The notation a ≠ b means that a is not equal to b.

It does not say that one is greater than the other, or even that they can be compared in size.

If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size.

The notation a < b means that a is less than b.
The notation a > b means that a is greater than b.

In either case, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".

In contrast to strict inequalities, there are two types of inequality relations that are not strict:

The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b, or at most b); "not greater than" can also be represented by the symbol for "greater than" bisected by a vertical line, "not." (The unicode for ≤ is "U+ 2264".)
The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b, or at least b),; "not less than" can also be represented by the symbol for "less than" bisected by a vertical line, "not." (The unicode for ≥ is "U+ 2265".)

In engineering sciences, a less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude.

The notation a ≪ b means that a is much less than b. (In measure theory, however, this notation is used for absolute continuity, an unrelated concept.)
The notation a ≫ b means that a is much greater than b.

Properties

Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and (in the case of applying a function) monotonic functions are limited to strictly monotonic functions.

Transitivity

The transitive property of inequality states:

For any real numbers a, b, c:
- If a ≥ b and b ≥ c, then a ≥ c.
- If a ≤ b and b ≤ c, then a ≤ c.
If either of the premises is a strict inequality, then the conclusion is a strict inequality:
- If a ≥ b and b > c, then a > c
- If a > b and b ≥ c, then a > c
Since a = b implies a ≥ b these imply:
- If a = b and b > c, then a > c
- If a > b and b = c, then a > c

Converse

The relations ≤ and ≥ are each other's converse:

For any real numbers a and b:
- If a ≤ b, then b ≥ a.
- If a ≥ b, then b ≤ a.

Addition and subtraction

If x < y, then x + a < y + a.

A common constant c may be added to or subtracted from both sides of an inequality:

For any real numbers a, b, c
- If a ≤ b, then a + c ≤ b + c and a − c ≤ b − c.
- If a ≥ b, then a + c ≥ b + c and a − c ≥ b − c.

i.e., the real numbers are an ordered group under addition.

Multiplication and division

If x < y and a > 0, then ax < ay.

If x < y and a < 0, then ax > ay.

The properties that deal with multiplication and division state:

For any real numbers, a, b and non-zero c:
- If c is positive, then multiplying or dividing by c does not change the inequality:
  - If a ≥ b and c > 0, then ac ≥ bc and a/c ≥ b/c.
  - If a ≤ b and c > 0, then ac ≤ bc and a/c ≤ b/c.
- If c is negative, then multiplying or dividing by c inverts the inequality:
  - If a ≥ b and c < 0, then ac ≤ bc and a/c ≤ b/c.
  - If a ≤ b and c < 0, then ac ≥ bc and a/c ≥ b/c.

More generally, this applies for an ordered field; see #Ordered fields.

Additive inverse

The properties for the additive inverse state:

For any real numbers a and b, negation inverts the inequality:
- If a ≤ b, then −a ≥ −b.
- If a ≥ b, then −a ≤ −b.

Multiplicative inverse

The properties for the multiplicative inverse state:

For any non-zero real numbers a and b that are both positive or both negative:
- If a ≤ b, then 1/a ≥ 1/b.
- If a ≥ b, then 1/a ≤ 1/b.
If one of a and b is positive and the other is negative, then:
- If a < b, then 1/a < 1/b.
- If a > b, then 1/a > 1/b.

These can also be written in chained notation as:

For any non-zero real numbers a and b:
- If 0 < a ≤ b, then 1/a ≥ 1/b > 0.
- If a ≤ b < 0, then 0 > 1/a ≥ 1/b.
- If a < 0 < b, then 1/a < 0 < 1/b.
- If 0 > a ≥ b, then 1/a ≤ 1/b < 0.
- If a ≥ b > 0, then 0 < 1/a ≤ 1/b.
- If a > 0 > b, then 1/a > 0 > 1/b.

Applying a function to both sides

The graph of y = ln x

Any monotonically increasing function may be applied to both sides of an inequality (provided they are in the domain of that function) and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.

If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. The rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.

A few examples of this rule are:

Exponentiating both sides of an inequality by n > 0 when a and b are positive real numbers:

a ≤ b ⇔ aⁿ ≤ bⁿ.

a ≤ b ⇔ a⁻ⁿ ≥ b⁻ⁿ.

Taking the natural logarithm to both sides of an inequality when a and b are positive real numbers:

a ≤ b ⇔ ln(a) ≤ ln(b).

a < b ⇔ ln(a) < ln(b).

This is true because the natural logarithm is a strictly increasing function.

Ordered fields

If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:

a ≤ b implies a + c ≤ b + c;
0 ≤ a and 0 ≤ b implies 0 ≤ a × b.

Note that both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of i and would therefore be positive.

The non-strict inequalities ≤ and ≥ on real numbers are total orders. That is, given arbitrary a,b∈R, at least one of a≤b and b≤a holds; at the same time, at least one of a≥b and b≥a holds. The strict inequalities < and > on real numbers are strict total orders. That is, < on R has trichotomy property: given arbitrary a,b∈R, exactly one of a<b, b<a and a=b is true; likewise, > on R has the trichotomy property.

Chained notation

The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.

This notation can be generalized to any number of terms: for instance, a₁ ≤ a₂ ≤ ... ≤ a_n means that a_i ≤ a_i+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to a_i ≤ a_j for any 1 ≤ i ≤ j ≤ n.

When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1/2.

Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python.

Sharp inequalities

An inequality is said to be sharp, if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀a ∈ ℝ. a² ≥ 0 is sharp, whereas the inequality ∀a ∈ ℝ. a² ≥ −1 is not sharp.

Inequalities between means

There are many inequalities between means. For example, for any positive numbers a₁, a₂, …, a_n we have H ≤ G ≤ A ≤ Q, where

$H={\frac {n}{1/a_{1}+1/a_{2}+\cdots +1/a_{n}}}$	(harmonic mean),
$G={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}$	(geometric mean),
$A={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}$	(arithmetic mean),
$Q={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}{n}}}$	(quadratic mean).

Power inequalities

A "power inequality" is an inequality containing terms of the form a^b, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.

Examples

For any real x,

e^{x}\geq 1+x.

If x > 0 and p > 0, then

(x^{p}-1)/p\geq \ln(x)\geq (1-{1}/{x^{p}})/p.

In the limit of p → 0, the upper and lower bounds converge to ln(x).

If x > 0, then

x^{x}\geq \left({\frac {1}{e}}\right)^{1/e}.

If x ≥ 1, then

x^{x^{x}}\geq x.

If x, y, z > 0, then

(x+y)^{z}+(x+z)^{y}+(y+z)^{x}>2.

For any real distinct numbers a and b,

{\frac {e^{b}-e^{a}}{b-a}}>e^{(a+b)/2}.

If x, y > 0 and 0 < p < 1, then

x^{p}+y^{p}>(x+y)^{p}.

If x, y, z > 0, then

x^{x}y^{y}z^{z}\geq (xyz)^{(x+y+z)/3}.

If a, b > 0, then

a^{a}+b^{b}\geq a^{b}+b^{a}.

This inequality was solved by I.Ilani in JSTOR,AMM,Vol.97,No.1,1990.

If a, b > 0, then

a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.

This inequality was solved by S.Manyama in AJMAA,Vol.7,Issue 2,No.1,2010 and by V.Cirtoaje in JNSA, Vol.4, Issue 2, 130–137, 2011.

If a, b, c > 0, then

a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.

If a, b > 0, then

a^{b}+b^{a}>1.

Well-known inequalities

Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:

Complex numbers and inequalities

The set of complex numbers $\mathbb {C}$ with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that $(\mathbb {C} ,+,\times ,\leq )$ becomes an ordered field. To make $(\mathbb {C} ,+,\times ,\leq )$ an ordered field, it would have to satisfy the following two properties:

if a ≤ b then a + c ≤ b + c
if 0 ≤ a and 0 ≤ b then 0 ≤ a b

Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a²; this means that $i^{2}>0$ and $1^{2}>0$ ; so $-1>0$ and $1>0$ , which means $(-1+1)>0$ ; contradiction.

However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:

$a \le b$ if $\mathrm {Re} (a)<\mathrm {Re} (b)$ or $\left(\mathrm {Re} (a)=\mathrm {Re} (b)\right.$ and $\left.\mathrm {Im} (a)\leq \mathrm {Im} (b)\right)$

It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.

Vector inequalities

Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors $x,y\in \mathbb {R} ^{n}$ (meaning that $x=\left(x_{1},x_{2},\ldots ,x_{n}\right)^{\mathsf {T}}$ and $y=\left(y_{1},y_{2},\ldots ,y_{n}\right)^{\mathsf {T}}$ where $x_{i}$ and $y_{i}$ are real numbers for $i=1,\ldots ,n$ ), we can define the following relationships.

$x=y$ if $x_{i}=y_{i}$ for $i=1,\ldots ,n$
$x<y$ if $x_{i}<y_{i}$ for $i=1,\ldots ,n$
$x\leq y$ if $x_{i}\leq y_{i}$ for $i=1,\ldots ,n$ and $x\neq y$
$x\leqq y$ if $x_{i}\leq y_{i}$ for $i=1,\ldots ,n$

Similarly, we can define relationships for $x>y$ , $x\geq y$ , and $x\geqq y$ . We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).

The trichotomy property (as stated above) is not valid for vector relationships. For example, when $x=\left[2,5\right]^{\mathsf {T}}$ and $y=\left[3,4\right]^{\mathsf {T}}$ , there exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.

General existence theorems

For a general system of polynomial inequalities, one can find a condition for a solution to exist. Firstly, any system of polynomial inequalities can be reduced to a system of quadratic inequalities by increasing the number of variables and equations (for example by setting a square of a variable equal to a new variable). A single quadratic polynomial inequality in n − 1 variables can be written as:

X^{T}AX\geq 0

where X is a vector of the variables $X=(x,y,z,\ldots ,1)^{T}$ and A is a matrix. This has a solution, for example, when there is at least one positive element on the main diagonal of A.

Systems of inequalities can be written in terms of matrices A, B, C, etc. and the conditions for existence of solutions can be written as complicated expressions in terms of these matrices. The solution for two polynomial inequalities in two variables tells us whether two conic section regions overlap or are inside each other. The general solution is not known but such a solution could be theoretically used to solve such unsolved problems as the kissing number problem. However, the conditions would be so complicated as to require a great deal of computing time or clever algorithms.

References

Hardy, G., Littlewood J.E., Pólya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.
Beckenbach, E.F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.
Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.
Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications", Advances in Applied Mathematics, 34 (1): 71–100, doi:10.1016/j.aam.2004.05.001
Murray S. Klamkin. "'Quickie' inequalities" (PDF). Math Strategies.
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
Harold Shapiro (2005, 1972–1985). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan. Check date values in: |date= (help)
"3rd USAMO". Archived from the original on 2008-02-03.
Pachpatte, B.G. (2005). Mathematical Inequalities. North-Holland Mathematical Library. 67 (first ed.). Amsterdam, The Netherlands: Elsevier. ISBN 0-444-51795-2. ISSN 0924-6509. MR 2147066. Zbl 1091.26008.
Ehrgott, Matthias (2005). Multicriteria Optimization. Springer-Berlin. ISBN 3-540-21398-8.
Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 978-0-521-54677-5.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Graph of Inequalities by Ed Pegg, Jr., Wolfram Demonstrations Project.
AoPS Wiki entry about Inequalities

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