Inequality (mathematics)

Properties on the number line

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.

Converse

The relations ≤ and ≥ are each other's converse, meaning that for any real numbers a and b:

a ≤ b and b ≥ a are equivalent.

Transitivity

The transitive property of inequality states that for any real numbers a, b, c:^[6]

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.

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.^[3] So, for any real numbers a, b, c:

If a ≤ b, then a + c ≤ b + c and a − c ≤ b − c.

In other words, the inequality relation is preserved under addition (or subtraction) and 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 that for any real numbers, a, b and non-zero c:

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.

In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered field. For more information, see § Ordered fields.

Additive inverse

The property for the additive inverse states that for any real numbers a and b:

If a ≤ b, then −a ≥ −b.

Multiplicative inverse

If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative):

If a ≤ b, then 1/a ≥ 1/b.

All of the cases for the signs of a and b can also be written in chained notation, as follows:

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, by its definition,^[7] may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. 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. In fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.

A few examples of this rule are:

• Raising both sides of an inequality to a power n > 0 (equiv., −n < 0), when a and b are positive real numbers:

0 ≤ a ≤ b ⇔ 0 ≤ aⁿ ≤ bⁿ.

0 ≤ a ≤ b ⇔ a⁻ⁿ ≥ b⁻ⁿ ≥ 0.

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

0 < a ≤ b ⇔ ln(a) ≤ ln(b).

0 < a < b ⇔ ln(a) < ln(b).

(this is true because the natural logarithm is a strictly increasing function.)

Formal definitions and generalizations

A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive.^[8] That is, for all a, b, and c in P, it must satisfy the three following clauses:

1. a ≤ a (reflexivity)
2. if a ≤ b and b ≤ a, then a = b (antisymmetry)
3. if a ≤ b and b ≤ c, then a ≤ c (transitivity)

A set with a partial order is called a partially ordered set.^[9] Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set P include:

1. For every a and b in P, a ≤ b or b ≤ a (total order).
2. For all a and b in P for which a < b, there is a c in P such that a < c < b (dense order).
3. Every non-empty subset of P with an upper bound has a least upper bound (supremum) in P (least-upper-bound property).

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.

Both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field,^[10] because −1 is the square of i and would therefore be positive.

Besides from being an ordered field, R also has the Least-upper-bound property. In fact, R can be defined as the only ordered field with that quality.^[11]

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 example, the defining condiction of a zigzag poset is written as a₁ < a₂ > a₃ < a₄ > a₅ < a₆ > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. 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. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning.^[12]

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.^{[citation needed]}

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

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

Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that

${\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space Rⁿ with the standard inner product, the Cauchy–Schwarz inequality is

${\displaystyle \left(\sum _{i=1}^{n}u_{i}v_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}u_{i}^{2}\right)\left(\sum _{i=1}^{n}v_{i}^{2}\right).}$

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,

${\displaystyle e^{x}\geq 1+x.}$

• If x > 0 and p > 0, then

${\displaystyle {\frac {x^{p}-1}{p}}\geq \ln(x)\geq {\frac {1-{\frac {1}{x^{p}}}}{p}}.}$

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

• If x > 0, then

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

• If x > 0, then

${\displaystyle x^{x^{x}}\geq x.}$

• If x, y, z > 0, then

${\displaystyle \left(x+y\right)^{z}+\left(x+z\right)^{y}+\left(y+z\right)^{x}>2.}$

• For any real distinct numbers a and b,

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

• If x, y > 0 and 0 < p < 1, then

${\displaystyle x^{p}+y^{p}>\left(x+y\right)^{p}.}$

• If x, y, z > 0, then

${\displaystyle x^{x}y^{y}z^{z}\geq \left(xyz\right)^{(x+y+z)/3}.}$

• If a, b > 0, then^[13]

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

• If a, b > 0, then^[14]

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

• If a, b, c > 0, then

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

• If a, b > 0, then

${\displaystyle a^{b}+b^{a}>1.}$

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 ℂ with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that (ℂ, +, ×, ≤) becomes an ordered field. To make (ℂ, +, ×, ≤) 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 ≤ ab.

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 i² > 0 and 1² > 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 ≤ b, if
  • Re(a) < Re(b), or
  • Re(a) = Re(b) and Im(a) ≤ Im(b)

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 ${\displaystyle x,y\in \mathbb {R} ^{n}}$ (meaning that ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})^{\mathsf {T}}}$ and ${\displaystyle y=(y_{1},y_{2},\ldots ,y_{n})^{\mathsf {T}}}$, where ${\displaystyle x_{i}}$ and ${\displaystyle y_{i}}$ are real numbers for ${\displaystyle i=1,\ldots ,n}$), we can define the following relationships:

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

Similarly, we can define relationships for ${\displaystyle x>y}$, ${\displaystyle x\geq y}$, and ${\displaystyle x\geqq y}$. 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 ${\displaystyle x=(2,5)^{\mathsf {T}}}$ and ${\displaystyle y=(3,4)^{\mathsf {T}}}$, 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.

Systems of inequalities

Systems of linear inequalities can be simplified by Fourier–Motzkin elimination.^[15]

The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.

See also

• Binary relation
• Bracket (mathematics), for the use of similar ‹ and › signs as brackets
• Inclusion (set theory)
• Inequation
• Interval (mathematics)
• List of inequalities
• List of triangle inequalities
• Partially ordered set
• Relational operators, used in programming languages to denote inequality

References

Sources

• Hardy, G., Littlewood J. E., Pólya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.CS1 maint: multiple names: authors list (link)
• Beckenbach, E. F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.CS1 maint: multiple names: authors list (link)
• Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.CS1 maint: multiple names: authors list (link)
• Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications", Advances in Applied Mathematics, 34 (1): 71–100, doi:
• Murray S. Klamkin. "'Quickie' inequalities" (PDF). Math Strategies.
• Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
• Harold Shapiro (2005). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan.
• "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

• "Inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Graph of Inequalities by Ed Pegg, Jr.
• AoPS Wiki entry about Inequalities