Inequality (mathematics)

Not to be confused with Inequation."Less than" and "Greater than" redirect here. For the use of the "<" and ">" signs as punctuation, see Bracket."More than" redirects here. For the UK insurance brand, see RSA Insurance Group.
In mathematics, an inequality is a statement how the relative size or order of two objects, or about whether they are the same or not (See also: equality).
 The notation a < b means that a is less than b.
 The notation a > b means that a is greater than b.
 The notation a ≠ b means that a is not equal to b, but does not say that one is greater than the other or even that they can be compared in size.
In each statement above, 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 statements that are not strict:
 The notation a ≤ b means that a is less than or equal to b (or, equivalently, not greater than b)
 The notation a ≥ b means that a is greater than or equal to b (or, equivalently, not less than b)
An additional use of the notation is to show 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, it denotes instead absolute continuity.)
 The notation a ≫ b means that a is much greater than b.
If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality.
Contents
Properties
Inequalities are governed by the following properties. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding nonstrict inequality sign (≤ and ≥).
Transitivity
The transitivity of inequalities 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 a > b and b = c; then a > c
 If a < b and b = c; then a < c'^{[1]}
Addition and subtraction
The properties that deal with addition and subtraction state:
 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
Multiplication and division
The properties that deal with multiplication and division state:
 For any real numbers, a, b, and nonzero c
 If c is positive and a < b, then ac < bc and a/c < b/c
 If c is negative and a < b, then ac > bc and a/c > b/c
More generally this applies for an ordered field, see below.
Additive inverse
The properties for the additive inverse state:
 For any real numbers a and b
 If a < b then −a > −b
 If a > b then −a < −b
Multiplicative inverse
The properties for the multiplicative inverse state:
 For any nonzero 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 either a or b is negative (but not both) then
 If a < b then 1/a < 1/b
 If a > b then 1/a > 1/b
Applying a function to both sides
Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function.
For a nonstrict inequality (a ≤ b, a ≥ b):
 Applying a monotonically increasing function preserves the relation (≤ remains ≤, ≥ remains ≥)
 Applying a monotonically decreasing function reverses the relation (≤ becomes ≥, ≥ becomes ≤)
As an example, consider the application of the natural logarithm to both sides of an inequality:
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 nonstrict inequalities ≤ and ≥ on real numbers are total orders. The strict inequalities < and > on real numbers are strict total orders.
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. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. 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, a_{1} ≤ a_{2} ≤ ... ≤ 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.
Inequalities between means
There are many inequalities between means. For example, for any positive numbers a_{1}, a_{2}, …, a_{n} we have H ≤ G ≤ A ≤ Q, where

(harmonic mean), (geometric mean), (arithmetic mean), (quadratic mean).
Power inequalities
Sometimes with notation "power inequality" understand inequalities that contain a^{b} type expressions where a and b are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations.
Examples
 If x > 0, then
 If x > 0, then
 If x, y, z > 0, then
 For any real distinct numbers a and b,
 If x, y > 0 and 0 < p < 1, then
 If x, y, z > 0, then
 If a, b > 0, then

 This inequality was solved by I.Ilani in JSTOR,AMM,Vol.97,No.1,1990.
 If a, b > 0, then

 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 2,130137,2011.
 If a, b > 0, then

 This result was generalized by R. Ozols in 2002 who proved that if a_{1}, ..., a_{n} > 0, then
 (result is published in Latvian popularscientific quarterly The Starry Sky, see references).
Wellknown inequalities
See also list of 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:
 Azuma's inequality
 Bernoulli's inequality
 Boole's inequality
 Cauchy–Schwarz inequality
 Chebyshev's inequality
 Chernoff's inequality
 Cramér–Rao inequality
 Hoeffding's inequality
 Hölder's inequality
 Inequality of arithmetic and geometric means
 Jensen's inequality
 Kolmogorov's inequality
 Markov's inequality
 Minkowski inequality
 Nesbitt's inequality
 Pedoe's inequality
 Poincaré inequality
 Triangle inequality
Complex numbers and inequalities
The set of complex numbers with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that becomes an ordered field. To make 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 ≤ a^{2}; 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 ≤ 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 vector. If we let the vectors (meaning that and where x_{i} and y_{i} are real numbers for ), we can define the following relationships.
 if for
 if for
 if for and
 if for
Similarly, we can define relationships for x > y, , and . We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
We observe that the property of Trichotomy (as stated above) is not valid for vector relationships. We consider the case where and . 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.
See also
 Binary relation
 Bracket for the use of the < and > signs as brackets
 FourierMotzkin elimination
 Inclusion (set theory)
 Inequation
 Interval (mathematics)
 List of inequalities
 Partially ordered set
 Relational operators, used in programming languages to denote inequality
References
 ^ [5,16]
 Hardy, G., Littlewood J.E., Polya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0521052068.
 Beckenbach, E.F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0394015592.
 Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. SpringerVerlag. ISBN 0387984046.
 Murray S. Klamkin. ""Quickie" inequalities" (PDF). Math Strategies. http://uamirror.pims.math.ca/pi/issue7/page2629.pdf.
 Arthur Lohwater (1982). "Introduction to Inequalities". Online ebook in PDF format. http://www.mediafire.com/?1mw1tkgozzu.
 Harold Shapiro (2005,1972–1985). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan. http://www.math.kth.se/math/TOPS/index.html.
 "3rd USAMO". Archived from the original on 20080203. http://web.archive.org/web/20080203070350/www.kalva.demon.co.uk/usa/usa74.html.
 Ehrgott, Matthias (2005). Multicriteria Optimization. SpringerBerlin. ISBN 3540213988.
 Steele, J. Michael (2004). The CauchySchwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 9780521546775. http://wwwstat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html.
External links
 Graph of Inequalities by Ed Pegg, Jr., Wolfram Demonstrations Project.
Categories: Inequalities
 Elementary algebra
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