Inequality (mathematics)

This article is about relations "greater than" and "less than". For the relation "not equal", seeInequation.

"Less than" and "Greater than" redirect here. For the symbol <, seeless-than sign.For the symbol >, seegreater-than sign.For the song, seeLess Than (song).

"≪" redirects here. Not to be confused withAbsolute continuity of measures.

Thefeasible regionsoflinear programmingare defined by a set of inequalities.

Inmathematics,aninequalityis a relation which makes a non-equal comparison between two numbers or other mathematical expressions.It is used most often to compare two numbers on thenumber lineby their size. The main types of inequality areless thanandgreater than.

Notation

There are several different notations used to represent different kinds of inequalities:

• The notationa<bmeans thataisless thanb.
• The notationa>bmeans thataisgreater thanb.

In either case,ais not equal tob.These relations are known asstrict inequalities,meaning thatais strictly less than or strictly greater thanb.Equality is excluded.

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

• The notationa≤bora⩽bora≦bmeans thataisless than or equal tob(or, equivalently, at mostb,or not greater thanb).
• The notationa≥bora⩾bora≧bmeans thataisgreater than or equal tob(or, equivalently, at leastb,or not less thanb).

In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities.For example, In 1670,John Wallisused a single horizontal baraboverather than below the < and >. Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or "less than (greater than) or equal to with double horizontal bars", first appeared inPierre Bouguer's work.After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽).

The relationnot greater thancan also be represented by$a\ngtr b,$the symbol for "greater than" bisected by a slash, "not". The same is true fornot less than,$a\nless b.$

The notationa≠bmeans thatais not equal tob;thisinequationsometimes is considered a form of strict inequality.It does not say that one is greater than the other; it does not even requireaandbto be member of anordered set.

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

• The notationa≪bmeans thataismuch less thanb.
• The notationa≫bmeans thataismuch greater thanb.

This implies that the lesser value can be neglected with little effect on the accuracy of anapproximation(such as the case ofultrarelativistic limitin physics).

In all of the cases above, any two symbols mirroring each other are symmetrical;a<bandb>aare equivalent, etc.

Properties on the number line

Inequalities are governed by the followingproperties.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 tostrictlymonotonic functions.

Converse

The relations ≤ and ≥ are each other'sconverse,meaning that for anyreal numbersaandb:

a≤bandb≥aare equivalent.

Transitivity

The transitive property of inequality states that for anyreal numbersa,b,c:

Ifa≤bandb≤c,thena≤c.

Ifeitherof the premises is a strict inequality, then the conclusion is a strict inequality:

Ifa≤bandb<c,thena<c.

Ifa<bandb≤c,thena<c.

Addition and subtraction

Ifx<y,thenx+a<y+a.

A common constantcmay beaddedto orsubtractedfrom both sides of an inequality.So, for anyreal numbersa,b,c:

Ifa≤b,thena+c≤b+canda−c≤b−c.

In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are anordered groupunder addition.

Multiplication and division

Ifx<yanda> 0, thenax<ay.

Ifx<yanda< 0, thenax>ay.

The properties that deal withmultiplicationanddivisionstate that for any real numbers,a,band non-zeroc:

Ifa≤bandc> 0, thenac≤bcanda/c≤b/c.

Ifa≤bandc< 0, thenac≥bcanda/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 anordered field.For more information, see§ Ordered fields.

Additive inverse

The property for theadditive inversestates that for any real numbersaandb:

Ifa≤b,then −a≥ −b.

Multiplicative inverse

If both numbers are positive, then the inequality relation between themultiplicative inversesis opposite of that between the original numbers. More specifically, for any non-zero real numbersaandbthat are bothpositive(or bothnegative):

Ifa≤b,then⁠1/a⁠≥⁠1/b⁠.

All of the cases for the signs ofaandbcan also be written inchained notation,as follows:

If 0 <a≤b,then⁠1/a⁠≥⁠1/b⁠> 0.

Ifa≤b< 0, then 0 >⁠1/a⁠≥⁠1/b⁠.

Ifa< 0 <b,then⁠1/a⁠< 0 <⁠1/b⁠.

Applying a function to both sides

The graph ofy= lnx

Anymonotonicallyincreasingfunction,by its definition,may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in thedomainof 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)andthe 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 astrictlymonotonically decreasing function.

A few examples of this rule are:

• Raising both sides of an inequality to a powern> 0 (equiv., −n< 0), whenaandbare positive real numbers:
  0 ≤a≤b⇔ 0 ≤aⁿ≤bⁿ.
  0 ≤a≤b⇔a⁻ⁿ≥b⁻ⁿ≥ 0.
• Taking thenatural logarithmon both sides of an inequality, whenaandbare 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 orderis abinary relation≤ over asetPwhich isreflexive,antisymmetric,andtransitive.That is, for alla,b,andcinP,it must satisfy the three following clauses:

• a≤a(reflexivity)
• ifa≤bandb≤a,thena=b(antisymmetry)
• ifa≤bandb≤c,thena≤c(transitivity)

A set with a partial order is called apartially ordered set.Those are the very basic axioms that every kind of order has to satisfy.

A strict partial order is a relation < that satisfies:

• a<⃥͏a(irreflexivity)
• ifa<b,thenb<⃥͏a(asymmetry)
• ifa<bandb<c,thena<c(transitivity)

Some types of partial orders are specified by adding further axioms, such as:

• Total order:For everyaandbinP,a≤borb≤a.
• Dense order:For allaandbinPfor whicha<b,there is acinPsuch thata<c<b.
• Least-upper-bound property:Every non-emptysubsetofPwith anupper boundhas aleastupper bound(supremum) inP.

Ordered fields

Main article:Ordered field

If (F,+, ×) is afieldand ≤ is atotal orderonF,then (F,+, ×, ≤) is called anordered fieldif and only if:

• a≤bimpliesa+c≤b+c;
• 0 ≤aand 0 ≤bimplies 0 ≤a×b.

Both⁠$(\mathbb {Q},+,\times,\leq )$⁠and⁠$(\mathbb {R},+,\times,\leq )$⁠areordered fields,but≤cannot be defined in order to make⁠$(\mathbb {C},+,\times,\leq )$⁠anordered field,because −1 is the square ofiand would therefore be positive.

Besides being an ordered field,Ralso has theLeast-upper-bound property.In fact,Rcan be defined as the only ordered field with that quality.

Chained notation

The notationa<b<cstands for "a<bandb<c",from which, by the transitivity property above, it also follows thata<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<cis equivalent toa−e<b<c−e.

This notation can be generalized to any number of terms: for instance,a₁≤a₂≤... ≤a_nmeans thata_i≤a_i+1fori= 1, 2,...,n− 1. By transitivity, this condition is equivalent toa_i≤a_jfor 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 isolatexin any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yieldingx<⁠1/2⁠andx≥ −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 thelogical conjunctionof the inequalities between adjacent terms. For example, the defining condition of azigzag posetis written asa₁<a₂>a₃<a₄>a₅<a₆>.... Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance,a<b=c≤dmeans thata<b,b=c,andc≤d.This notation exists in a fewprogramming languagessuch asPython.In contrast, in programming languages that provide an ordering on the type of comparison results, such asC,even homogeneous chains may have a completely different meaning.

Sharp inequalities

An inequality is said to besharpif it cannot berelaxedand still be valid in general. Formally, auniversally quantifiedinequalityφis called sharp if, for every valid universally quantified inequalityψ,ifψ⇒φholds, thenψ⇔φalso holds. For instance, the inequality∀a∈R.a²≥ 0is sharp, whereas the inequality∀a∈R.a²≥ −1is not sharp.

Inequalities between means

See also:Inequality of arithmetic and geometric means

There are many inequalities between means. For example, for any positive numbersa₁,a₂,...,a_nwe haveH≤G≤A≤Q,where they represent the following means of the sequence:

• Harmonic mean:$H={\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}$
• Geometric mean:$G={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}$
• Arithmetic mean:$A={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}$
• Quadratic mean:$Q={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}{n}}}$

Cauchy–Schwarz inequality

See also:Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectorsuandvof aninner product spaceit is true that $$|\langle \mathbf {u},\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u},\mathbf {u} \rangle \cdot \langle \mathbf {v},\mathbf {v} \rangle,$$ where$\langle \cdot,\cdot \rangle$is theinner product.Examples of inner products include the real and complexdot product;InEuclidean spaceRⁿwith the standard inner product, the Cauchy–Schwarz inequality is $$\left(\sum _{i=1}^{n}u_{i}v_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}u_{i}^{2}\right)\left(\sum _{i=1}^{n}v_{i}^{2}\right).$$

Power inequalities

Apower inequalityis an inequality containing terms of the forma^b,whereaandbare real positive numbers or variable expressions. They often appear inmathematical olympiadsexercises.

Examples:

• For any realx,$$e^{x}\geq 1+x.$$
• Ifx> 0 andp> 0, then$${\frac {x^{p}-1}{p}}\geq \ln(x)\geq {\frac {1-{\frac {1}{x^{p}}}}{p}}.$$In the limit ofp→ 0, the upper and lower bounds converge to ln(x).
• Ifx> 0, then$$x^{x}\geq \left({\frac {1}{e}}\right)^{\frac {1}{e}}.$$
• Ifx> 0, then$$x^{x^{x}}\geq x.$$
• Ifx,y,z> 0, then$$\left(x+y\right)^{z}+\left(x+z\right)^{y}+\left(y+z\right)^{x}>2.$$
• For any real distinct numbersaandb,$${\frac {e^{b}-e^{a}}{b-a}}>e^{(a+b)/2}.$$
• Ifx,y> 0 and 0 <p< 1, then$$x^{p}+y^{p}>\left(x+y\right)^{p}.$$
• Ifx,y,z> 0, then$$x^{x}y^{y}z^{z}\geq \left(xyz\right)^{(x+y+z)/3}.$$
• Ifa,b> 0, then$$a^{a}+b^{b}\geq a^{b}+b^{a}.$$
• Ifa,b> 0, then$$a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.$$
• Ifa,b,c> 0, then$$a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.$$
• Ifa,b> 0, then$$a^{b}+b^{a}>1.$$

Well-known inequalities

See also:List of inequalities

Mathematiciansoften 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
• Bell'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
• Samuelson's inequality
• Sobolev inequality
• Triangle inequality

Complex numbers and inequalities

The set ofcomplex numbers$\mathbb {C}$with its operations ofadditionandmultiplicationis afield,but it is impossible to define any relation≤so that$(\mathbb {C},+,\times,\leq )$becomes anordered field.To make$(\mathbb {C},+,\times,\leq )$anordered field,it would have to satisfy the following two properties:

• ifa≤b,thena+c≤b+c;
• if0 ≤aand0 ≤b,then0 ≤ab.

Because ≤ is atotal order,for any numbera,either0 ≤aora≤ 0(in which case the first property above implies that0 ≤ −a). In either case0 ≤a²;this means thati²> 0and1²> 0;so−1 > 0and1 > 0,which means (−1 + 1) > 0; contradiction.

However, an operation ≤ can be defined so as to satisfy only the first property (namely, "ifa≤b,thena+c≤b+c"). Sometimes thelexicographical orderdefinition is used:

• a≤b,if
  • Re(a) < Re(b),or
  • Re(a) = Re(b)andIm(a) ≤ Im(b)

It can easily be proven that for this definitiona≤bimpliesa+c≤b+c.

Systems of inequalities

Systems oflinear inequalitiescan be simplified byFourier–Motzkin elimination.

Thecylindrical algebraic decompositionis 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 isdoubly exponentialin 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 asbrackets
• 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

Sources

• Hardy, G., Littlewood J. E., Pólya, G. (1999).Inequalities.Cambridge Mathematical Library, Cambridge University Press.ISBN0-521-05206-8.{{cite book}}:CS1 maint: multiple names: authors list (link)
• Beckenbach, E. F., Bellman, R. (1975).An Introduction to Inequalities.Random House Inc.ISBN0-394-01559-2.{{cite book}}:CS1 maint: multiple names: authors list (link)
• Drachman, Byron C., Cloud, Michael J. (1998).Inequalities: With Applications to Engineering.Springer-Verlag.ISBN0-387-98404-6.{{cite book}}: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:10.1016/j.aam.2004.05.001
• Murray S. Klamkin."'Quickie' inequalities "(PDF).Math Strategies.Archived(PDF)from the original on 2022-10-09.
• 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 fromthe originalon 2008-02-03.
• Pachpatte, B. G. (2005).Mathematical Inequalities.North-Holland Mathematical Library. Vol. 67 (first ed.). Amsterdam, the Netherlands:Elsevier.ISBN0-444-51795-2.ISSN0924-6509.MR2147066.Zbl1091.26008.
• Ehrgott, Matthias (2005).Multicriteria Optimization.Springer-Berlin.ISBN3-540-21398-8.
• Steele, J. Michael(2004).The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities.Cambridge University Press.ISBN978-0-521-54677-5.

External links

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