A function is said to have a lower bound if for all in its domain. The greatest lower bound is called the infimum.
A function is said to have a upper bound if for all in its domain. The least upper bound is called the supremum. A set is said to be bounded from above if it has an upper bound.
Let be a real continuous monotonic strictly increasing function on the interval with and , thenwhere is the inverse function. Equality holds iff .
If is a monotonically increasing integrable function on with , then if is a real function integrable on ,
Let be a real-valued, continuous, and strictly increasing function on with . If , , and , then(1)where is the inverse function of . Equality holds iff .Taking the particular function gives the special case(2)which is often written in the symmetric form(3)where , , and(4)
A necessary and sufficient condition that should be comparable with for all positive values of the is that one of () and () should be majorized by the other. If , thenwith equality only when () and () are identical or when all the are equal. See Hardy et al. (1988) for a definition of notation.
The conjecture that, for any triangle,(1)where , , and are the vertex angles of the triangle and is the Brocard angle. The Abi-Khuzam inequality states that(2)(Yff 1963, Le Lionnais 1983, Abi-Khuzam and Boghossian 1989), which can be used to prove the conjecture (Abi-Khuzam 1974).The maximum value of occurs when two angles are equal, so taking , and using , the maximum occurs at the maximum of(3)which occurs when(4)Solving numerically gives (OEIS A133844), corresponding to a maximum value of approximately 0.440053 (OEIS A133845).
If , then Minkowski's integral inequality states thatSimilarly, if and , , then Minkowski's sum inequality states thatEquality holds iff the sequences , , ... and , , ... are proportional.
Let be a matrix and and vectors. Then the systemhas no solution iff the systemhas a solution, where is a vector (Fang and Puthenpura 1993, p. 60). This lemma is used in the proof of the Kuhn-Tucker theorem.
If has period , is , and(1)then(2)unless(3)(Hardy et al. 1988).Another inequality attributed to Wirtinger involves the Kähler form, which in can be written(4)Given vectors in , let denote the oriented -dimensional parallelepiped and its -dimensional volume. Then(5)with equality iff the vectors span a -dimensional complex subspace of , and they are positively oriented. Here, is the th exterior power for , and the orientation of a complex subspace is determined by its complex structure.
A function satisfies the Lipschitz condition of order at iffor all , where and are independent of , , and is an upper bound for all for which a finite exists.
The cylindrical parts of a system of real algebraic equations and inequalities in variables are the terms(1)(2)(3)(4)where '' is one of , , or , and and are or algebraic expressions in variables that are real-valued for all -tuples of real numbers satisfying(5)(6)(7)(8)The conjunction of a finite number of disjoint cylindrical parts is called a cylindrical algebraic decomposition.
A quantity is said to be less than if is smaller than , written . If is less than or equal to , the relationship is written . In the Wolfram Language, this is denoted Less[a, b], or a < b.If is much less than , this is written . Statements involving greater than and less than symbols are called inequalities.
Define a cell in as an open interval or a point. A cell in then has one of two forms,(1)or(2)where , is a cell in , and are either (1) continuous functions on such that for some polynomials and , and , or (2) , and for all .A cylindrical algebraic decomposition of is a representation of as a finite union of disjoint cells. Let be finite set of polynomials in variables. A cylindrical algebraic decomposition of is said to be -invariant if each of the polynomials from has a constant sign on each cell of the decomposition.The cylindrical algebraic decomposition (CAD) algorithm, given a finite set of polynomials in variables, computes an -invariant cylindrical algebraic decomposition of . Given a logical combination of polynomial equations and inequalities in real unknowns, one can use the CAD algorithm to find a cylindrical algebraic decomposition of its solution set. For example, the decomposition of(3)is given by(4)The command CylindricalDecomposition[ineqs,..
Suppose are given positive numbers. Let , ..., and . Then(1)where(2)(3)are the arithmetic and geometric mean, respectively, of the first and last numbers. The Kantorovich inequality is central to the study of convergence properties of descent methods in optimization (Luenberger 1984).
Let be a nonnegative sequence and a nonnegative integrable function. Define(1)(2)and(3)(4)and take . For integrals,(5)(unless is identically 0). For sums,(6)(unless all ).
For a set of positive , , 1, 2..., Turán's inequalities are given byfor , 2, ....
If , ..., are positive numbers which sum to 1 and is a real continuous function that is convex, then(1)If is concave, then the inequality reverses, giving(2)The special case of equal with the concave function gives(3)which can be exponentiated to give the arithmeticmean-geometric mean inequality(4)Here, equality holds iff .
where , , ..., are nonnegative integrable functions on which are all either monotonic increasing or monotonic decreasing.
Let and be vectors. Then the triangle inequality is given by(1)Equivalently, for complex numbers and ,(2)Geometrically, the right-hand part of the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.A generalization is(3)
Given a convex plane region with area and perimeter , thenwhere is the number of enclosed lattice points.
Apply Markov's inequality with to obtain(1)Therefore, if a random variable has a finite mean and finite variance , then for all ,(2)(3)
Let a triangle have angles , , and , then inequalities include (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(Siddons and Hughes 1929, p. 283), and(13)(Weisstein).
The isoperimetric quotient of a closed curve is defined as the ratio of the curve area to the area of a circle () with same perimeter () as the curve,(1)(2)(3)where is the area of the plane figure and is its perimeter. The isoperimetric inequality gives , with equality only in the case of the circle.For a regular -gon with inradius , the area is given by(4)edge length by(5)and the perimeter is given by(6)Thus,(7)which converges to 1 for .The isoperimetric quotient can similarly be defined for a polyhedron, where it is defined as the dimensionless quantity obtained using the volume () and surface area () of the sphere as a reference,(8)(9)(10)
A special case of Hölder's sum inequality with ,(1)where equality holds for . The inequality is sometimes also called Lagrange's inequality (Mitrinović 1970, p. 42), and can be written in vector form as(2)In two-dimensions, it becomes(3)It can be proven by writing(4)If is a constant , then . If it is not a constant, then all terms cannot simultaneously vanish for real , so the solution is complex and can be found using the quadratic equation(5)In order for this to be complex, it must be truethat(6)with equality when is a constant. The vector derivation is much simpler,(7)where(8)and similarly for .
Let a plane figure have area and perimeter . Thenwhere is known as the isoperimetric quotient. The equation becomes an equality only for a circle.
Let be a set of positive numbers. Then(which is given incorrectly in Gradshteyn and Ryzhik 2000). Here, the constant e is the best possible, in the sense that counterexamples can be constructed for any stricter inequality which uses a smaller constant. The theorem is suggested by writing in Hardy's inequalityand letting .
A mathematical statement that one quantity is greater than or less than another. " is less than " is denoted , and " is greater than " is denoted . " is less than or equal to " is denoted , and " is greater than or equal to " is denoted . The symbols and are used to denote " is much less than " and " is much greater than ," respectively.Solutions to the inequality consist of the set , or equivalently .Solutions to the inequality consist of the set , or equivalently . If and are both positive or both negative and , then .The portions of the -plane satisfying a number of specific inequalities are illustrated above. Inequalities in two dimensions can be plotted using RegionPlot[ineqs, x, xmin, xmax, y, ymin, ymax].Similarly, the portions of three-space satisfying a number of specific inequalities in the three Cartesian coordinates are illustrated above. Inequalities in three dimensions can..
If, in the Gershgorin circle theorem for a given ,for all , then exactly one eigenvalue of lies in the disk .
Let(1)with , . Then Hölder's inequality for integrals states that(2)with equality when(3)If , this inequality becomes Schwarz's inequality.Similarly, Hölder's inequality for sums states that(4)with equality when(5)If , this becomes Cauchy's inequality.
If and is nonincreasing on the interval [0, 1], then for all possible values of and ,
A function satisfies the Hölder condition on two points and on an arc whenwith and positive real constants.In some literature, functions satisfying the Hölder condition are sometimes said to be (locally) -Hölder continuous; moreover, and are sometimes called the Hölder exponent and Hölder constant of , respectively.The Hölder condition comes up frequently in several branches of mathematics, notable among which is the study of Brownian motion in probability.
The Bernoulli inequality states(1)where is a real number and an integer.This inequality can be proven by taking a Maclaurin series of ,(2)Since the series terminates after a finite number of terms for integral , the Bernoulli inequality for is obtained by truncating after the first-order term.When , slightly more finesse is needed. In this case, let so that , and take(3)Since each power of multiplies by a number and since the absolute value of the coefficient of each subsequent term is smaller than the last, it follows that the sum of the third order and subsequent terms is a positive number. Therefore,(4)or(5)completing the proof of the inequality over all rangesof parameters.For , the following generalizations of Bernoulli inequality are valid for real exponents:(6)and(7)(Mitrinović 1970)...
Let be a nonnegative and monotonic decreasing function in and such that in , thenwhere
Let be an inner product space and let . Hlawka's inequality states thatwhere the norm denotes the norm induced by the inner product.
If is continuous on a closed interval , then there is at least one number in such thatThe average value of the function on this interval is then given by .
Given a positive sequence ,(1)where the s are real and "square summable."Another inequality known as Hilbert's applies to nonnegative sequences and ,(2)unless all or all are 0. If and are nonnegative integrable functions, then the integral form is(3)The constant is the best possible, in the sense that counterexamples can be constructed for any smaller value.
Extend Hilbert's inequality by letting and(1)so that(2)Levin (1937) and Stečkin (1949) showed that(3)and(4)Mitrinovic et al. (1991) indicate that this constant is the best possible.
Let be a nonnegative sequence and a nonnegative integrable function. Define(1)and(2)and take . For sums,(3)(unless all ), and for integrals,(4)(unless is identically 0).
Let be an arbitrary nonsingular matrix with real elements and determinant , then
A quantity is said to be greater than if is larger than , written . If is greater than or equal to , the relationship is written . In the Wolfram Language, this is denoted Greater[a, b], or a > b.If is much greater than , this is written . Statements involving greater than and less than symbols are called inequalities.
Let and be sequences with for , 2, ..., thenwhere