Editing Integral (section)

=== Inequalities ===
A number of general inequalities hold for Riemann-integrable [[Function (mathematics)|functions]] defined on a [[Closed set|closed]] and [[Bounded set|bounded]] [[Interval (mathematics)|interval]] {{closed-closed|''a'', ''b''}} and can be generalized to other notions of integral (Lebesgue and Daniell).

* ''Upper and lower bounds.'' An integrable function {{mvar|f}} on {{closed-closed|''a'', ''b''}}, is necessarily [[Bounded function|bounded]] on that interval. Thus there are [[real number]]s {{mvar|m}} and {{mvar|M}} so that {{math|''m'' ≤ ''f'' (''x'') ≤ ''M''}} for all {{mvar|x}} in {{closed-closed|''a'', ''b''}}. Since the lower and upper sums of {{mvar|f}} over {{closed-closed|''a'', ''b''}} are therefore bounded by, respectively, {{math|''m''(''b'' − ''a'')}} and {{math|''M''(''b'' − ''a'')}}, it follows that <math display="block"> m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). </math>
* ''Inequalities between functions.''<ref>{{Harvnb|Apostol|1967|p=81}}.</ref> If {{math|''f''(''x'') ≤ ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}} then each of the upper and lower sums of {{mvar|f}} is bounded above by the upper and lower sums, respectively, of {{mvar|g}}. Thus <math display="block"> \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. </math> This is a generalization of the above inequalities, as {{math|''M''(''b'' − ''a'')}} is the integral of the constant function with value {{mvar|M}} over {{closed-closed|''a'', ''b''}}. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if {{math|''f''(''x'') < ''g''(''x'')}} for each {{mvar|x}} in {{closed-closed|''a'', ''b''}}, then <math display="block"> \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. </math>
* ''Subintervals.'' If {{closed-closed|''c'', ''d''}} is a subinterval of {{closed-closed|''a'', ''b''}} and {{math|''f''&hairsp;(''x'')}} is non-negative for all {{mvar|x}}, then <math display="block"> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
* ''Products and absolute values of functions.'' If {{mvar|f}} and {{mvar|g}} are two functions, then we may consider their [[pointwise product]]s and powers, and [[absolute value]]s: <math display="block">
 (fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.</math> If {{mvar|f}} is Riemann-integrable on {{closed-closed|''a'', ''b''}} then the same is true for {{math|{{abs|''f''}}}}, and <math display="block">\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. </math> Moreover, if {{mvar|f}} and {{mvar|g}} are both Riemann-integrable then {{math|''fg''}} is also Riemann-integrable, and <math display="block">\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). </math> This inequality, known as the [[Cauchy–Schwarz inequality]], plays a prominent role in [[Hilbert space]] theory, where the left hand side is interpreted as the [[Inner product space|inner product]] of two [[Square-integrable function|square-integrable]] functions {{mvar|f}} and {{mvar|g}} on the interval {{closed-closed|''a'', ''b''}}.
* ''Hölder's inequality''.<ref name=":4">{{Harvnb|Rudin|1987|p=63}}.</ref> Suppose that {{mvar|p}} and {{mvar|q}} are two real numbers, {{math|1 ≤ ''p'', ''q'' ≤ ∞}} with {{math|1={{sfrac|1|''p''}} + {{sfrac|1|''q''}} = 1}}, and {{mvar|f}} and {{mvar|g}} are two Riemann-integrable functions. Then the functions {{math|{{abs|''f''}}<sup>''p''</sup>}} and {{math|{{abs|''g''}}<sup>''q''</sup>}} are also integrable and the following [[Hölder's inequality]] holds: <math display="block">\left|\int f(x)g(x)\,dx\right| \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.</math> For {{math|1=''p'' = ''q'' = 2}}, Hölder's inequality becomes the Cauchy–Schwarz inequality.
* ''Minkowski inequality''.<ref name=":4" /> Suppose that {{math|''p'' ≥ 1}} is a real number and {{mvar|f}} and {{mvar|g}} are Riemann-integrable functions. Then {{math|{{abs| ''f'' }}<sup>''p''</sup>, {{abs| ''g'' }}<sup>''p''</sup>}} and {{math|{{abs| ''f'' + ''g'' }}<sup>''p''</sup>}} are also Riemann-integrable and the following [[Minkowski inequality]] holds: <math display="block">\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} +
\left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.</math> An analogue of this inequality for Lebesgue integral is used in construction of [[Lp space|L<sup>p</sup> spaces]].