# INEQUALITIES FOR INTEGRALS

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A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell).

• Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that mf (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(ba) and M(ba), it follows that
• Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus
This is a generalization of the above inequalities, as M(ba) is the integral of the constant function with value M over [a, b].
In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if f(x) < g(x) for each x in [a, b], then

<img src="http://upload.wikimedia.org/math/b/a/1/ba10dd13faa9bc7cba96bcf4dcba480b.png&quot; alt=" \int_a^b f(x) \, dx
• Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then
• Products and absolute values of functions. If f and g are two functions then we may consider their pointwise products and powers, and absolute values:
If f is Riemann-integrable on [a, b] then the same is true for |f|, and

Moreover, if f and g are both Riemann-integrable then f 2, g 2, and fgare also Riemann-integrable, and

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 of two square-integrable functions f and g on the interval [a, b].
• Hölder’s inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemann-integrable functions. Then the functions |f|p and |g|q are also integrable and the following Hölder’s inequality holds:
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.” />
For p = q = 2, Hölder’s inequality becomes the Cauchy–Schwarz inequality.
• Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then |f|p, |g|p and |f + g|p are also Riemann integrable and the following Minkowski inequality holds:
<img src="http://upload.wikimedia.org/math/b/5/6/b56649a576f626f925d84a0ac045602c.png&quot; alt="\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}.” />
An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.

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Posted on September 1, 2011, in education and tagged , , , , , , , . Bookmark the permalink. 3 Comments.