What are integral inequalities?
Integral inequalities are often a very important tool in mathematical analysis, number theory, partial differential equations, differential geometry, probability, statistics, etc. The most basic integral inequality is given by the following: Given a continuous function (Riemann integrable is sufficient) , we have. .
How do you prove a Holder’s inequality?
Proof of Hölder’s inequality
- If ||f ||p = 0, then f is zero μ-almost everywhere, and the product fg is zero μ-almost everywhere, hence the left-hand side of Hölder’s inequality is zero.
- If ||f ||p = ∞ or ||g||q = ∞, then the right-hand side of Hölder’s inequality is infinite.
What are the properties of integrals?
Definite Integrals Properties
| Properties | Description |
|---|---|
| Property 1 | p∫q f(a) da = p∫q f(t) dt |
| Property 2 | p∫q f(a) d(a) = – q∫p f(a) d(a), Also p∫p f(a) d(a) = 0 |
| Property 3 | p∫q f(a) d(a) = p∫r f(a) d(a) + r∫q f(a) d(a) |
| Property 4 | p∫q f(a) d(a) = p∫q f( p + q – a) d(a) |
What are definite integrals used for?
Definite integrals can be used to find the area under, over, or between curves. If a function is strictly positive, the area between it and the x axis is simply the definite integral. If it is simply negative, the area is -1 times the definite integral.
Does integral preserve inequality?
Integration is linear, additive, and preserves inequality of functions.
What is power mean inequality?
The Power Mean Inequality states that for all real numbers and , if . In particular, for nonzero and. , and equal weights (i.e. ), if , then. Considering the limiting behavior, we also have , and .
What is the significance of gronwall Bellman inequality?
In mathematics, Grönwall’s inequality (also called Grönwall’s lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation.
What are the rules of integration?
Integration Rules
| Common Functions | Function | Integral |
|---|---|---|
| Power Rule (n≠−1) | ∫xn dx | xn+1n+1 + C |
| Sum Rule | ∫(f + g) dx | ∫f dx + ∫g dx |
| Difference Rule | ∫(f – g) dx | ∫f dx – ∫g dx |
| Integration by Parts | See Integration by Parts |