Young's inequality for products
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In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers.[1] The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
Young's inequality for products can be used to prove Hölder's inequality. It is also widely used to estimate the norm of nonlinear terms in PDE theory, since it allows one to estimate a product of two terms by a sum of the same terms raised to a power and scaled.
Standard version for conjugate Hölder exponents
The standard form of the inequality is the following, which can be used to prove Hölder's inequality.
Theorem — If and are nonnegative real numbers and if and are real numbers such that then
Equality holds if and only if
Since A graph on the -plane is thus also a graph From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines and the fact that is always increasing for increasing and vice versa, we can see that upper bounds the area of the rectangle below the curve (with equality when ) and upper bounds the area of the rectangle above the curve (with equality when ). Thus, with equality when (or equivalently, ). Young's inequality follows from evaluating the integrals. (See below for a generalization.)
A second proof is via Jensen's inequality.
The claim is certainly true if or so henceforth assume that and Put and Because the logarithm function is concave, with the equality holding if and only if Young's inequality follows by exponentiating.
Yet another proof is to first prove it with an then apply the resulting inequality to . The proof below illustrates also why Hölder conjugate exponent is the only possible parameter that makes Young's inequality hold for all non-negative values. The details follow:
Let and . The inequality holds if and only if (and hence ). This can be shown by convexity arguments or by simply minimizing the single-variable function.
To prove full Young's inequality, clearly we assume that and . Now, we apply the inequality above to to obtain: It is easy to see that choosing and multiplying both sides by yields Young's inequality.
Young's inequality may equivalently be written as
Where this is just the concavity of the logarithm function. Equality holds if and only if or This also follows from the weighted AM-GM inequality.
Generalizations
Theorem[4] — Suppose and If and are such that then
Using and replacing with and with results in the inequality: which is useful for proving Hölder's inequality.
Define a real-valued function on the positive real numbers by for every and then calculate its minimum.
Theorem — If with then Equality holds if and only if all the s with non-zero s are equal.
Elementary case
An elementary case of Young's inequality is the inequality with exponent which also gives rise to the so-called Young's inequality with (valid for every ), sometimes called the Peter–Paul inequality. [5] This name refers to the fact that tighter control of the second term is achieved at the cost of losing some control of the first term – one must "rob Peter to pay Paul"
Proof: Young's inequality with exponent is the special case However, it has a more elementary proof.
Start by observing that the square of every real number is zero or positive. Therefore, for every pair of real numbers and we can write: Work out the square of the right hand side: Add to both sides: Divide both sides by 2 and we have Young's inequality with exponent
Young's inequality with follows by substituting and as below into Young's inequality with exponent
Matricial generalization
T. Ando proved a generalization of Young's inequality for complex matrices ordered by Loewner ordering.[6] It states that for any pair of complex matrices of order there exists a unitary matrix such that where denotes the conjugate transpose of the matrix and
Standard version for increasing functions
For the standard version[7][8] of the inequality, let denote a real-valued, continuous and strictly increasing function on with and Let denote the inverse function of Then, for all and with equality if and only if
With and this reduces to standard version for conjugate Hölder exponents.
For details and generalizations we refer to the paper of Mitroi & Niculescu.[9]
Generalization using Fenchel–Legendre transforms
By denoting the convex conjugate of a real function by we obtain This follows immediately from the definition of the convex conjugate. For a convex function this also follows from the Legendre transformation.
More generally, if is defined on a real vector space and its convex conjugate is denoted by (and is defined on the dual space ), then where is the dual pairing.
Examples
The convex conjugate of is with such that and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case.
The Legendre transform of is , hence for all non-negative and This estimate is useful in large deviations theory under exponential moment conditions, because appears in the definition of relative entropy, which is the rate function in Sanov's theorem.
See also
- Convex conjugate – Generalization of the Legendre transformation
- Integral of inverse functions – Mathematical theorem, used in calculus
- Legendre transformation – Mathematical transformation
- Young's convolution inequality – Mathematical inequality about the convolution of two functions
Notes
- ^ Young, W. H. (1912), "On classes of summable functions and their Fourier series", Proceedings of the Royal Society A, 87 (594): 225–229, Bibcode:1912RSPSA..87..225Y, doi:10.1098/rspa.1912.0076, JFM 43.1114.12, JSTOR 93236
- ^ Pearse, Erin. "Math 209D - Real Analysis Summer Preparatory Seminar Lecture Notes" (PDF). Retrieved 17 September 2022.
- ^ Bahouri, Chemin & Danchin 2011.
- ^ a b Jarchow 1981, pp. 47–55.
- ^ Tisdell, Chris (2013), The Peter Paul Inequality, YouTube video on Dr Chris Tisdell's YouTube channel,
- ^ T. Ando (1995). "Matrix Young Inequalities". In Huijsmans, C. B.; Kaashoek, M. A.; Luxemburg, W. A. J.; et al. (eds.). Operator Theory in Function Spaces and Banach Lattices. Springer. pp. 33–38. ISBN 978-3-0348-9076-2.
- ^ Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) [1934], Inequalities, Cambridge Mathematical Library (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-05206-8, MR 0046395, Zbl 0047.05302, Chapter 4.8
- ^ Henstock, Ralph (1988), Lectures on the Theory of Integration, Series in Real Analysis Volume I, Singapore, New Jersey: World Scientific, ISBN 9971-5-0450-2, MR 0963249, Zbl 0668.28001, Theorem 2.9
- ^ Mitroi, F. C., & Niculescu, C. P. (2011). An extension of Young's inequality. In Abstract and Applied Analysis (Vol. 2011). Hindawi.
References
- Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
- Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128.
External links
- Young's Inequality at PlanetMath
- Weisstein, Eric W. "Young's Inequality". MathWorld.
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