Young's inequality for products

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.

The area of the rectangle a,b can't be larger than sum of the areas under the functions (red) and (yellow)

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

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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  

Proof[2]

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.

Proof[3]

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:

Proof

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

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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.

Proof[4]

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

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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

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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

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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

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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

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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

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Notes

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  1. ^ 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
  2. ^ Pearse, Erin. "Math 209D - Real Analysis Summer Preparatory Seminar Lecture Notes" (PDF). Retrieved 17 September 2022.
  3. ^ Bahouri, Chemin & Danchin 2011.
  4. ^ a b Jarchow 1981, pp. 47–55.
  5. ^ Tisdell, Chris (2013), The Peter Paul Inequality, YouTube video on Dr Chris Tisdell's YouTube channel,
  6. ^ 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.
  7. ^ 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
  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
  9. ^ Mitroi, F. C., & Niculescu, C. P. (2011). An extension of Young's inequality. In Abstract and Applied Analysis (Vol. 2011). Hindawi.

References

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