Young's inequality (product)

General form of Young's inequality: The green bordered rectangle cannot be larger than the sum of the yellow and red areas.

When Young's inequality - named after William Henry Young - are in mathematics may refer to inequalities. This article describes three inequalities named after Young that are closely related to each other. The second and third inequations listed here are each a special case of the previous one. All three versions make it possible to estimate a product against a sum.

1. In its general form , the inequality has a simple and easily understandable geometric meaning.
2. A special case that is used, for example, to prove Hölder's inequality , is of practical importance . This special case is also an important generalization of the inequality between the geometric and the arithmetic mean .
3. For concrete estimates, for example in connection with partial differential equations , one often needs a scaled special form .

statement

General form

Let be a continuous, strictly monotonically increasing and unlimited function with , and be its (thus existing) inverse function , which has the same properties. ${\displaystyle f\colon \mathbb {R} _{\geq 0}\to \mathbb {R} _{\geq 0}}$ ${\displaystyle f(0)=0}$ ${\displaystyle f^{-1}}$

Then applies to all : ${\displaystyle a,b\geq 0}$

${\displaystyle ab\leq \int _{0}^{a}f(x)\,{\rm {d}}x+\int _{0}^{b}f^{-1}(y)\,{\rm {d}}y}$.

The equality applies if and only if is. ${\displaystyle f(a)=b}$

Special case

If with and , then the following applies: ${\displaystyle p,q>1}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle a,b\geq 0}$

${\displaystyle ab\leq {\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}}$

with equality if and only if . ${\displaystyle a^{p}=b^{q}}$

You get this from the general case by betting. The inverse function is then . ${\displaystyle f(x)=x^{p-1}}$ ${\displaystyle f^{-1}(y)=y^{q-1}}$

On the other hand, this inequality is obtained as an application of the inequality of the weighted arithmetic and geometric mean for the two summands and and the weights and . ${\displaystyle a^{p}}$ ${\displaystyle b^{q}}$ ${\displaystyle {\tfrac {1}{p}}}$ ${\displaystyle {\tfrac {1}{q}}}$

The special case can also be derived directly (see evidence archive ).

Scaled version of the special case

For everyone with : ${\displaystyle x,y\in \mathbb {R} ,\;\varepsilon >0,\;p,q>1}$ ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}$

${\displaystyle |xy|\leq \varepsilon |x|^{p}+{\frac {(p\varepsilon )^{1-q}}{q}}|y|^{q}.}$

This is obtained from the previous special case for and . ${\displaystyle a:=(\varepsilon p)^{\frac {1}{p}}|x|}$ ${\displaystyle b:=(\varepsilon p)^{-{\frac {1}{p}}}|y|}$

literature

• R. Cooper: Notes on certain inequalities I , J. London Math. Soc. 2 , 17-21 (1927)
• WH Young: On classes of summable functions and their Fourier series , Proc. Roy. Soc. (A) 87 , 225-229 (1912).
• Alfred Witkowski: On Young's inequality (PDF; 104 kB). In: Journal of Inequalities in Pure and Applied Mathematics Vol. 7, No. 5, November 2006

Web links

Young's Inequality. Archived from the original on March 22, 2009 ; Retrieved July 29, 2015 . 

• Proof of Young's inequality in the evidence archive