In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. 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.
A second proof is via Jensen's inequality:
Yet another proof is to first prove it with b = 1 {\displaystyle b=1} and then apply the resulting inequality to a / b q {\displaystyle a/b^{q}} . 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:
Young's inequality may equivalently be written as
a α b β ≤ α a + β b , 0 ≤ α , β ≤ 1 , α + β = 1. {\displaystyle a^{\alpha }b^{\beta }\leq \alpha a+\beta b,\qquad \,0\leq \alpha ,\beta \leq 1,\quad \ \alpha +\beta =1.}
Where this is just the concavity of the logarithm function. Equality holds if and only if a = b {\displaystyle a=b} or { α , β } = { 0 , 1 } . {\displaystyle \{\alpha ,\beta \}=\{0,1\}.} This also follows from the weighted AM-GM inequality.
Generalizations
Elementary case An elementary case of Young's inequality is the inequality with exponent 2 , {\displaystyle 2,}
a b ≤ a 2 2 + b 2 2 , {\displaystyle ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}},}
which also gives rise to the so-called Young's inequality with ε {\displaystyle \varepsilon } (valid for every ε > 0 {\displaystyle \varepsilon >0} ), sometimes called the Peter–Paul inequality. 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"
a b ≤ a 2 2 ε + ε b 2 2 . {\displaystyle ab~\leq ~{\frac {a^{2}}{2\varepsilon }}+{\frac {\varepsilon b^{2}}{2}}.}
Proof: Young's inequality with exponent 2 {\displaystyle 2} is the special case p = q = 2. {\displaystyle p=q=2.} 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 a {\displaystyle a} and b {\displaystyle b} we can write:
0 ≤ ( a − b ) 2 {\displaystyle 0\leq (a-b)^{2}} Work out the square of the right hand side:
0 ≤ a 2 − 2 a b + b 2 {\displaystyle 0\leq a^{2}-2ab+b^{2}} Add 2 a b {\displaystyle 2ab} to both sides:
2 a b ≤ a 2 + b 2 {\displaystyle 2ab\leq a^{2}+b^{2}}
Divide both sides by 2 and we have Young's inequality with exponent 2 : {\displaystyle 2:}
a b ≤ a 2 2 + b 2 2 {\displaystyle ab\leq {\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}}
Young's inequality with ε {\displaystyle \varepsilon } follows by substituting a ′ {\displaystyle a'} and b ′ {\displaystyle b'} as below into Young's inequality with exponent 2 {\displaystyle 2} :
a ′ = a / ε , b ′ = ε b . {\displaystyle a'=a/{\sqrt {\varepsilon }},\;b'={\sqrt {\varepsilon }}b.}
Young's inequality with ε {\displaystyle \varepsilon } is an instance of the arithmetic mean–geometric mean inequality with two numbers.
Matricial generalization T. Ando proved a generalization of Young's inequality for complex matrices ordered by Loewner ordering. It states that for any pair A , B {\displaystyle A,B} of complex matrices of order n {\displaystyle n} there exists a unitary matrix U {\displaystyle U} such that
U ∗ | A B ∗ | U ⪯ 1 p | A | p + 1 q | B | q , {\displaystyle U^{*}|AB^{*}|U\preceq {\tfrac {1}{p}}|A|^{p}+{\tfrac {1}{q}}|B|^{q},}
where
∗ {\displaystyle {}^{*}} denotes the conjugate transpose of the matrix and | A | = A ∗ A . {\displaystyle |A|={\sqrt {A^{*}A}}.}
Standard version for increasing functions For the standard version of the inequality, let f {\displaystyle f} denote a real-valued, continuous and strictly increasing function on [ 0 , c ] {\displaystyle [0,c]} with c > 0 {\displaystyle c>0} and f ( 0 ) = 0. {\displaystyle f(0)=0.} Let f − 1 {\displaystyle f^{-1}} denote the inverse function of f . {\displaystyle f.} Then, for all a ∈ [ 0 , c ] {\displaystyle a\in [0,c]} and b ∈ [ 0 , f ( c ) ] , {\displaystyle b\in [0,f(c)],}
a b ≤ ∫ 0 a f ( x ) d x + ∫ 0 b f − 1 ( x ) d x {\displaystyle ab~\leq ~\int _{0}^{a}f(x)\,dx+\int _{0}^{b}f^{-1}(x)\,dx}
with equality if and only if b = f ( a ) . {\displaystyle b=f(a).}
With f ( x ) = x p − 1 {\displaystyle f(x)=x^{p-1}} and f − 1 ( y ) = y q − 1 , {\displaystyle f^{-1}(y)=y^{q-1},} this reduces to standard version for conjugate Hölder exponents. For details and generalizations we refer to the paper of Mitroi & Niculescu.
Generalization using Fenchel–Legendre transforms By denoting the convex conjugate of a real function f {\displaystyle f} by g , {\displaystyle g,} we obtain
a b ≤ f ( a ) + g ( b ) . {\displaystyle ab~\leq ~f(a)+g(b).}
This follows immediately from the definition of the convex conjugate. For a convex function f {\displaystyle f} this also follows from the Legendre transformation. More generally, if f {\displaystyle f} is defined on a real vector space X {\displaystyle X} and its convex conjugate is denoted by f ⋆ {\displaystyle f^{\star }} (and is defined on the dual space X ⋆ {\displaystyle X^{\star }} ), then
⟨ u , v ⟩ ≤ f ⋆ ( u ) + f ( v ) . {\displaystyle \langle u,v\rangle \leq f^{\star }(u)+f(v).}
where ⟨ ⋅ , ⋅ ⟩ : X ⋆ × X → R {\displaystyle \langle \cdot ,\cdot \rangle :X^{\star }\times X\to \mathbb {R} } is the dual pairing.
Examples The convex conjugate of f ( a ) = a p / p {\displaystyle f(a)=a^{p}/p} is g ( b ) = b q / q {\displaystyle g(b)=b^{q}/q} with q {\displaystyle q} such that 1 p + 1 q = 1 , {\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1,} and thus Young's inequality for conjugate Hölder exponents mentioned above is a special case. The Legendre transform of f ( a ) = e a − 1 {\displaystyle f(a)=e^{a}-1} is g ( b ) = 1 − b + b ln b {\displaystyle g(b)=1-b+b\ln b} , hence a b ≤ e a − b + b ln b {\displaystyle ab\leq e^{a}-b+b\ln b} for all non-negative a {\displaystyle a} and b . {\displaystyle b.} This estimate is useful in large deviations theory under exponential moment conditions, because b ln b {\displaystyle b\ln b} 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
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 for products at PlanetMath. Weisstein, Eric W. "Young's Inequality". MathWorld.