In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.
The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 {\displaystyle p=q=2} gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if ‖ f g ‖ 1 {\displaystyle \|fg\|_{1}} is infinite, the right-hand side also being infinite in that case. Conversely, if f is in L p ( μ ) {\displaystyle L^{p}(\mu )} and g is in L q ( μ ) {\displaystyle L^{q}(\mu )} , then the pointwise product f g {\displaystyle fg} is in L 1 ( μ ) {\displaystyle L^{1}(\mu )} . Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L p ( μ ) {\displaystyle L^{p}(\mu )} , and also to establish that L q ( μ ) {\displaystyle L^{q}(\mu )} is the dual space of L p ( μ ) {\displaystyle L^{p}(\mu )} for p ∈ [ 1 , ∞ ) {\displaystyle p\in [1,\infty )} . Hölder's inequality (in a slightly different form) was first found by Leonard James Rogers (1888). Inspired by Rogers' work, Hölder (1889) gave another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named for work of Johan Jensen building on Hölder's work.
Remarks
Conventions The brief statement of Hölder's inequality uses some conventions.
In the definition of Hölder conjugates, 1/∞ means zero. If p, q ∈ [1, ∞), then ‖f ‖p and ‖g‖q stand for the (possibly infinite) expressions
( ∫ S | f | p d μ ) 1 p ( ∫ S | g | q d μ ) 1 q {\displaystyle {\begin{aligned}&\left(\int _{S}|f|^{p}\,\mathrm {d} \mu \right)^{\frac {1}{p}}\\&\left(\int _{S}|g|^{q}\,\mathrm {d} \mu \right)^{\frac {1}{q}}\end{aligned}}}
If p = ∞, then ‖f ‖∞ stands for the essential supremum of |f |, similarly for ‖g‖∞. The notation ‖f ‖p with 1 ≤ p ≤ ∞ is a slight abuse, because in general it is only a norm of f if ‖f ‖p is finite and f is considered as equivalence class of μ-almost everywhere equal functions. If f ∈ Lp(μ) and g ∈ Lq(μ), then the notation is adequate. On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means 0. Multiplying a > 0 with ∞ gives ∞.
Estimates for integrable products As above, let f and g denote measurable real- or complex-valued functions defined on S. If ‖fg‖1 is finite, then the pointwise products of f with g and its complex conjugate function are μ-integrable, the estimate
| ∫ S f g ¯ d μ | ≤ ∫ S | f g | d μ = ‖ f g ‖ 1 {\displaystyle {\biggl |}\int _{S}f{\bar {g}}\,\mathrm {d} \mu {\biggr |}\leq \int _{S}|fg|\,\mathrm {d} \mu =\|fg\|_{1}}
and the similar one for fg hold, and Hölder's inequality can be applied to the right-hand side. In particular, if f and g are in the Hilbert space L2(μ), then Hölder's inequality for p = q = 2 implies
| ⟨ f , g ⟩ | ≤ ‖ f ‖ 2 ‖ g ‖ 2 , {\displaystyle |\langle f,g\rangle |\leq \|f\|_{2}\|g\|_{2},}
where the angle brackets refer to the inner product of L2(μ). This is also called Cauchy–Schwarz inequality, but requires for its statement that ‖f ‖2 and ‖g‖2 are finite to make sure that the inner product of f and g is well defined. We may recover the original inequality (for the case p = 2) by using the functions |f | and |g| in place of f and g.
Generalization for probability measures If (S, Σ, μ) is a probability space, then p, q ∈ [1, ∞] just need to satisfy 1/p + 1/q ≤ 1, rather than being Hölder conjugates. A combination of Hölder's inequality and Jensen's inequality implies that
‖ f g ‖ 1 ≤ ‖ f ‖ p ‖ g ‖ q {\displaystyle \|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}}
for all measurable real- or complex-valued functions f and g on S.
Notable special cases For the following cases assume that p and q are in the open interval (1,∞) with 1/p + 1/q = 1.
Counting measure For the n {\displaystyle n} -dimensional Euclidean space, when the set S {\displaystyle S} is { 1 , … , n } {\displaystyle \{1,\dots ,n\}} with the counting measure, we have
∑ k = 1 n | x k y k | ≤ ( ∑ k = 1 n | x k | p ) 1 p ( ∑ k = 1 n | y k | q ) 1 q for all ( x 1 , … , x n ) , ( y 1 , … , y n ) ∈ R n or C n . {\displaystyle \sum _{k=1}^{n}|x_{k}\,y_{k}|\leq \left(\sum _{k=1}^{n}|x_{k}|^{p}\right)^{\frac {1}{p}}\left(\sum _{k=1}^{n}|y_{k}|^{q}\right)^{\frac {1}{q}}{\text{ for all }}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})\in \mathbb {R} ^{n}{\text{ or }}\mathbb {C} ^{n}.}
Often the following practical form of this is used, for any ( r , s ) ∈ R + {\displaystyle (r,s)\in \mathbb {R} _{+}} :
( ∑ k = 1 n | x k | r | y k | s ) r + s ≤ ( ∑ k = 1 n | x k | r + s ) r ( ∑ k = 1 n | y k | r + s ) s . {\displaystyle \left(\sum _{k=1}^{n}|x_{k}|^{r}\,|y_{k}|^{s}\right)^{r+s}\leq \left(\sum _{k=1}^{n}|x_{k}|^{r+s}\right)^{r}\left(\sum _{k=1}^{n}|y_{k}|^{r+s}\right)^{s}.}
For more than two sums, the following generalisation (Lohwater (1982), Chen (2014)) holds, with real positive exponents λ i {\displaystyle \lambda _{i}} and λ a + λ b + ⋯ + λ z = 1 {\displaystyle \lambda _{a}+\lambda _{b}+\cdots +\lambda _{z}=1} :
∑ k = 1 n | a k | λ a | b k | λ b ⋯ | z k | λ z ≤ ( ∑ k = 1 n | a k | ) λ a ( ∑ k = 1 n | b k | ) λ b ⋯ ( ∑ k = 1 n | z k | ) λ z . {\displaystyle \sum _{k=1}^{n}|a_{k}|^{\lambda _{a}}\,|b_{k}|^{\lambda _{b}}\cdots |z_{k}|^{\lambda _{z}}\leq \left(\sum _{k=1}^{n}|a_{k}|\right)^{\lambda _{a}}\left(\sum _{k=1}^{n}|b_{k}|\right)^{\lambda _{b}}\cdots \left(\sum _{k=1}^{n}|z_{k}|\right)^{\lambda _{z}}.}
Equality holds iff | a 1 | : | a 2 | : ⋯ : | a n | = | b 1 | : | b 2 | : ⋯ : | b n | = ⋯ = | z 1 | : | z 2 | : ⋯ : | z n | {\displaystyle |a_{1}|:|a_{2}|:\cdots :|a_{n}|=|b_{1}|:|b_{2}|:\cdots :|b_{n}|=\cdots =|z_{1}|:|z_{2}|:\cdots :|z_{n}|} . If S = N {\displaystyle S=\mathbb {N} } with the counting measure, then we get Hölder's inequality for sequence spaces:
∑ k = 1 ∞ | x k y k | ≤ ( ∑ k = 1 ∞ | x k | p ) 1 p ( ∑ k = 1 ∞ | y k | q ) 1 q for all ( x k ) k ∈ N , ( y k ) k ∈ N ∈ R N or C N . {\displaystyle \sum _{k=1}^{\infty }|x_{k}\,y_{k}|\leq \left(\sum _{k=1}^{\infty }|x_{k}|^{p}\right)^{\frac {1}{p}}\left(\sum _{k=1}^{\infty }|y_{k}|^{q}\right)^{\frac {1}{q}}{\text{ for all }}(x_{k})_{k\in \mathbb {N} },(y_{k})_{k\in \mathbb {N} }\in \mathbb {R} ^{\mathbb {N} }{\text{ or }}\mathbb {C} ^{\mathbb {N} }.}
Lebesgue measure If S {\displaystyle S} is a measurable subset of R n {\displaystyle \mathbb {R} ^{n}} with the Lebesgue measure, and f {\displaystyle f} and g {\displaystyle g} are measurable real- or complex-valued functions on S {\displaystyle S} , then Hölder's inequality is
∫ S | f ( x ) g ( x ) | d x ≤ ( ∫ S | f ( x ) | p d x ) 1 p ( ∫ S | g ( x ) | q d x ) 1 q . {\displaystyle \int _{S}{\bigl |}f(x)g(x){\bigr |}\,\mathrm {d} x\leq {\biggl (}\int _{S}|f(x)|^{p}\,\mathrm {d} x{\biggr )}^{\frac {1}{p}}{\biggl (}\int _{S}|g(x)|^{q}\,\mathrm {d} x{\biggr )}^{\frac {1}{q}}.}
Probability measure For the probability space ( Ω , F , P ) , {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ),} let E {\displaystyle \mathbb {E} } denote the expectation operator. For real- or complex-valued random variables X {\displaystyle X} and Y {\displaystyle Y} on Ω , {\displaystyle \Omega ,} Hölder's inequality reads
E [ | X Y | ] ⩽ ( E [ | X | p ] ) 1 p ( E [ | Y | q ] ) 1 q . {\displaystyle \mathbb {E} [|XY|]\leqslant \left(\mathbb {E} {\bigl [}|X|^{p}{\bigr ]}\right)^{\frac {1}{p}}\left(\mathbb {E} {\bigl [}|Y|^{q}{\bigr ]}\right)^{\frac {1}{q}}.}
Let 1 < r < s < ∞ {\displaystyle 1<r<s<\infty } and define p = s r . {\displaystyle p={\tfrac {s}{r}}.} Then q = p p − 1 {\displaystyle q={\tfrac {p}{p-1}}} is the Hölder conjugate of p . {\displaystyle p.} Applying Hölder's inequality to the random variables | X | r {\displaystyle |X|^{r}} and 1 Ω {\displaystyle 1_{\Omega }} we obtain
E [ | X | r ] ⩽ ( E [ | X | s ] ) r s . {\displaystyle \mathbb {E} {\bigl [}|X|^{r}{\bigr ]}\leqslant \left(\mathbb {E} {\bigl [}|X|^{s}{\bigr ]}\right)^{\frac {r}{s}}.}
In particular, if the sth absolute moment is finite, then the r th absolute moment is finite, too. (This also follows from Jensen's inequality.)
Product measure For two σ-finite measure spaces (S1, Σ1, μ1) and (S2, Σ2, μ2) define the product measure space by
S = S 1 × S 2 , Σ = Σ 1 ⊗ Σ 2 , μ = μ 1 ⊗ μ 2 , {\displaystyle S=S_{1}\times S_{2},\quad \Sigma =\Sigma _{1}\otimes \Sigma _{2},\quad \mu =\mu _{1}\otimes \mu _{2},}
where S is the Cartesian product of S1 and S2, the σ-algebra Σ arises as product σ-algebra of Σ1 and Σ2, and μ denotes the product measure of μ1 and μ2. Then Tonelli's theorem allows us to rewrite Hölder's inequality using iterated integrals: If f and g are Σ-measurable real- or complex-valued functions on the Cartesian product S, then
∫ S 1 ∫ S 2 | f ( x , y ) g ( x , y ) | μ 2 ( d y ) μ 1 ( d x ) ≤ ( ∫ S 1 ∫ S 2 | f ( x , y ) | p μ 2 ( d y ) μ 1 ( d x ) ) 1 p ( ∫ S 1 ∫ S 2 | g ( x , y ) | q μ 2 ( d y ) μ 1 ( d x ) ) 1 q . {\displaystyle \int _{S_{1}}\int _{S_{2}}|f(x,y)\,g(x,y)|\,\mu _{2}(\mathrm {d} y)\,\mu _{1}(\mathrm {d} x)\leq \left(\int _{S_{1}}\int _{S_{2}}|f(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\,\mu _{1}(\mathrm {d} x)\right)^{\frac {1}{p}}\left(\int _{S_{1}}\int _{S_{2}}|g(x,y)|^{q}\,\mu _{2}(\mathrm {d} y)\,\mu _{1}(\mathrm {d} x)\right)^{\frac {1}{q}}.}
This can be generalized to more than two σ-finite measure spaces.
Vector-valued functions Let (S, Σ, μ) denote a σ-finite measure space and suppose that f = (f1, ..., fn) and g = (g1, ..., gn) are Σ-measurable functions on S, taking values in the n-dimensional real- or complex Euclidean space. By taking the product with the counting measure on {1, ..., n}, we can rewrite the above product measure version of Hölder's inequality in the form
∫ S ∑ k = 1 n | f k ( x ) g k ( x ) | μ ( d x ) ≤ ( ∫ S ∑ k = 1 n | f k ( x ) | p μ ( d x ) ) 1 p ( ∫ S ∑ k = 1 n | g k ( x ) | q μ ( d x ) ) 1 q . {\displaystyle \int _{S}\sum _{k=1}^{n}|f_{k}(x)\,g_{k}(x)|\,\mu (\mathrm {d} x)\leq \left(\int _{S}\sum _{k=1}^{n}|f_{k}(x)|^{p}\,\mu (\mathrm {d} x)\right)^{\frac {1}{p}}\left(\int _{S}\sum _{k=1}^{n}|g_{k}(x)|^{q}\,\mu (\mathrm {d} x)\right)^{\frac {1}{q}}.}
If the two integrals on the right-hand side are finite, then equality holds if and only if there exist real numbers α, β ≥ 0, not both of them zero, such that
α ( | f 1 ( x ) | p , … , | f n ( x ) | p ) = β ( | g 1 ( x ) | q , … , | g n ( x ) | q ) , {\displaystyle \alpha \left(|f_{1}(x)|^{p},\ldots ,|f_{n}(x)|^{p}\right)=\beta \left(|g_{1}(x)|^{q},\ldots ,|g_{n}(x)|^{q}\right),}
for μ-almost all x in S. This finite-dimensional version generalizes to functions f and g taking values in a normed space which could be for example a sequence space or an inner product space.
Proof of Hölder's inequality There are several proofs of Hölder's inequality; the main idea in the following is Young's inequality for products.
Alternative proof using Jensen's inequality:
We could also bypass use of both Young's and Jensen's inequalities. The proof below also explains why and where the Hölder exponent comes in naturally.
Extremal equality
Statement Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then for every f ∈ Lp(μ),
‖ f ‖ p = max { | ∫ S f g d μ | : g ∈ L q ( μ ) , ‖ g ‖ q ≤ 1 } , {\displaystyle \|f\|_{p}=\max \left\{\left|\int _{S}fg\,\mathrm {d} \mu \right|:g\in L^{q}(\mu ),\|g\|_{q}\leq 1\r