Let 1p+1q=1\frac{1}{p}+\frac{1}{q} = 1, with p,q>1p,q >1, then

‖fg‖1≤‖f‖p‖g‖q\|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}

more generally for sums,

∑k=1n|akbk|≤(∑k=1n|ak|p)1/p(∑k=1n|bk|q)1/q\sum_{k=1}^n \lvert a_k b_k \rvert \leq \left( \sum_{k=1}^n \lvert a_k \rvert^p \right)^{1/p} \left( \sum_{k=1}^n \lvert b_k \rvert^q \right)^{1/q}

with equality when |bk|=c|ak|p−1\lvert b_k \rvert = c \lvert a_k \rvert^{p-1}. p=q=2p=q=2 yields Cauchy's inequality.

For integrals,

∫ab|f(x)g(x)|dx≤[∫ab|f(x)|pdx]1/p[∫ab|g(x)|qdx]1/q\int_a^b \lvert f(x) g(x) \rvert dx \leq \left[ \int_a^b \lvert f(x) \rvert^p dx \right]^{1/p} \left[ \int_a^b \lvert g(x) \rvert^q dx \right]^{1/q}

with equality when |g(x)|=c|f(x)|p−1\lvert g(x) \rvert = c \lvert f(x) \rvert^{p-1}. p=q=2p=q=2 yields Schwarz's inequality

For vector pp-norms: the dual of the ℓp\ell_p norm is the ℓq\ell_q norm.
e.g. dual of ℓ2\ell_2 norm is ℓ2\ell_2 norm, dual of ℓ1\ell_1 norm is ℓ∞\ell_\infty norm.

See also: Jensen's inequality (note that Hölder's inequality can be derived from this)

References

  1. https://mathworld.wolfram.com/HoeldersInequalities.html
  2. https://math.stackexchange.com/questions/211633/hölder-inequality-from-jensen-inequality
  3. https://artofproblemsolving.com/wiki/index.php/Hölder's_Inequality
  4. https://en.wikipedia.org/wiki/Hölder's_inequality