Holder inequality ($1/p+1/q=1$)

For conjugate exponents $p,q$,with the analogous inequality ${\sum }_n|x_n{y}_n|\le \Vert x{\Vert }_p\Vert y{\Vert }_q$ for sequences.

For a random variable with finite fourth moment, apply Holder inequality with conjugate exponents $3/2$ and $3$ to $|X{|}^{2/3}|X{|}^{4/3}$. This gives $\mathbb{E}|X{|}^2\le (\mathbb{E}|X|{)}^{2/3}(\mathbb{E}|X{|}^4{)}^{1/3}$. Raising to the power $3/2$ proves the displayed Lp norm interpolation inequality.

If $f_n\rightharpoonup f$ in $L^p$, $g_n\to g$ in $L^q$, and $1/p+1/q=1$, thenIndeed, split the difference into the weak pairing with $g$ and a term bounded by the Holder inequality using $g_n-g$.

Exponents $p,q\in [1,\infty ]$ are conjugate when $1/p+1/q=1$, with the convention $1/\infty =0$.

For nonzero $y\in {\ell }^q$, a normalized sequence proportional to $\mathrm{sgn}(y_n)|y_n{|}^{q-1}$ attains equality in Hölder's inequality.

If $p_1,\dots ,p_n\in [1,\infty ]$ satisfy ${\sum }_{i}1/p_i=1$, thenIt follows by induction from the two-factor Holder inequality.

This is the integral version of the Minkowski inequality.

For suitable measurable $F(x,y)$ and $1\le p<\infty$,Pair the left side with a unit vector in the dual $L^q$ space, use Tonelli's theorem, and apply Hölder's inequality in $y$.