HÖLDER'S INEQUALITY:
If \(\mathcal{S}\) is a measurable subset of \(\mathbf{R}^n\) with the Lebesgue measure, and \(\eta\) and \(\xi\) are measurable real- or complex-valued functions on \(\mathcal{S}\), then

\[\displaystyle\left(\int_\mathcal{S}|\eta(x)\xi(x)|\ \mathrm{d}x\right)^{ab}\leq\left(\int_\mathcal{S}|\eta(x)|^a\ \mathrm{d}x\right)^b\left(\int_\mathcal{S}|\xi(x)|^b\ \mathrm{d}x\right)^a\]
where, \((a,b)\in(1,\infty)^2\) with \(1/a+1/b=1\).
#HolderInequality #LebesgueMeasure #Integrals

HÖLDER'S INEQUALITY:
If \(\mathcal{S}\) is a measurable subset of \(\mathbf{R}^n\) with the Lebesgue measure, and \(\eta\) and \(\xi\) are measurable real- or complex-valued functions on \(\mathcal{S}\), then

\[\displaystyle\left(\int_\mathcal{S}|\eta(x)\xi(x)|\ \mathrm{d}x\right)^{ab}=\left(\int_\mathcal{S}|\eta(x)|^a\ \mathrm{d}x\right)^b\left(\int_\mathcal{S}|\xi(x)|^b\ \mathrm{d}x\right)^a\]
where, \((a,b)\in(1,\infty)^2\) with \(1/a+1/b=1\).
#HolderInequality #LebesgueMeasure #Integrals

HÖLDER'S INEQUALITY:
If \(\mathcal{S}\) is a measurable subset of \(\mathbf{R}^n\) with the Lebesgue measure, and \(\eta\) and \(\xi\) are measurable real- or complex-valued functions on \(\mathcal{S}\), then

\[\displaystyle\int_\mathcal{S}|\eta(x)\xi(x)|\ \mathrm{d}x=\left(\int_\mathcal{S}|\eta(x)|^a\ \mathrm{d}x\right)^{\frac{1}{a}}\left(\int_\mathcal{S}|\xi(x)|^b\ \mathrm{d}x\right)^{\frac{1}{b}}\]
where, \((a,b)\in(1,\infty)^2\) with \(1/a+1/b=1\).
#HolderInequality #LebesgueMeasure

HÖLDER'S INEQUALITY:
If \(\mathcal{S}\) is a measurable subset of \(\mathbf{R}^n\) with the Lebesgue measure, and \(\eta\) and \(\xi\) are measurable real- or complex-valued functions on \(\mathcal{S}\), then

\[\displaystyle\int_\mathcal{S}|\eta(x)\xi(x)|\ \mathrm{d}x=\left(\int_\mathcal{S}|\eta(x)|^a\ \mathrm{d}x\right)^{\frac{1}{a}}\left(\int_\mathcal{S}|\xi(x)|^b\ \mathrm{d}x\right)^{\frac{1}{b}}\]
where, \((a,b)\in(0,\infty)^2\) with \(1/a+1/b=1\).
#HolderInequality #LebesgueMeasure