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Some important integral inequalities in mathematical analysis and beyond

Integral inequalities are often a very important tool in mathematical analysis, number theory, partial differential equations, differential geometry, probability, statistics, etc.

The most basic integral inequality is given by the following: Given a continuous function (Riemann integrable is sufficient) h:[a, b]\to\mathbb{R}, we have

\displaystyle\int_a^bh^2(x)dx\geq 0.

For any real number \lambda, consider h(x)=f(x)-\lambda g(x), h\in C([a, b]). After computations, this us give us the celebrated Cauchy-Schwarz’s inequality,


Moreover, the Cauchy-Schwarz’s inequality can be obtained by integrating over [a, b]\times [a, b] the symmetrizing functions u\mapsto f(x)g(y) and v\mapsto f(y)g(x) together with the elementary inequality 2uv\leq u^2+v^2 and Fubini’s theorem.

A generalization of Cauchy-Schwarz’s inequality was given by Rogers (1888) and Holder (1889),


where p, q>1 such that \frac{1}{p}+\frac{1}{q}=1. This follows from Young’s inequality, \displaystyle \frac{a^p}{p}+\frac{b^q}{q}\geq ab for all a, b\geq 0 applied for a=\frac{|f(x)|}{||f||_{p}} and b=\frac{|g(x)|}{||g||_{q}} and an integration afterwards. Here ||f||_{p} is the p-norm and it is defined as \displaystyle ||f||_{p}:=\left(\int_a^b|f(x)|^pdx\right)^{1/p} .  An self-extension of Holder’s inequality reads as follows:

\displaystyle\left(\int_a^b|f(x)g(x)|dx\right)^{1/r}\leq\left(\int_a^b|f(x)|^pdx\right)^{1/p}\left(\int_a^b|g(x)|^qdx\right)^{1/q}-\textbf{(extended Holder}),

where p, q, r>0 such that \displaystyle \frac{1}{p}+\frac{1}{q}=\frac{1}{r}. This last inequality follows easily from the first Holder inequality applied for x\mapsto |f(x)|^r and x\mapsto |g(x)|^r with exponents \displaystyle p_{1}=\frac{p}{r}, q_{1}=\frac{q}{r}. By an easy induction, the above inequality can be generalized as follows:

\displaystyle\left(\int_a^b |f_{1}(x)f_{2}(x)\ldots f_{k}(x)|^rdx\right)^{1/r}\leq \left(\int_a^b|f_{1}(x)|^{p_{1}}\right)^{1/p_{1}}\left(\int_a^b|f_{2}(x)|^{p_{2}}\right)^{1/p_{2}}\ldots \left(\int_a^b|f_{k}(x)|^{p_{k}}\right)^{1/p_{k}}-\textbf{(generalized Holder}),

where \displaystyle p_{i}, r\geq 1 and \displaystyle\sum_{i=1}^k\frac{1}{p_{i}}=\frac{1}{r}. Applications of Holder’s inequality are the following inequalities due to Minkovki:

\displaystyle\left(\int_a^b|f(x)+g(x)|^pdx\right)^{1/p}\leq \left(\int_a^b|f(x)|^pdx\right)^{1/p}+\left(\int_a^b|g(x)|^pdx\right)^{1/p}-\textbf{(Minkowski}),


\displaystyle\left(\int_a^b\left|\int_c^d f(x, y)dy\right|^pdx\right)^{1/p}\leq \int_c^d\left(\int_a^b|f(x, y)|^pdx\right)^{1/p}dy-\textbf{(generalized Minkowski})

All these inequalities presented above are discussed in details here. Some of these inequalities such as Cauchy-Schwarz, Holder and Minkowski (with their proofs) are also presented in the video below.


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