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) , we have
For any real number , consider , . After computations, this us give us the celebrated Cauchy-Schwarz’s inequality,
Moreover, the Cauchy-Schwarz’s inequality can be obtained by integrating over the symmetrizing functions and together with the elementary inequality and Fubini’s theorem.
A generalization of Cauchy-Schwarz’s inequality was given by Rogers (1888) and Holder (1889),
where such that . This follows from Young’s inequality, for all applied for and and an integration afterwards. Here is the -norm and it is defined as . An self-extension of Holder’s inequality reads as follows:
where such that . This last inequality follows easily from the first Holder inequality applied for and with exponents . By an easy induction, the above inequality can be generalized as follows:
where and . Applications of Holder’s inequality are the following inequalities due to Minkovki:
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.