Abstract
This chapter is an invitation to the systematic study of algebraic inequalities. More precisely, our main purpose here is to discuss some interesting examples of inequalities, for whose derivation we can use the simple mathematics we developed so far. Later, when we have the tools of Calculus at our disposal, we shall return to the study of algebraic inequalities, largely generalizing some of those we will study here.
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Notes
- 1.
For the elementary facts on the notion of volume of solids, we refer the reader to [4].
- 2.
- 3.
For a thorough discussion of the facts that follow, see Sect. 3. 4 of [4], for instance.
- 4.
Augustin Louis Cauchy , one of the greatest mathematicians of the XIX century, and maybe of History. Cauchy was one of the precursors of Mathematical Analysis, an extremely important area of higher Mathematics. He also has his name attached to several important results in Differential Equations and Mathematical Physics.
- 5.
Jacob and Johann Bernoulli, Swiss mathematicians of the XVIII century.
- 6.
After Pafnuty Chebyshev , Russian mathematician of the XIX century.
- 7.
In the more precise language of functions (cf. Chap. 6), we say that (x 1, x 2, …, x n ) is a permutation of (a 1, a 2, …, a n ) if there is a bijection φ: { 1, …, n} → { 1, …, n}, so that x i = a φ(i), for 1 ≤ i ≤ n.
- 8.
Actually, it is easy to show that there are exactly n! such permutations. For a proof, see [5], or provide one yourself, by making an induction argument.
Bibliography
A. Caminha, An Excursion Through Elementary Mathematics II - Euclidean Geometry (Springer, New York, 2018)
A. Caminha, An Excursion Through Elementary Mathematics III - Discrete Mathematics and Polynomial Algebra (Springer, New York, 2018)
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Caminha Muniz Neto, A. (2017). Elementary Inequalities. In: An Excursion through Elementary Mathematics, Volume I. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-53871-6_5
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