Analysis to rank
Improve, gain a little here, avoid losing too much there, optimize… all of this falls under the purview of analysis! If geometric intuition helps us compare, it is mathematical analysis that will allow us to quantify variations and thus to bound from above or below many expressions. Among the stars of the field, we find the arithmetic-geometric mean inequality, well known to high school students, and the Cauchy–Schwarz inequality, more advanced but of formidable power.
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The usual inequality on real numbers | Tangente
Once their values are known, any two numbers can be compared—whether they are integers, fractions, or real numbers—and we can say which is smaller.
Solving inequalities: methods and examples | Tangente
Some of our readers no doubt have unpleasant memories of inequalities.
A history of ordered means: AM, GM, HM | Tangente
“The” mean of two numbers is defined “naturally” according to the context. It is not always the familiar arithmetic mean! Several different notions coexist and are closely interconnected, as Liouville, Cauchy and Jensen clearly understood.
Tchebychev's inequalities for sequences | Tangente
Elementary results can sometimes prove astonishingly fertile, opening the way to a host of developments and applications. Chebyshev's inequality is a case in point.
Five variants of the Cauchy–Schwarz inequality | Tangente
The Cauchy–Schwarz inequality appears in several branches of mathematics: analysis, arithmetic, geometry, probability... It is so important that many widely differing proofs have been devised!
Cauchy–Schwarz inequality: proofs | Tangente
The Cauchy–Schwarz inequality takes many forms: arithmetic, integral and geometric; it even appears in probability theory. Let us trace its development, from Cauchy's numerical formulation around 1820 to its general form a century later.
Order amid disorder: rearrangement | Tangente
Take a random sequence of numbers. It is unlikely that they will all be in increasing or decreasing order from the outset. Can we nevertheless hope to extract perfectly ordered subsequences? Yes—but they may not be as long as we would like!
More means: harmonic and arithmetic | Tangente
The arithmetic, quadratic, geometric and harmonic means (among many others) are all examples of a general family: the power means. They satisfy a well-known chain of inequalities.
A showcase of mathematical inequalities | Tangente
Mathematics abounds in inequalities, often drawing on differential or integral calculus. Many problems can be solved with their help. So let's explore them!