Dmitri Mendeleev is best known for creating the first periodic table of chemical elements. He also discovered an interesting mathematical theorem. Empirical research led him to a question about interpolation, which in turn led him to a theorem about polynomials and their derivatives.
I ran across Mendeleev’s theorem via a paper by Boas [1]. The opening paragraph describes what Mendeleev was working on.
Some years after the chemist Mendeleev invented the periodic table of the elements he made a study of the specific gravity of a solution as a function of the percentage of the dissolved substance. This function is of some practical importance: for example, it is used in testing beer and wine for alcoholic content, and in testing the cooling system of an automobile for concentration of anti-freeze; but present-day physical chemists do not seem to find it as interesting as Mendeleev did.
Mendeleev fit his data by patching together quadratic polynomials, i.e. he used quadratic splines. A question about the slopes of these splines lead to the following.
Theorem (Mendeleev): Let P(x) be a quadratic polynomial on [−1, 1] such that |P(x)| ≤ 1. Then |P′(x)| ≤ 4.
Mendeleev showed his result to mathematician Andrey Markov who generalized it to the following.
Theorem (Markov): If P(x) is a real polynomial of degree n, and |P(x)| ≤ 1 on [−1, 1] then |P′(x)| ≤ n² on [−1, 1].
Both inequalities are sharp with equality if and only if P(x) = ±Tn(x), the nth Chebyshev polynomial. In the special case of Mendeleev’s inequality, equality holds for
T2(x) = 2x² − 1.
Andrey Markov’s brother Vladimir proved an extension of Andrey’s theorem to higher derivatives,
Related posts
[1] R. P. Boas, Jr. Inequalities for the Derivatives of Polynomials. Mathematics Magazine, Vol. 42, No. 4 (Sep., 1969), pp. 165–174