Turán's inequalities

Turán's inequalities
Name	Turán's inequalities
Field	Mathematics
Introduced	1950s
Introduced by	Pál Turán

Turán's inequalities are a family of inequalities originally discovered by Pál Turán relating values of sequences and functions, notably orthogonal polynomials and special functions. They provide bounds of the form f(x)^2 − f(x−h)f(x+h) ≥ 0 or ≤ 0 in various settings and serve as tools in analysis, approximation theory, and combinatorics. The inequalities connect with the work of many mathematicians and institutions across the twentieth and twenty-first centuries and have been adapted to numerous contexts in pure and applied mathematics.

Definition and Statement

Turán's original inequality concerns classical orthogonal polynomials: for the sequence of polynomials P_n associated to a weight on an interval, the inequality asserts P_n(x)^2 − P_{n−1}(x)P_{n+1}(x) ≥ 0 (or with sign variants) for x in specified domains. Precise formulations appear for sequences such as the Legendre polynomials, Chebyshev polynomials, Hermite polynomials, and Laguerre polynomials, and for entire functions including Bessel functions and hypergeometric functions. Statements of the inequality are often parameterized by indices n, arguments x, and parameters drawn from families studied by Émile Picard, David Hilbert, and John von Neumann.

Historical Background and Motivation

The inequality was introduced by Pál Turán in the mid-twentieth century during investigations linked to problems studied by Paul Erdős and Paul Turán, with antecedents in work of Joseph Fourier and Gaston Darboux. Early motivations arose from approximation problems treated at institutions such as the Hungarian Academy of Sciences and in correspondence with mathematicians including John Littlewood and Gábor Szegő. Subsequent development involved contributions from Srinivasa Ramanujan, Constantin Carathéodory, and Norbert Wiener as researchers explored connections to entire function theory and to questions posed at conferences like the International Congress of Mathematicians and meetings at Princeton University.

Examples and Applications

Concrete instances include inequalities for Legendre polynomials appearing in studies by Henri Poincaré and Émile Borel, and for Bessel functions that relate to work by Gustav Kirchhoff and Lord Rayleigh on wave propagation. Applications span bounds used in analytic number theory in research connected to Atle Selberg, Donald Knuth, and Yuri Manin, and in probability theory where links to Andrey Kolmogorov and Paul Lévy arise. Computational applications link to algorithms developed at Bell Labs and IBM Research and appear in signal analysis frameworks influenced by Claude Shannon and Norbert Wiener. In mathematical physics the inequalities inform spectral estimates studied at institutions like the Institute for Advanced Study and in texts by Roger Penrose and Freeman Dyson.

Proofs and Methods

Proof techniques draw upon orthogonality relations established by Émile Borel and properties of three-term recurrence relations linked to Carl Gustav Jacob Jacobi. Methods include use of Sturm sequences as in the work of Jacques Sturm, spectral theory approaches influenced by David Hilbert, and integral representations following methods from Bernhard Riemann and Augustin-Louis Cauchy. Analytic proofs often employ complex analysis tools developed by Henri Lebesgue and Lars Ahlfors, while algebraic proofs use determinant identities reminiscent of those studied by Carl Friedrich Gauss and Arthur Cayley.

Generalizations and Related Inequalities

Generalizations extend to Turán-type inequalities for q-analogues investigated in contexts associated with Srinivasa Ramanujan and Richard Askey, to matrix-valued functions studied in relation to John von Neumann and Issai Schur, and to multivariate polynomials considered by Alexander Grothendieck and Olga Taussky-Todd. Related inequalities include the Hadamard inequality, Cauchy’s interlacing theorem, and inequalities studied by G. H. Hardy and John Littlewood. Further extensions connect to majorization results from Alfred Marshall and inequalities in convexity theory linked to Hermann Minkowski and Leonid Kantorovich.

Connections to Orthogonal Polynomials and Special Functions

Turán-type bounds are central in the theory of orthogonal polynomials as developed by Gábor Szegő and Walter Gautschi, and they interact with the Askey scheme of hypergeometric orthogonal polynomials studied by Richard Askey and James A. Wilson. Connections to special functions include classical Bessel functions (studied by Friedrich Bessel and Lord Rayleigh), confluent hypergeometric functions linked to Carl Gustav Jacobi and Ernest Rutherford, and parabolic cylinder functions with applications traced through the work of Paul Dirac and Erwin Schrödinger. These relationships underpin stability estimates used in numerical analysis at institutions such as the Courant Institute and in monographs by E. T. Whittaker and G. N. Watson.

Category:Inequalities in analysis