Askey–Gasper inequality

A positivity result for certain sums of Jacobi polynomials. Its proof supplied a key ingredient in de Branges’s proof of the Bieberbach conjecture.

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Jacobi polynomials: A family of orthogonal polynomials that generalizes the Legendre and Gegenbauer polynomials. They are the polynomials whose weighted sum the inequality proves positive.
Bieberbach conjecture: The conjecture that the nth Taylor coefficient of a normalized univalent function has absolute value at most n. The inequality furnished the positivity needed for de Branges’s proof of its coefficient bounds.
Richard Askey: An American mathematician whose work helped shape the modern theory of special functions. His collaboration with George Gasper produced the inequality’s original proof.
Turán’s inequality: An inequality comparing a polynomial sequence’s squared middle term with the product of its neighbors. It is another positivity inequality for orthogonal polynomials, but concerns adjacent degrees rather than a weighted sum.
Orthogonal polynomials: Polynomial families orthogonal with respect to a specified weight function on an interval or other domain. The inequality exploits the structure and identities of an orthogonal-polynomial family.
Louis de Branges: An American mathematician known for proving the Bieberbach conjecture in 1984. He used the inequality as part of the chain of estimates completing the conjecture’s proof.
George Gasper: An American mathematician known for research on hypergeometric series and orthogonal polynomials. He coauthored the positivity result that later entered de Branges’s proof.
Christoffel–Darboux formula: An identity expressing a sum of products of orthogonal polynomials as a quotient involving two neighboring degrees. It gives a kernel identity for polynomial sums, unlike the Askey–Gasper positivity bound.
Hypergeometric function: A function defined by a power series whose successive coefficients have a rational ratio. Jacobi polynomials can be expressed hypergeometrically, enabling transformations used in positivity proofs.
Univalent function: A holomorphic function that is one-to-one on its domain. The conjecture proved with this inequality concerns Taylor coefficients of normalized univalent functions.