Jensen's Theorem

There are at least three theorems known as Jensen's theorem.

The first states that, for a fixed vector , the function

is a decreasing function of  (Cheney 1999).

The second states that if  is a real polynomial not identically constant, then all nonreal zeros of  lie inside the Jensen disks determined by all pairs of conjugate nonreal zeros of  (Walsh 1955, 1961; Householder 1970; Trott 2004, p. 22). This theorem is a sharpening of Lucas's root theorem.

The third theorem considers  a function defined and analytic throughout a disk  and supposes that has no zeros on the bounding circle , that inside the disk it has zeros , , ...,  (where a zero of order  is included  times in the list, and that . Then

(Edwards 2001, p. 40).

REFERENCES:

Cheney, E. W. Introduction to Approximation Theory, 2nd ed. Providence, RI: Amer. Math. Soc., 1999.

Edwards, H. M. "Jensen's Theorem." §2.2 in Riemann's Zeta Function. New York: Dover, pp. 40-41, 2001.

Householder, A. S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970.

Rahman, Q. I. and Schmeisser, G. Analytic Theory of Polynomials. Oxford, England: Oxford University Press, 2002.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Walsh, J. L. "A Generalization of Jensen's Theorem on the Zeros of the Derivative of a Polynomial." Amer. Math. Monthly 62, 91-93, 1955.

Walsh, J. L. "A New Generalization of Jensen's Theorem on the Zeros of the Derivative of a Polynomial." Amer. Math. Monthly68, 978-983, 1961.