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Jensen's Theorem


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

The first states that, for a fixed vector v=(v_1,...,v_m), the function

 |v|_p=(sum_(i=1)^m|v_i|^p)^(1/p)

is a decreasing function of p (Cheney 1999).

JensensTheorem

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

The third theorem considers f(z) a function defined and analytic throughout a disk {|z|<=R} and supposes that f(z) has no zeros on the bounding circle |z|=R, that inside the disk it has zeros z_1, z_2, ..., z_n (where a zero of order k is included k times in the list, and that f(0)!=0. Then

 ln|f(0)R/(z_1)R/(z_2)...R/(z_n)|=1/(2pi)int_0^(2pi)ln|f(Re^(itheta))|dtheta

(Edwards 2001, p. 40).


See also

Gauss's Root Theorem, Lucas's Root Theorem

This entry contributed by Ronald M. Aarts

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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. Monthly 68, 978-983, 1961.

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Jensen's Theorem

Cite this as:

Aarts, Ronald M. "Jensen's Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/JensensTheorem.html

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