9
$\begingroup$

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post.

Most of the details below are fairly elementary, they are mostly included for completeness.


1. The first inequality is quite easy to verify: Suppose that $f$ is a real-valued polynomial in one variable, all of whose roots are real. Then $$ f f''\le (f')^2, $$ with equality holding only at the double roots of $f$. I've seen this referred to as Turan inequality [a], but do not know of any places where it is discussed, or what a primary reference would be. It is pretty enough that I would imagine it well-known.

Also, I would be interested in knowing about generalizations, by which I mean either more relaxed conditions on the roots of $f$ that still suffice for the inequality to hold, or a discussion of intervals where the inequality is ensured in general, or a combination of both. Some assumption is needed, as $f(x)=x^2+1$ verifies.

I would also be curious to hear of applications outside of the context of Newton's dynamics.

[a]. MR0729188 (85a:58060). Saari, Donald G.(1-NW); Urenko, John B. Newton's method, circle maps, and chaotic motion. Amer. Math. Monthly 91 (1984), no. 1, 3–17.


2. The second inequality is more specialized. It is due to Barna [b] and the proof is straightforward but I would not call it trivial. Since [b] may not be easily accessible, I wrote a short note with the details, which can be found here.

Some notation is needed before I can state the result: Given a differentiable function $f$, the Newton function for $f$ is $N=N_f$ given by $$ N_f(x)=x-\frac{f(x)}{f'(x)}. $$ This is the function that results from applying the familiar Newton's method for approximating the zeros of $f$. (Note that $x^*$ is a zero of $f$ iff it is a fixed-point of $N$.)

Under reasonable assumptions on $f$, if $x^*$ is a zero of $f$, then there is an interval $I$ about $x^*$ such that for any $x_0\in I$, the Newton sequence $x_0,x_1,\dots,x_{n+1}=N(x_n),\dots$ is well defined and converges to $x^*$. Perhaps the simplest proof of this goes by observing that $$ N'=\frac{f f''}{(f')^2}, $$ so $N'(x^*)=0$, and there is a small interval $J$ centered at $x^*$ and a positive $\rho<1$ such that $|N'(t)|<\rho$ for all $t\in J$. But then we see that $$ |N(t)-x^*|=|N(t)-N(x^*)|\le\rho|t-x^*| $$ by the mean-value theorem, from which it readily follows that the Newton sequence starting at $t$ indeed converges to $x^*$.

The largest interval $I$ as above is the immediate basin of attraction of $x^*$. One easily checks that it is open and that, if bounded, say $I=(a,b)$, then $N(a)=b$ and $N(b)=a$.

Barna's inequality is as follows: Suppose that $f$ is a real-valued polynomial, all of whose roots are real. If $r$ is a root of $f$ and its immediate basin of attraction $I=(a,b)$ is bounded, then $|N'(a)|>1$ and $|N'(b)|>1$.

I would like to know of references other than [b] presenting details of a proof or some discussion of it ([a] refers to [b] in a brief remark, without details). I would also like to hear of generalizations or strengthenings.

[b]. MR0135224 (24 #B1274). Barna, Béla. Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen. III. (German) Publ. Math. Debrecen 8 1961, 193–207.

$\endgroup$
4
$\begingroup$

One generalization of (1), which is not quite an inequality, is the so-called Hawaiian conjecture. It states that the number of real zeroes of $(f'/f)'$ does not exceed the number of nonreal zeroes of $f$. This paper claims a proof: Mikhail Tyaglov, On the number of real critical points of logarithmic derivatives and the Hawaii conjecture, arXiv:0902.0413v3.

$\endgroup$
  • 1
    $\begingroup$ Very nice, thank you! The paper appeared as MR2837080 (2012k:30017). Tyaglov, Mikhail(D-TUB-IM), On the number of real critical points of logarithmic derivatives and the Hawaii conjecture. (English summary) J. Anal. Math. 114 (2011), 1–62. $\endgroup$ – Andrés E. Caicedo Mar 4 '16 at 22:02
  • $\begingroup$ Does the conjecture require simplicity of zeros of $f$? I found papers claiming both ways. $\endgroup$ – joro Mar 5 '16 at 12:35
  • $\begingroup$ @joro I'm aware of one paper where it is proved. There, the simplicity does not seem to be required. $\endgroup$ – Alex Degtyarev Mar 5 '16 at 12:57
2
$\begingroup$

Regarding Newton's method for polynomials with all zeros real, there is some discussion in the paper by J. H. Hubbard, D. Schleicher and S. Sutherland:http://www.math.cornell.edu/~hubbard/NewtonInventiones.pdf, especially in Section 7. There is a reference to another paper by Barna there, and lots of other references, which may be useful.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.