SOME APPLICATIONS OF THE GAUSS-LUCAS THEOREM

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1 SOME APPLICATIONS OF THE GAUSS-LUCAS THEOREM Autor(en): Rubel, L. A. Objekttyp: Article Zeitschrift: L'Enseignement Mathématique Band (Jahr): 12 (1966) Heft 1-2: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Persistenter Link: Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 SOME APPLICATIONS OF THE GAUSS-LUCAS THEOREM 1 ) par L. A. Rubel The Gauss-Lucas Theorem, that the zéros of the derivative of a non-constant polynomial lie in the convex hull of the set of zéros of the polynomial, is a surprisingly powerful and versatile tool in classical analysis, despite its simplicity. To illustrate this point, we prove several results, using the Gauss-Lucas Theorem as our principal tool. To begin with, we présent a short proof of the Gauss-Lucas Theorem. As applications, we first give a new lower bound on the largest modulus of the zéros of a poly nomial, in terms of the coefficients of the polynomial. Next, we strengthen slightly a resuit of Edrei [2] on the zéros of the partial sums of a power séries. Finally, we reformulate a method of Fejér, and use it to strengthen some classical results on lacunary polynomials and entire functions with lacunary power séries. The Gauss-Lucas Theorem. The zéros of the derivative of a non-constant polynomial P lie in the convex hull of the set of zéros of P. Proof. It is enough to prove that any open half-plane that contains the zéros of P contains the zéros of P'/P. Without loss of generality, we may suppose that ail the zéros of P lie in the open right half-plane. Writing. with z n =xn + iy n, x n >o,we hâve i) This study was partially supported by the United States Air Force Office of Scientiflc Research Grant AF OSR

3 so that Hence, if x < 0, then Re (P f (z)/p (z)) < 0, and P'jP conse quently has no zéros in the open left half-plane. Theorem. // P (z) =ao+atz+... +an z n, a B 0, then P has a zéro of modulus al least Proof. First, considering the v-th derivative of P, we see that the product of the zéros of P (v) is so that P (v) must hâve a zéro of modulus at least since there are n? v roots. But by the Gauss-Lucas Theorem, the modulus of the largest root of P cannot be smaller than this, and the resuit is proved on interchanging v and n? v. Corollary. // Q (z) =b0 +bt z b tt has a zéro of modulus at most z", b 0 ï 0, then Q Proof. Apply the preceding theorem to P (z) = z n Q (l/z). The first part of the next resuit was proved, in a différent way, by Edrei [2].

4 Theorem. Suppose that the formai power séries, not a polynomial, has the property that for infinitely many n, there is a closed half-plane T n that contains the origin and that contains ail the n... + a n z. Then no two consécutive coefficients of f (z) may vanish. one coefficient vanishes, then there is a Une through the origin that contains ail the zéros of ail the partial sums. zéros of the partial sum S n (z) = a 0 -\~ a l z + If Proof. Suppose, by way of contradiction, that two consécutive coefficients of / do vanish. They are contained in a block of zéro coefficients, flanked left and right by non-zero coefficients, say a p and a q, respectively. Choose n q, and differentiate S n successively p times, to get S*n = S n^ p), where a*p 0, a* q o,r=q?p,risa polynomial, and n ~ p S* S*n has degree zz Now define S*n * by S*n * (z) n (1/z), * * S* S* is so that again a polynomial. Differentiate n successi n ** vely m times, where m = n? q, to get a polynomial S*n, n-p. = 3. Now if S n has 0, and 0, a** ail its zéros in a closed half-plane T n that contains the origin, then repeated applications of the Gauss-Lucas Theorem show that S*n also has ail its zéros in T n. Then S** has ail its zéros in the closed half-plane T* = { 1/z: zs T n ) u {o}. Again repeatedly applying the Gauss-Lucas Theorem, we see that S*** has ail its zéros in T*n. But the zéros of S*** are just s-th roots of fa*, and since s 3,we hâve a contradiction. In the sequel, the word «set» will dénote subsets of the finite complex plane. where a** s=q?p -a* Définition. // P zéros of P. is a polynomial, then Z (P) dénotes the set of

5 Définition. // E is a finite set, then K (E) dénotes the convex huit of E, and X* (E) dénotes K (E) u {o}. Définition. If E is a set, then ije dénotes the set Définition: If P is a polynomial, then P # dénotes the polynomial that is, In other words, P # is just the arithmetic mean of the proper partial sums of P. The next resuit is latent in the paper of Fejér [3]. Theorem. If aq 0 and a n 0, then or equivalently, Addendum. It will be clear from the proof that if a 0 0, then Z (P # ) s 1/K* (1/Z (P)). Further, Z {P*) s {0} v ijk (IJZ (P)) if a n 0 and P (z) # a n z n. In any event, so long as P (z) #an z", Z (P # ) s {0} u!/*? (1/Z (P)).

6 I reciprocals Proof. A straightforward computation shows that if and then Hence while Since R const., we may apply the Gauss-Lucas Theorem to get Now while Combining thèse results, the theorem is proved. Corollary. // a dise with center al the origin is free of zéros of P, then it is free of zéros of P #.! Corollary. // a 0 a n 0 and if P* has al least three zéros whose are non-collinear, then so does P. The first corollary is the basis of the proofs of the next i results. Thèse results are quite classical, except that the first! is somewhat elaborated.

7 Theorem. Suppose that where o<ki<k2<... < k n, and a } 0 for j= 0,1..., ra Lel TV&era P has at least one zéro in the dise If ki 3, then P must hâve at least two distinct zéros in this dise, and at least three distinct zéros whose reciprocals lie on or outside the regular polygon whose vertices are the k th roots r of - 1/A. Proof. Apply the opération # repeatedly, taking into account the degrees of the resulting polynomials, to get the polynomial Applying the fîrst corollary, we obtain the first part of the theorem, since the zéros of P* are the roots of z kl =?A. The other parts follow from simple géométrie considérations and the fact that Theorem. Suppose that f is a transcendental entire function with power séries expansion and suppose that Then the range of f contains each complex number.

8 Proof. It is enough to prove that / has a zéro, and we may = 0. From the proceding resuit, we see clearly suppose that n Q that the n^-th partial sum of the power séries for / has a zéro in the dise But since 1/% < the» 00 product f\(i?(n 1 ln k )) converges, so that there is a fixed dise with center at the origin that contains a zéro of the n k -th partial sum for Je = 2, 3, 4,.... It follows that / has a zéro in this dise, and the resuit is proved. It should be pointed out that Biernacki [1] proved, under the same hypothèses, and using a stronger form of the Gauss-Lueas Theorem, that / takes each complex value infinitely often. It is likely that our method can give a slight improvement of the preeeding resuit, but not to the full strength of Biernacki's resuit. A récent resuit of G. and M. Weiss [5] gives a partial analogue for functions regular in the unit dise. REFERENCES [1] Biernacki, M., Sur les zéros des polynômes et sur les fonctions entières dont le développement taylorien présente des lacunes. Bull. Soc. Math. France (2), 69 (1945), pp [2] Edrei, A., Power séries having partial sums with zéros in a half plane. Proc. Amer. Math. Soc, 9 (1958), pp [3] Fejér, L., Ueber die Wurzel von kleinsten absolu ten Betrage einer algebraischer Gleichung. Math. Ann., 65 (1908), pp [4] Marden, M., The geometry of the zéros of a polynomial in a complex variable. Math. Surveys, No. 111, American Mathematical Society, [5] Weiss, G. and Weiss M., On the Picard property of lacunary power séries. Studia Math., 22 (1962/63), pp L. A. Rubel, Dept. of Math. University of Illinois Urbana, (Reçu le 30 août 1964.)

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