PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS IGOR E. PRITSKER. polynomials with the norm of their product. This subject includes the well

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1 TRANSACTIONS OF TH AMRICAN MATHMATICAL SOCITY Volume 353, Number, Pages 397{3993 S -9947()856- Article electronically published on May, PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS IGOR. PRITSKR Abstract. We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment [; ]. Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases. It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.. Introduction Let be a compact set in the complex plane C.For a continuous function f on, we dene the uniform norm on as follows: kfk =max z jf(z)j: Consider algebraic polynomials fp k (z)g m k= of one complex variable and their product p(z) := my k= p k (z): This paper is devoted to a study of polynomial inequalities of the form (.) my k= kp k k Ckpk : In particular, if deg p = n is the degree of the product polynomial p(z), then we are interested in the asymptotic behavior of the constant C, as n!. While the inequality opposite to (.) is obvious with C =, (.) itself has been studied in a number of papers, by considering various cases of the set and Received by the editors December 7, Mathematics Subject Classication. Primary 3C, C8, 3C5 Secondary 3A5, 3A5. Key words and phrases. Polynomials, uniform norm, zero counting measure, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points. Research supported in part by the National Science Foundation grants DMS and DMS c American Mathematical Society

2 397 IGOR. PRITSKR dierent aspects of (.). Apparently, one of the rst results in this direction is due to Kneser [5], for =[; ] and m = (see also Aumann []), who proved that (.) where (.3) K` n := n; kp k [; ] kp k [; ] K` n kp p k [; ] Ỳ k= +cos k ; n n;` Y k= + cos k ; n deg p = ` and deg(p p )=n. Note that (.) becomes an equality for the Chebyshev polynomial t(z) = cos n arccos z = p (z)p (z), with a proper choice of the factors p (z) andp (z). P. B.Borwein [7] has recently given a new proof of (.)- (.3) and generalized this to the multifactor inequality (.4) my k= He has also shown that (.5) Y kp k k [; ] n; [ n ] Y n; [ n ] k= k= + cos k ; n kpk [; ] : +cos k ; n (3:99 :::) n as n!: Another case of the inequality (.)was considered by Gel'fond [3, p. 35]in connection with the theory of transcendental numbers, for = D, where D := fw : jwj < g is the unit disk, (.6) my k= kp k k D e n kpk D : The latter inequality was improved by Mahler [8], who replaced e by : (.7) my k= kp k k D n kpk D : It is easy to see that the base cannot be decreased, if m = n and n!. However, (.7) has recently been further improved in two directions. D. W. Boyd [8, 9] showed that, by taking into account the number of factors m in (.7), one has (.8) where (.9) my k= C m := exp kp k k D (C m ) n kpk D m =m! log cos t dt is asymptotically best possible for each xed m, as n!. Kroo and Pritsker [6] showed that, for any m n (.) my k= kp k k D n; kpk D where equality holds in (.) for each n N, with m = n and p(z) =z n ;. The above-mentioned results represent only a selection, which is directly related to the subject of this paper, from the existing literature on inequalities for products of polynomials in various norms. Another particularly important direction is

3 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3973 related to polynomials in many variables. We do not discuss it here, but give the references to the results of Mahler [9] for the polydisk, of Avanissian and Mignotte [] for the unit ball in C k, of Beauzamy and no [4], and of Beauzamy, Bombieri, no and Montgomery [3] for an extensive study of products of polynomials in several variables, using dierent norms. We would also like tomention the research on norms of products of polynomials in abstract Banach spacesby Benitez, Sarantopoulos and Tonge [5]. The rest of this paper is organized as follows. Our general results are stated below in Section. Their applications for the unit disk, the segment [; ] and a circular arc are given in Section 3. We discuss a question about the possibility of improvement for a xed number of factors in Section 4. The proofs of the results stated in Sections, 3 and 4 can be found in Section 5.. General results The known results discussed in the Introduction indicate that the constant C in (.) typically grows exponentially fast with n, which is the degree of the product, and it also depends on the set, as one might expect. Therefore, we consider a general problem of nding the smallest constant M > such that (.) my k= kp k k M n kpk for arbitrary Q algebraic polynomials fp k (z)g m k= with complex coecients, where m p(z) = k= p k(z) and n = deg p, asbefore. In order to give a solution of the above problem,wehave tointroduce certain notions from the logarithmic potential theory (cf. [4]). Let cap() bethelog- arithmic capacity of a compact set C. For with cap() >, denote the equilibrium measure of (in the sense of the logarithmic potential theory) by. We remark that is a positive unit Borel measure supported on, supp (see [4, p. 55]). Dene (.) d (z) :=maxjz ; tj z t C which is clearly a positive and continuous function on C. Theorem.. Let C be acompact set, cap() >. Then the best constant M in (.) is given by exp;r log d (z)d (z) (.3) M = : cap() We show in the next section that this general result gives the inequality (.7) of Mahler and the asymptotic version of Borwein-Kneser inequality (.4)-(.5). Note that the restriction cap() > excludes only very thin sets from our consideration (see [4, pp ]), e.g., nite sets in the plane. But if consists of nitely many points (more than one) then the inequality (.) (or (.)) cannot be true at all, which is easy to see for a polynomial p(z) with zeros at every point of and for its linear factors fp k (z)g n k=. On the other hand, Theorem. is applicable to any compact set with a connected component consisting of more than one point (cf. [4, p. 56]). In particular, if is a continuum, then we obtain the following crude but interesting estimate.

4 3974 IGOR. PRITSKR Corollary.. Let C be a bounded continuum (not a single point). Then we have (.4) M diam() cap() 4 where diam() is the uclidean diameter of the set. An interesting question connected with our problem is about the nature of the extremal polynomials for (.), i.e., about those polynomials, for which (.) becomes an asymptotic equality, asn!. The classical cases = D and =[; ] suggest that the extremal polynomials have certain very special properties, and special zero distributions, in particular. We show that this is true in general. Namely, we say that a sequence of polynomials fq n (z)g n=,degq n = n, is asymptotically extremal for (.) if (.5) Q n k= kq =n k nk = M n! kq n k where fq k n (z)g n k= are the linear factors of Q n(z), i.e., (.6) Q n (z) =: ny k= q k n (z) n N: Observe that there is no loss of generality ifwe consider only monic polynomials fp k (z)g m k= in (.), so that their product p(z) is also monic. In particular, if we have a sequence of asymptotically extremal polynomials fq n (z)g n= then the corresponding sequence of monic polynomials, obtained by dividing each Q n (z) by its leading coecient, is also asymptotically extremal. Recall that for any monic polynomial P (z) of degree n, wehave (.7) kp k (cap()) n where C is an arbitrary compact set (cf. Theorem 5.5.4(a) in [, p. 55]). Thus, if a sequence of monic polynomials fp n (z)g n= deg P n = n, satises (.8) n! kp nk =n =cap() then it is customary to say that such polynomials have asymptotically minimal norms on. Sequences of polynomials with asymptotically minimal norms have been studied in many papers, e.g., see Faber [], Fekete and Walsh [], Widom [5, 6], Blatt, Sa and Simkani [6], Mhaskar and Sa [], Stahl and Totik [3]. Our next result relates monic asymptotically extremal polynomials for (.) to monic polynomials satisfying (.8). Theorem.3. Let fq n (z)g n= be a sequence of monic asymptotically extremal polynomials for (.) on a compact set C cap() >. If all zeros of Q n (z) are uniformly bounded for all n N, then (.9) kq nk =n n! =cap(): We remark that if the condition that all zeros be uniformly bounded is dropped, then (.9) may not always be true. This is easy to see for = D and ~ Q n (z) :=

5 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3975 (z n; +)(z + n ), n N. Since M D = (see [8] or Section 3. of this paper), equation (.5) is readily checked for this sequence. However, k Qn ~ k =n n! =6= cap(d) =: Also, note that polynomials with asymptotically minimal norms need not be asymptotically extremal for (.), in general. To verify this, it is sucient to consider P n (z) =z n, n N and = D. We next turn to the study of asymptotic zero distributions of asymptotically extremal polynomials (not necessarily monic). Assume that fz k n g n k= C are the zeros of Q n(z) and dene the normalized zero counting measure for Q n (z) by (.) n := n nx k= zk n n N where z denotes the Dirac measure at z. Note that n is a unit Borel measure and supp n = fz k n g n k= n N. A sequence of Borel measures f ng n= is said to converge to a Borel measure in the weak* topology if (.) n! f(z)d n (z) = f(z)d(z) for any continuous function f(z) with compact support in C.We write in this case that n!, asn!. Theorem.4. Let C be acompact set and let be the unbounded component of C n. Suppose that satises one of the following: (i) has empty interior, C n =and is regular (ii) = G where G is a bounded domain, and C n is connected. If fq n (z)g n= is an asymptotically extremal sequence ofpolynomials, then we have for the normalized zero counting measures n of (.) that (.) n! as n!: It is well known (cf. [4, p. 79]) that the equilibrium measure is supported on the outer boundary of, i.e., on the boundary of the unbounded component of C n. Thus, Theorem.4 says that the zeros of asymptotically extremal polynomials are asymptotically equidistributed along the outer boundary of, according to,asn!. Classical examples of the asymptotically extremal polynomials with the above zero distributions include Fekete polynomials, Leja polynomials, Chebyshev polynomials for sets with empty interior, etc. In fact, Theorem.4 allows a converse stated below. Theorem.5. Let C be a compact set, cap() >, and let be the unbounded component of C n. If is regular, then (.) implies that the sequence of polynomials fq n (z)g n= is asymptotically extremal. For the notion of regularity in the sense of Dirichlet problem, we refer to [, Ch. 4]. One can see that the regularity of is essential in the above theorem, by considering the following example. Let = D [fag, where a > 3isreal, so that a is the isolated irregular point. Then cap(d [fag) = cap(d) = and

6 3976 IGOR. PRITSKR D[fag = D = d (see Theorems III.3 and III.37 in [4]), where d is the arclength This gives M D[fag = exp log d D[fag (e i )d = exp log je i ; ajd = a: For the polynomials ~ Qn (z) =z n ;, n N, wehave that n ( Qn ~ )! d = D = D[fag as n!: On the other hand, using the above weak* convergence, we obtain n! Q n! k= kz ; ei n k =n k D[fag = kz n ; k D[fag a n! = a exp n! log ja ; zjd n ( ~ Q n )(z) ny k= = a exp ja ; e i n k j! =n log ja ; e i jd i.e., the sequence f Qn ~ (z)g n= is not asymptotically extremal on = D [fag. = 3. Applications We consider here three special cases of the general results of Section, where the measure is known explicitly, and obtain the explicit values of M. Those are the cases of the unit disk, of the segment [; ] and of a circular arc. Further, we give several known representations of for general sets and discuss how to obtain the explicit form of from them. 3.. Unit Disk. For the unit disk D = fw : jwj < g, wehave thatcap(d) = [4, p. 84] and that (3.) D = d where d is the arclength Thus, Theorem. yields (3.) M D = exp log d D (e i ) d =exp log d = so that we immediately obtain Mahler's inequality (.7). The results of Theorems.4 and.5 are apparently new even in this classical case. Corollary 3.. Asequence ofpolynomials fq n (z)g n= is asymptotically extremal for (.) on = D if and only if (3.3) n! d as n! where n is dened by(.) and where d is the arclength

7 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS Segment [; ]. If =[; ], then cap([; ]) = = and (3.4) dx [; ] = p x [; ] ; x which is the Chebyshev (or arcsin) distribution (see [4, p. 84]). Using Theorem., we obtain (3.5) M [; ] = exp = exp ; = log d [; ] (x) p dx =exp ; x! log( + sin t)dt 3:993 log( + x) p dx ; x which gives the asymptotic version of Borwein's inequality (.4)-(.5). Again, the description of the extremal polynomials appears to be new in this case. Corollary 3.. Asequence ofpolynomials fq n (z)g n= is asymptotically extremal for (.) on =[; ] if and only if (3.6) dx n! p x [; ] as n! ; x where n is dened by(.) Circular Arc. We now consider the case =, where := fe i : jj = <g. It is known that (3.7) cap() =sin(=4) and that (3.8) (z) = z ; +p (z ; e i= )(z ; e ;i= ) sin(=4) z C n is the conformal mapping of C n onto C n D, such that() = and () = sin(=4) (cf. [, p. 37]). Note that every point z = e i except for the endpoints of, hastwo images on fw : jwj =g: (3.9) and (3.) + (e i )= ei ; +e i=p (cos ; cos(=)) sin(=4) ; (e i )= ei ; ; e i=p (cos ; cos(=)) sin(=4) which correspond to the two branches of the root in (3.8). We show in Section 5 that (3.) = j +(e i ) + sin(=4)j p + j ; (e i ) + sin(=4)j d e i : (cos ; cos(=))

8 3978 IGOR. PRITSKR Figure. M () for the circular arc = fe i : jj = g. Corollary 3.3. Let = = fe i : jj = <g. Then(.) holds with (3.) M () = 8 >< >: R sin(=4) exp = log(sin(=+=4)) d () R sin(=4) exp ;= log ( sin(=+=4)) d () + R = ;= log d () [ ] [ ]: Furthermore, a sequence ofpolynomials fq n (z)g n= is asymptotically extremal for (.) on = if and only if (3.3) n! as n! where n is dened by(.) and is dened by(3.). The graph of M () [ ] is given in Figure General Sets. We give several well known representations for the equilibrium measure in this subsection. First, consider the case of an arbitrary compact set C, with cap() > and with the unbounded component of its complement C n denoted by. Then (3.4) =!( ) where!( ) is the harmonic measure of at (cf. [, p. 5]). This useful identity implies that is invariant under certain conformal mappings of and gives the following form of, especially important for our applications. Assume that is a closure of Jordan domain with rectiable boundary. Since is simply connected in this case, there exists a conformal mapping :! D :=

9 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3979 fw : jwj > g normalized by (3.5) (z) () = and = z! z cap() (see [, p. 33]). Furthermore, for any Borel set B C we have from (3.4) that (3.6) (B) =m((b where dm = d=() is the normalized arclength on fjwj =g. Thisgives that (3.7) (z) = j (z)jjdzj so that both cap() and can be found from the conformal mapping by (3.5) and (3.7). Thus, if an explicit form of is known, then the constant M of (.3) is also known explicitly. In fact, the cases of the unit disk, of the segment [; ] and of a circular arc, considered in the previous three subsections, are handled in a similar way. Yet nding the capacity and the equilibrium measure of a general set is a hard classical problem, often without an explicit solution. Fortunately, onecan approach the problem of nding the constant M numerically, using sequences of asymptotically extremal polynomials. Many examples of suitable sequences of polynomials can be derived with the help of Theorem.5, by generating polynomials with asymptotically uniformly distributed zeros, such asfekete polynomials, Leja polynomials, etc. We shall consider the numerical aspect of the problem elsewhere. 4. Possibility of improvement in a fixed number of factors case We already mentioned a result of Boyd [9] (see (.8)-(.9) in the Introduction), that one can improve the constant in (.3) for = D, by considering a xed number of factors m in (.). On the other hand, comparing the results of Kneser (.)- (.3) and of Borwein (.4), one can immediately observe that the constant isthe same for any m, i.e., there is no improvement for =[; ]. The nature of this phenomenon can be easily explained by the presence of two endpoints in the case =[; ]. We give a more general result below. It follows from the proof of Theorem. (cf. (5.6)), that my R exp( um n (z)d (z)) (4.) kp k k kpk cap() where (4.) and (4.3) k= u m (z) := max km log jz ; c kj jp k (c k )j = kp k k k = ::: m: This is a generalization of Boyd's ideas in [9]. In fact, the value of the constant C m in (.9) is obtained in this way, seetheproofoftheoremin[9]. However, if =[; ] then log d [; ] (z) = max(log jz ; j logjz +j) z [; ]: Consequently, u m (z) =logd [; ] (z) forany z [; ] if the set fc k g m k= contains the endpoints f ;g. Thus, the dierence between the constants in (4.) and in (.3) is essentially the dierence between u m (z) and log d (z).

10 398 IGOR. PRITSKR Consider the Fekete polynomials ff n (z)g n= deg F n = n for the set (cf. [, p. 55]). Theorem 4.. Let C be acompact set, cap() >. Suppose that there exist points f l g s l= such that (4.4) d (z) = max ls jz ; lj for any If m s then we can nd such factoring for the sequence offekete polynomials (4.5) that (4.6) F n (z) = my k= F k n (z) n N Q m k= kf =n k nk = M : n! kf n k Consequently, no improvement is possible in (.), for a xed number of factors m s as n!: It is easy to see from the above theorem that there is no improvement in constant, for any m for such sets as a circular arc of angular measure at most and a segment. Also, there is no improvement forany polygon with s vertices, if m s: 5. Proofs We start with several lemmas necessary for the proofs of our general results from Section. The following lemma is a generalization of Lemma in [9]. Lemma 5.. Let F C be acompact set (not a single point) and let d F (z) :=maxjz ; tj z C : tf Then log d F (z) is a subharmonic function in C and (5.) log d F (z) = log jz ; tjd(t) z C where is a positive unit Borel measure in C with unbounded support, i.e., (5.) (5.3) (C )= and supp: Furthermore, if F = G, where G is a bounded domain, then supp \ G 6= : Note that Lemma 5. reduces back to Lemma of Boyd [9] inthecasef = fc k g m k=, where c k k= ::: m (m ), are complex numbers and (5.4) Proof. Note that log d fckg m (z) = max log jz ; c kj =: u k= m (z): km u(z) := log d F (z) =suplog jz ; tj z C : tf Since log jz ; tj is upper semicontinuous on C F and is subharmonic in z for each t F, Theorem.4.7 of [, p. 38] implies that u(z) is subharmonic in C.

11 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 398 Let fc k g be a sequence of points dense On dening (5.5) u m (z) := max km log jz ; c kj z C m we observe that fu m (z)g m= is a monotonically increasing sequence of continuous subharmonic functions in C, such that (5.6) u(z) = m! u m(z) z C where the above convergence is uniform on compact subsets of C. It follows from Lemma of [9] that, for any m (5.7) u m (z) = log jz ; tjd m (t) z C where m is a probability measure in C.Let D R := fz : jzj <Rg R>: Then (5.8) where (5.9) u m (z) = h m (z) = D R log jz ; tjd m (t)+h m (z) CnD R log jz ; tjd m (t) z C z C is harmonic in D R for any m : It follows from (5.8) that inf h m (z) inf u m (z) ; sup log jz ; tjd m (t) zd R zd R zd R D R inf zd R u (z) ; log(r) =:A R m because m (C )=: Since fh m (z) ; A R g m= is a sequence of positive harmonic functions in D R,wehave that either h m (z)! + locally uniformly in D R or there is a subsequence N N such thath m (z) m N converges locally uniformly to a harmonic function ~ h R (z) ind R (see Theorem.3. in [, p. 6]). We show that the rst case is impossible, in fact. Indeed, if that were possible, then (5.) m! D R log jz ; tjd m (t) =; z D R by (5.6) and (5.8), where the above divergence is locally uniform in D R : Using (5.), submean inequality andfubini's theorem, we have ; = m! m! D R m! m(d R )log R D R log jre i = ; tjd m (t)d log jre i = ; tjd d m (t) which isanobvious contradiction. Next, we choose a subsequence N N such that (5.) m j DR! ~R as m! m N

12 398 IGOR. PRITSKR by Helley's selection theorem (see Theorem.. in []). Note that supp ~ R D R ~ R (C ), and sup m! mn D R log jz ; tjd m (t) = log jz ; tjd~ R quasi everywhere (with a possible exception of a set of zero capacity) in C, by Theorem I.6.9 of []. Thus, passing to the it in (5.8), as m! mn we have (5.) u(z) = log jz ; tjd~ R (t)+ ~ h R (z) holds quasi everywhere in D R : Using Riesz Decomposition Theorem [, p. 76] for u(z) ind R,we obtain (5.3) u(z) = log jz ; tjd R (t)+h R (z) z D R where h R (z) isharmonicind R and R is supported in D R. Hence, the Unicity Theorem (cf. Theorem II.. in []), (5.) and (5.3) imply that (5.4) R =~ R j DR and R (C ) for any R>: Clearly, the sequence of measures R is monotonically increasing, as R!, to the associated Riesz measure for u(z) inc (cf. [, p. 5]). Observe that (C ), by (5.4) and (5.), and that (u(z) ; log jzj) = log d F (z) =: z! z! jzj Let us show that (C )=: Integrating (5.3), we obtain = = D R u((r ; )e i )d = (D R; )log(r ; )+ which gives on letting! that D R log j(r ; )e i ; tjd(t)+h R ((R ; )e i ) log j(r ; )e i ; tjd D RnD R; d(t)+h R () log jtjd(t)+h R () u(re i )d ; (D R ) log R = h R (): If (C ) < then, passing to the it in the above equation, as R!,wehave that (5.5) R h R() = +: R! Note that log jtjd R (t) is increasing with R, aslogjtj > forjtj > : Considering (5.3) for z =,wededuce that h R () is decreasing when R! whichcontradicts (5.5). Thus, (C ) = and sup R! u(re i )d ; (D R )logr : d

13 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3983 It follows from the above inequality and (9.3) of Theorem.IV.9(d) in [, p. 53] that u(z) = log jz ; tjd(t)+h(z) z C where h(z) is a harmonic function in C given by h(z) = u(z + Re i )d ; (C )logr R! = R! log d F (z + Re i ) d = z C : R This completes the proof of (5.). Let us show that supp is unbounded. Assume to the contrary that it is compact, then u(z) must be harmonic near innity by (5.). We can also assume, without loss of generality, that F and dene the function v(z) :=u(z) ; log jzj = log d F (z) ; log jzj z C : Observe that v(z) is continuous for any z C with jzj R>. Moreover, by assigning v() =by continuity, wehave thatv(z) is harmonic in C n D R for R suciently large. It follows by the mean value property that v(re i )d = v() = r>r: On the other hand, d F (z) jzjfor any z C and d F (re i ) >rfor some [ ), because F has other points beside z =. This implies that v(re i )d > which is a direct contradiction. We now turn to the proof of (5.3) in the case F = G, where G is a bounded domain. Again we assume to the contrary that supp \ G =, which implies that u(z) is harmonic in G by (5.). Let z G and be such that Then, for the harmonic function we have d G (z) =jz ; z j: w(t) :=logjt ; z j;log d G (t) t G w(z) = and w(t) t F: Consequently, w(t) attains its maximum at t = z G, sothatw(t) ing by the maximum principle (see Theorem..8 in [, p. 6]). But d G (t) =jt ; z j is clearly impossible for t! z, t G. t G Lemma 5.. (Bernstein-Walsh) Let C be a compact set, cap() >, with the unbounded component of C n denoted by. Then, for any polynomial p(z) of degree n, we have (5.6) jp(z)j kpk e ng(z ) z C

14 3984 IGOR. PRITSKR where g (z ) is the Green function of, with pole at, satisfying (5.7) g (z ) =log cap() + log jz ; tjd (t) z C : This is a well known result about the upper bound (5.6) for the growth of p(z) o the set (see [, p. 56], for example). The representation (5.7) for g (z ) is also classical (cf. Theorem III.37 in [4, p. 8]). Consider the n-th Fekete points fa k n g n k= for a compact set C (cf. [, p. 5]). Let (5.8) F n (z) := ny k= (z ; a k n ) be the Fekete polynomial of degree n, and dene the normalized counting measures in Fekete points by (5.9) n := n nx k= ak n n N: Lemma 5.3. For a compact set C, cap() >, we have that (5.) and (5.) n! kf nk =n =cap() n! as n!: quation (5.) is standard (see Theorems and 5.5. in [, pp ]), while (5.) follows from (5.) and Theorem. of [6]. Proof of Theorem.. First we show that the best constant M in (.) is at most the right-hand side of (.3). We give two proofs of this fact. The rst one is a generalization of the proof of Theorem in [9], whose by-product is needed in Section 4. Clearly, it is sucient to prove an inequality of the type (.) for monic polynomials only. Thus, we assume that p k (z) k m, are all monic, so that p(z) is monic too. (i) For any k = ::: m, there exists c such that (5.) kp k k = jp k (c k )j: Applying Lemma 5. (or Lemma of [9]) to the set F = fc k g m k=,we obtain that the function (5.3) is subharmonic in C, and that (5.4) u m (z) := max km log jz ; c kj u m (z) = log jz ; tjd m (t) z C z C where m is a probability measure on C.If k is the set of zeros of p k (z) (counted according to multiplicities), k = ::: m,then mx k= (5.5) log kp k k = = mx mx X mx X log jp k (c k )j = log jc k ; zj u m (z) k= k= z k k= z k X z S log jz ; tjd m (t) = log jp(t)jd m (t): m k= k

15 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3985 Using (5.6) and (5.7) of Lemma 5., we proceed further as follows: mx k= log kp k k log kpk + n log cap() + n = log kpk + n log = log kpk + n log = log kpk + n log cap() + n cap() + n cap() + n log jz ; tjd (z) d m (t) log jz ; tjd (z)d m (t) log jz ; tjd m (t)d (z) u m (z)d (z) where we changed the order of integration by Fubini's theorem. It follows from the above estimate that my R exp( um n (z)d (z)) (5.6) kp k k kpk : cap() k= Since u m (z) log d (z) forany z C, we immediately obtain that (5.7) exp(r log d (z)d (z)) M : cap() Note that we could proceed in a more direct way toprove (5.7), which isdone below. However, the inequality (5.6) is needed for our analysis in Section 4. (ii) Let fz k n g n k= be the zeros of p(z) and let n be the normalized zero counting measure for p(z). Then, we use (5.) with F =, Fubini's theorem and Lemma 5. in the following estimate: Q m n log k= kp kk kpk Q n n log k= kz ; z k nk kpk = log kpk =n + log d (z)d n (z) = log + log kpk =n jz ; tjd n (z)d(t) = log jp(t)j=n d(t) kpk =n g (t )d(t) = log cap() + log jz ; tjd(t)d (z) = log cap() + log d (z)d (z): This completes the second proof of (5.7). (iii) To show that equality holds in (5.7), we consider the n-th Fekete points fa k n g n k= n N, for, and dene the Fekete polynomials as in (5.8): Observe that F n (z) = ny k= (z ; a k n ) n N: kz ; a k n k = d (a k n ) k n n N:

16 3986 IGOR. PRITSKR Since cap() 6=, the set consists of more than one point and, therefore, d (z) is a strictly positive continuous function in C. Consequently, logd (z) isalsocontinuous in C, and we obtain by (5.) of Lemma 5.3 that (5.8) n! = exp ny k= n!! =n kz ; a k n k = exp n! n log d (z)d n (z) = exp Finally, we have from the above and (5.) that Q n ( k= M kz ; a k nk ) =n n! kf n k =n nx k= log d (a k n ) log d (z)d (z)! exp(r log d (z)d (z)) = : cap() : Proof of Corollary.. Since is a bounded continuum, we obtain from Theorem 5.3.(a) of [, p. 38] that cap() diam() : 4 Thus, the corollary follows by combining this estimate with the obvious inequality d (z) diam() and by using that (C )= supp : z Proof of Theorem.3. Let D R = fz : jzj <Rg be a disk containing and all zeros of the sequence fq n (z)g n=, where R> is suciently large. Note that! Y n =n kq n k =n kz ; z k n k R n N k= where fz k n g n k= are the zeros of Q n(z), as before. Consider a subsequence N N such that (see (.7)) (5.9) supkq n k =n n! = n! kq n k =n =: C cap(): nn Since the normalized counting measures n for Q n (z), dened in (.), are supported on D R for any n N, wehave by Helley's theorem (see []) that there exists a subsequence N N satisfying (5.3) n! as n! n N N for a probability measure supported on D R. Using the continuity oflogd (z) in C,we obtain from (5.3) that (5.3) n! nn ny k= = exp n! nn kz ; z k n k! =n = n! log d (z)d n (z) nn exp = exp n nx k= log d (z k n ) log d (z)d(z) :!

17 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3987 Recall that fq n (z)g n= is a sequence of extremal polynomials, so that exp(r log d (z)d(z)) M = C by (5.9) and (5.3). Hence we have by (.3) that log C + log d (z)d(z) =log cap() + log d (z)d (z): The integral representation (5.) for log d (z) andfubini's theorem give log C + log jz ; tjd(z)d(t) = log cap() + log jz ; tjd (z)d(t) and (5.3) log C + = log jz ; tjd(z) d(t) log cap() + log jz ; tjd (z) On the other hand, we have by Lemma 5. that log + log kq n k =n jz ; tjd n (z) g (t ) =log cap() + log jz ; tjd (z) t C : d(t): Since supp n D R for any n N,we can pass to the it in the above inequality, as n! nn and obtain by (5.9), (5.3) and the Lower nvelope Theorem (see Theorem I.6.9 in []) that log C + (5.33) log jz ; tjd(z) log cap() + log jz ; tjd (z) holds quasi everywhere (q.e.) in C i.e., with a possible exception of a set of zero capacity. Observe that supp D R and supp D R, and that (5.33) holds - almost everywhere, because has nite energy. This implies that (5.33) holds for any t C by the Principle of Domination (see Theorem II.3. in []). Furthermore, the strict inequality is impossible in (5.33) for any t (C n D R ) \ supp, asthis would immediately violate (5.3), because both functions on the left and on the right of (5.33) are harmonic and continuous in C n D R.Thus, we have that (5.34) log C + log jz ; tjd(z) = log cap() + log jz ; tjd (z) for any t (C n D R ) \ supp. It follows from (5.34) and Theorem 3.. of [, p. 53] that log cap() = log jz ; tjd (z) ; log jz ; tjd(z) = C t! tsupp where we can pass to the above it because supp is unbounded by Lemma 5.. Since (5.9) holds with C = cap() by the above proof, then (.9) now follows from (.7).

18 3988 IGOR. PRITSKR Proof of Theorem.4. Note that any of the assumptions (i) or (ii) implies that cap() > : Assume that fq n (z)g n= is a sequence of monic asymptotically extremal polynomials. On taking log of (.5) and using (.3), we have! nx log kz ; z k n k + log n! n kq n k or n! k= = log d (z)d (z) + log log d (z)d n (z)+ n log kq n k = log d (z)d (z) + log cap() cap(): It follows from (5.)-(5.) and Fubini's theorem that log jz ; tjd n (z)+ n! n log d(t) kq n k = log jz ; tjd (z) + log d(t) = g (t )d(t) cap() where we used (5.7) on the last step. The above equation can be also written as (5.35) n! Recall that, for any n N (5.36) n log jq n(t)j kq n k ; g (t ) n log jq n(t)j kq n k ; g (t ) D r(z) d(t) =: t C by (5.6). Hence, for any disk D r (z) :=ft : jz ; tj <rg z supp, wehave inf n! n log jq n(t)j ; g (t ) d(t) kq n k as the opposite inequality would violate (5.35), because of (5.36). It follows that inf sup n! n log jq n(t)j (5.37) ; g (t ) z supp: kq n k D r(z)\supp Note that for the polynomials P n (z) :=Q n (z)=kq n k, n N, wehave (5.38) kp n k = n N: If has empty interior and connected and regular complement, then (.) now follows from Theorem of [4], (5.37) and (5.38). This proves Theorem.4 in the case (i). However, the case (ii) requires an additional argument to show that (5.39) n! n(b) = for any compact set B C n. Indeed, if we apply Theorem of [4] tothe sequence fp n (z)g n= on ~ := C n in this case, then (.) is implied by (5.37), (5.38) and (5.39). Thus, our current goal is to prove (5.39). Observe that the open

19 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3989 set ~ G := C n is a simply connected domain, which contains the original domain G. Clearly, ~ G is the interior of ~. Note that g (t )= t ~ G by (5.7) and Theorem III.4 of [4, p. 6]. Lemma 5. gives by (5.3) that there exists a disk D r () G G ~ such thats := Dr () \ supp 6=, supp, and that (5.4) inf n! sup ts n log jq n(t)j kq n k by (5.37). One can see that (5.4) holds with S replaced by any closed disk K G ~ such thatk \ S =. Indeed, if this is not true, then we dene a harmonic function u(z) in G ~ n K, with boundary values u(t) = 8 < : ~ G inf n! sup tk n log jq n(t)j < kq n k Using the maximum principle for subharmonic functions, we obtain inf n! sup zs n log jq n(z)j kq n k sup u(z) < n! zs inf which contradicts (5.4). Thus, we can consider any compact set B ~ G assuming for the proof of (5.39) that B \ S =. Dene the subharmonic in ~ G function h n (z) := n log jq n(z)j kq n k + n X z k nb g ~G (z z k n ) z ~ G where the sum is over all zeros z k n of Q n (z) inb, and where g ~G (z z k n )isthe Green function of ~ G with pole at z k n. It follows by the maximum principle that (5.4) sup n! sup zs h n (z) sup n! sup h n (z) = z@ G ~ because g ~G (z z k n G,as ~ G ~ is simply connected and regular. We continue by estimating X A sup g ~G (z z k n ) n! zs n z k nb = sup inf h n (z) ; n! zs n log jq n(z)j kq n k sup sup h n (z) ; sup n! zs zs n log jq n(z)j sup n! sup zs h n (z) ; inf n! sup zs kq n k n log jq n(z)j kq n k where the last inequality follows from (5.4) and (5.4). Hence, (5.4) inf n! n X z k nb g ~G (z z k n ) A =:

20 399 IGOR. PRITSKR Since B \ S =, wehave This inequality and (5.4) yield = n! inf n a := inf zs inf tb g ~ G (z t) > : X z k nb g ~G (z z k n ) A a sup n(b) n! so that (5.39) is proved. Thus, the proof of the case (ii) is also completed. Proof of Theorem.5. Let D R := fz : jzj <Rg be large enough, so that D R. Since (.) is valid, we have that there are only o(n) zeros of Q n (z) outside of D R,asn!. Assume that Q n (z) is monic for all n N and dene a monic polynomial ~ Qn (z), whose zeros are the zeros of Q n (z) contained in D R.We denote the set of zeros of ~ Q n (z) by ~ n and consider the zero counting measures for ~ Q n (z): It is clear from (.) that X n := zk n n N: n z k n n ~ (5.43) n! as n!: Observe that supp n D R nn, Y =n kz ; z k n k A = X (5.44) log d (z k n ) n! n! n z k n ~ z k n ~ = exp log d (z)d n (z) =exp log d (z)d (z) n! by the weak* convergence in (5.43). Moreover, Theorem. of [6] and (5.43) imply that (5.45) k Q ~ n k =n n! = cap() because is regular by our assumptions and all zeros of ~ Qn (z) are uniformly bounded. Since the norm of product is at most the product of norms, we have inf n! Q n k= kz ; z =n k nk kq n k inf n! A Q! =n z k n n ~ kz ; z k n k k Qn ~ k exp(r log d (z)d (z)) = cap() by (5.44) and (5.45). The opposite inequality follows from Theorem.: sup n! Q n k= kz ; z =n k nk kq n k exp(r log d (z)d (z)) : cap() Hence, (.5) holds true for the sequence fq n (z)g n= and the proof is nished.

21 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 399 Proof of Corollary 3.3. Observe thatany Borel set B has two images + (B) and ; (B) on the unit circle. Thus, we have for the equilibrium measure that (B) =m( + (B)) + m( ; (B)) where dm = jdwj=() is the normalized arclength on fjwj = g, by (3.4). It follows that = ; j + (e i )j + j ;(e i )j d e i where one nds by a straightforward calculation that j (e i )j = j (e i ) + sin(=4)j p e i : (cos ; cos(=)) Hence, (3.) is proved. If then d (e i )=je i ; e ;i= j = sin(=+=4) However, if then d (e i sin(=+=4) )= =: ; = ; = =: Using the symmetry of, we obtain (3.) from (.3), (3.7), (3.) and the above formulae for d (e i ). The second part of Corollary 3.3 follows from Theorems.4 and.5. Proof of Theorem 4.. We shall adjust part (iii) of the proof of Theorem., to show that (4.5)-(4.6) are true. For the n-th Fekete points fa k n g n n N, consider the Fekete polynomials (see (5.8)) F n (z) = ny k= (z ; a k n ) n N: A proper factoring in (4.5) can be achieved by grouping Fekete points as follows. We dene a subset F l n fa k n g n k= associated with each point l l = ::: s so that a k n F l n if (5.46) d (a k n )=ja k n ; l j k n: In the case that (5.46) holds for more than one l,we refer a k n to only one set F l n,toavoid an overlap of these sets. It is clear from (4.4) that, for any n N, s[ l= F l n = fa k n g n k= and F l n \F l n = l 6= l : The desired factors of F n (z) in (4.5) are dened as follows: (5.47) F l n (z) := Y a k nf l n (z ; a k n ) l = ::: s: If m>sthen we letf l n (z) forl = s + ::: m: Using (5.46), we obtain that kf l n k = Y a k nf l n j l ; a k n j = Y a k nf l n d (a k n ) l = ::: s

22 399 IGOR. PRITSKR which gives by (5.8) that n! my l= kf l n k! =n = n! ny k= =n d (a k n )! = exp log d (z)d (z) : Hence, (4.6) follows by combining (5.) with the above equation. References [] G. Aumann, Satz uber das Verhalten von Polynomen auf Kontinuen, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. (933), [] V. Avanissian and M. Mignotte, A variant of an inequality of Gel'fond and Mahler, Bull. London Math. Soc. 6 (994), MR 94j:3 [3] B. Beauzamy,. Bombieri, P. no and H. L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (99), [4] B. Beauzamy andp. no, stimations de produits de polyn^omes, J. Number Theory (985), MR 9m:5 [5] C. Benitez, Y. Sarantopoulos and A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 4 (998), 395{48. MR 99h:4677 [6] H.-P. Blatt,. B. Sa and M. Simkani, Jentzsch-Szeg}o type theorems for the zeros of best approximants, J. London Math. Soc. 38 (988), MR 9a:34 [7] P. B. Borwein, xact inequalities for the norms of factors of polynomials, Can. J. Math. 46 (994), MR 95k:65 [8] D. W. Boyd, Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39 (99), MR 94a:6 [9] D. W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 6 (994), MR 95m:38 [] J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York, 984. MR 85k:3 [] G. Faber, Uber Tschebyschesche Polynome, J. Reine Angew. Math. 5 (9), [] M. Fekete and J. L. Walsh, On the asymptotic behavior of polynomials with extremal properties, and of their zeros, J. Analyse Math. 4 (955), MR 7:354f [3] A. O. Gel'fond, Transcendental and Algebraic Numbers, Dover, New York, 96. MR :598 [4] R. Grothmann, On the zeros of sequences of polynomials, J. Approx. Theory 6 (99), MR 9h:46 [5] H. Kneser, Das Maximum des Produkts zweies Polynome, Sitz. Preuss. Akad. Wiss. Phys.- Math. Kl. (934), [6] A. Kroo and I.. Pritsker, A sharp version of Mahler's inequality for products of polynomials, Bull. London Math. Soc. 3 (999), MR 99m:38 [7] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 97. MR 5:5 [8] K. Mahler, An application of Jensen's formula to polynomials, Mathematika 7 (96), 98-. MR 3:A779 [9] K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (96), MR 5:36 [] H. N. Mhaskar and. B. Sa, The distribution of zeros of asymptotically extremal polynomials, J. Approx. Theory 65 (99), 79{3. MR 9d:35 [] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 995. MR 96e:3 []. B. Sa and V. Totik, Logarithmic Potentials with xternal Fields, Springer-Verlag, Heidelberg, 997. MR 99h:3 [3] H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, New York, 99. MR 93d:49 [4] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, 975. MR 54:99

23 PRODUCTS OF POLYNOMIALS IN UNIFORM NORMS 3993 [5] H. Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 6 (967), MR 35:346 [6] H. Widom, xtremal polynomials associated with a system of curves in the complex plane, Adv. Math. 3 (969), 7-3. MR 39:48 Department of Mathematics, 4 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma mail address: igor@math.okstate.edu