JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematics and Mathematical Physics Vol. 66B, No. 2, April- June 1962

An Extension of Jensen1s Theorem for the Derivative of a Polynomial and for Infrapolynomials

O . Shishu

(February 20, 1962)

The purpose of t he paper is to genera lize J ensen's theorem on t he zeros of t he derivative of a real polynomia l. Results are first es tablished for i nfrapolynomi als a nd t herefrom derived for expressions of t he form G' (z) and aG(z) + bzG' (z), where G( z) is a (complex) polynomial and a,b are (complex) constant s. An implicit a im the paper t ries to ser ve is to further show how invest igation of infrapoly nomi a ls may be of help to t he classical st udy of the geometrical relation between t he ze ros of a poly nomial and those of its derivativ e (or related polynomials).

1. Let S be a subset or t he (op en) complex pl a ne. An irifrapolyn()mial I 0 11 S is It pol.\' ll omi aI A (z)=z"+ an_,z"- ' + . .. + ao (n;::=: 1) h aving the proper ty : t here docs not exist a pol.vnomial B (z)=z"+ bn_ 1zn - l + ... + bo(;;iI1( z)) such t haL

/B (z) /<i A (z) / wh ellever z eS and A (z) ~ O, (1)

B (z)= O when ever ZeS a nd A (z)= O. (2)

For exa mple [4], if S is infin ite, closed a nd bound ed, t hen for n = 1, 2 , . . . t he 'fch ebychefI poly nomifl1 2

of dcgree n for S is an infrftpolyno mial on S. 2. Another example of an infrapoly nomial is the fol­

lowing [10 , sec. 1, a nd 2 , sec. 10 , p . 100. Of. also 4 , sec. 5]. Let G(z) be a nonconstant polynomial, II, 12, . .. , 1m its distinct zeros (say G(z) =aII~n_ , (z - I,) 1',), and assume that m;::=: 2. Then

A () -(""m )- IG' (z) rrln ( r) Z = L...J" - Ipv G(z) ' - I z- ~ ,

is an injrapolynomiaL on {1' ,12 , ... , 1m },

3. Various r esul ts haNe b een obtained co ncerning tbe location of the zeros o[ infra poly nomi als . The first of th ese was given (implicitly) by L . F ejer [1] and others wer e proved by M. Fekete <md J . von Neu­malln [4] , J. L . Walsh [27], J. L . Walsh and T. S. YIotzkin [31, 11 , 12, 13, 14], M. Fekete [2], M. Marden [9], and O. Shish a [16, 17] . One of the most typ ici:Ll th eor ems in this direction is t he fonowing (a special case or [5], Theorem X and th e cnd or sec. 10, p . 66 . Or. also [4] , Th eorem 1). L et ~ and 1/ be distinct zeros oj an injrapolynomial on a closed and bounded set S. Let L be the p erpendicular bisector oj the segmen t

IO ri g- inally called " extremal polynom ial. " 'J'his con cept was in trod uced by M" . .F'eketc and 1. von Neumaml (4) . '1' 11 (' term " infrapolynomial" is due to Professors '1'. S.l\(oLzkin and J . L . Walsh [12J. Figures in brackets indicate t he litrraturc refrrcnccs ()t the end of this paper.

' This is the un ique polynomial T( z) of the form (*) Z'+C, _IZ.-l+ ... +co sll ch that tho TIlax II 1'(0)1 , Z Oil SJ::; max IIP(z)l, Z on SJ for every pol ynomial P (z) of the form (*).

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(E, 1/)' POI' every z , Let Oz denote the closed clilik havi ng the liegment joining z to i ts mirror image in L as a diameter (0. = [ z} 'if zeL), and let C; be the closure of the compleme11i: oj Oz' Then there exists a point p oj S such that ~ and 1/ belong to Op. There exists also a point q oj S such that ~ and 1/ belong to O~.

4. Tn Th eor em 1 we generalize th e lasL result by substituLing in tbe htter It circle [or the line L.

TlmoREM 1 : Let ~ and 1/ be distinct zero oj A (z), an inJrapolynomial on a cLosed and bounded set S. L et K be a circle whose center I does not belong 10 S , such thai ~ and 1/ are symmetric U.e ., inverse) to each other with respect to K . For every z( ~ n Let ('Z denote the closed disk having the segment joining z to i ts inverse with respect to K as a diameter (('z= { z} 'if zd() , and let ('; be the cLosure oj the complement oj ('z. Then there exists a point p oj S such that ~ and 1/ beLong to Cpo There exists also a point q oj S such that ~ and 1/ belong to O~ .

5. T o pro ve Lhis Lheorem , we eslabli h first Lh e following

LEMMA: L et K: /z- I /= r be a circle and Zo, 6 , ~2, 1/1, 1/2 points such that Zo ~ 1,0</ ~i-I/ < 1',1/ ; i symmetric to ~ ; with respect to J{ (i= 1, 2) and 6 lies on the (open ) segment (h,1/1) ' Then (using notati ons oj Theorem 1):

(a) J} ~2EC;0 ' then

/ZO-~2 //Z0- 1/ 2 /</ZO-~ I //Zo- 1/ d. (3)

(b) Jj ~leCZo then

/ ZO- ~2/ /Z0- 1/ 2 />/Zo- ~d /zo- 1/" .

PROOF OF THE L EMMA: IVe in Lrod uce a polar coord i­ll ftte system. in wbich Lhe pole is I a nd the fixed r a.\· is t he one em anating from I a nd p ass ing through zo ° L et (Pl,CP) , (P2, CP ) be, respectively, polar coo rdi­nates or ~J a nd ~2(0<P I<p2<r) , a nd set a=/zo- I /' For every positive x let

Then ( I ZO- ~ i ll zo - 11 i l )2=F(P i) (1'= 1, 2) . Thus in order to prove t h e Lemma it suffices to show th at if Pl<t< P2, th en F' (t) < O in case ~2e(';0 ' and F' (t» O in case ~l e ('Zo '

Let t b e an arbit rary poin t of the open interval (Pl,P2) and d eno te by T th e poillt with polar cOOI'cli­nates (t ,ip). Set

(5)

where z~ is the inverse of Zo with respect to K . Jj' 6 eC;0' then T is an in terior point of C;o li nd D> O. Similarly, if ~le CZo then T is an interior point of Czo and t herefor e D< O.

VVe h ave

(6)

From (4) , a str aightforward computation y ields

F' (t) = 2a2t - 3 (t 4- r4) - 2at - 2(t 2-r2) (a2+ r2) cos ip,

and in view of (6),

Thus, if' ~2 eC; 0' thell F' (t) < O, and if' ~lECzO' thell F' (t) > 0.

6. PROOF OF TH EOREM 1: If one of the t wo points ~, 11 b elongs to so m e C" so does th e other. Lct ~1 b e t hat one of ~,111~Ting within K , and l et 111 b e the other one. So in order to prove th e first conclusion of TheoI'f'm 1, i t suffices to show th at ~leC, where C= U CZ' Suppose ~l~C, Since C is closed, we can

uS find a poin t ~2 within K and on the open segmen t (h ,111) su ch that ~2iO. Let 11 2 be the inverse of ~2 with respect to K. Consider the polynomial

By conclusion (a) of' th e L emma, for ever y zoeS, (3) holds. Thus (1) and (2) hold , con tradictin g our h y pothesis th at A (z) is an infrapolynomial on S. Similarly one derives the second conclusion of Theorem 1 from (b) of the L emma.

7. The notion of an infr a polynomial was general­ized by J. L . Walsh a nd M. Zedek [34] to include more gcn er al polynomials, a nd further work in th at direction was done by J. L. Walsh [27, 28, 29], M. Fekete and J . L. Walsh [5, 6], M. Zedek [36, 37 , 38], O . Shish a a nd J . L. Walsh [23], and O . Shish a [16, 17, 20,21,22]. (O th er papers related to th e subj ect a r e [3, 10, 11 , 12, 15, 18, 19, 31, 32, 33, 35]). A sp ecial case studied in much detail b y J. L . Walsh [27] is obtained b~- adding to th e proper­ties of B (z) in sec. 1 t h e equ ality bo=ao.

54

Repeating here a particular instance of a general defin ition , we mention t h e fo llowing conven tio n . L et n(22) b e an integer and S a set in t h e (open) complex plan e. An n -th in f'rapol,vnomial on S with resp ect to (O,n) is a polynomial A(z) == :Z;~_oa,z' having t he property: th er e does not exist a polynomial B(z)==:Z;~_ o b pzp(¢o A(z) sat i s fy in g (1) , (2) and the equ ali t ies bo=ao, bn=an.

8. Conesponding to sec. 2, we h ave th e followin g r esul t which is a special casc of a morc general t heorem [23, Thcorem 3]. Let G (z) be a polynomial (G(O) ~O), 1t,\'2, . .. , 1m (m?::, 2) its distinct zeros, and let a and b be arbitrary complex numbers . Then

is an m-th injrapolynomial on {II , 12, ... , 1m} with respect to (O,m).

9. THEOREM 2 : Theorem 1 remains true iJ "irifra­polynomial" is replaced by "n-th (n?::, 2) injra.poly· nomial (¢o O) with reslJect to (O,n)," provided that the center .I oj J{ is the origin.

Indeed , we can use again th e proof of Theorem 1 s ince B (z) d efined there will have th e sam e co­effici ent of zn as A(z) h as, ancl (since 1 ~2 11 2 1 = squRre of' t he r adius of K = lh11d, which implies t h at ~2112 = ~1 111) the co nstan t terms of B (z) and .I1( z) will be equ al.

10. As an application of theorems 1 and 2 we h ave the following

THEOREM 3: I. Let ~ and 11 be distinct zeros oj G I (z) where G (z) is a non constant polynomial. Let ~ and 11 be symmetric with respect to a circle K whose center is not a zero oj G (z). Then there exists a zero p oj G (z) such that (using a notation oj Theorem 1) ~ and 11 belong to 0 11 , There exists also a zero q oj G(z) such that ~ and 11 belong to C~ . II. Let ~ and 11 be distinct zeros oj aG (z)+ bz G' (z) (¢0 0) where G(z) is a polynomial (G (0) ~ O ) and (t and b are complex numbers. Let ~ and 11 be symmetric with respect to a circle K. Then the conclusions oj part I hold, provided that the cerlfer of K is the origin.

PROOF: We set G(z) =aII~n~ l (Z- I ,)P p where I j ~lk whenevcr j ~k and where Pl, ])2, ... , Pm are positive. To prove part I we define .I1(z) as in sec. 2 and ob­ser ve that m?::,2 since G' (Z) vanish es at th e distinct points ~ and 11. We may assume th at G(~)G(11 ) ~ O , for otherwise we can take p= q= ~ or p= q= !) . Thus ~ and 11 a re zeros of .I1(z) , which by sec. 2 is an infra­polynomial on th e set 01' zeros of G(z). The d esired conclusions follow now 1'rom Theorem 1. '1'0 prove part II , wc assume th at m22, as otherwise ~ or 11 is .II and one can take P= q= \l' Define now .I1(z) by (7). As before, wc may also assume that G(~)G(11 ) ~ O. Then ~ and 11 are zeros of A (z), which by scc. 8 is au m-th infrapolynomial on LIb 12, .. ' ,Im } with re­spect to (O,m ). The desired conclusions follow from Theorem 2 .

11. For every z, let r z and r; b e, respectively, Cz

and C; of sec. 3, with L taken as th e real axis. Theorem 3, I genera lizes th e 1'ollowing well known thcorem, enunciated without proof by J . L . "iV. V .

Jen sen [7]lln cl proved b~' J . L. W alsh [24]. L et ~ be a 1wl1l'eal ZeN) (f AI (z ), where A(z) is a l1onCOl1siani polynomial whose coefficients are real. Then there exists (L zero p (d A (z) :such that ~Er p . It was proved also b.\' J. L. Wa,lsh th at there exists a ze ro q of A(z) s uch that ~Er ~. -. 12: The lils t results readily impl~' th e followin g

lJJlllLJn g ("lise of Theorem 3, 1. L et ~ and 71 be distinct zero.'! oj G' (z\ where G(z) is a nonconstant p olynomial. L et L be the perpendicular bisector of the segment a ,71 ). Then (using the notati on of sec. 3) there exists a zero p oj G (z) such that ~ and 71 belong to Cpo There exists also a zero qojG (z) suchthat~and 71belonglo C;. Indeeci , we l11 a~' assum e (usi ng a simple transformcttion ) t h,) t 71 =~. Set G (z)= aII~_l(z-Z,), A (z)= II~'_ l(z-z,)(z -2,) . Th en AI W = (II ~- 1 (z-z,»~ _ ~II~_ l (~-z,)+ II~'_ , ( ~ - z,) (II~ _ l (z- z,»' Z_71 = O. Th e desircd co nclu­siolls follow now 3 from sec. 11. Th ese samc conclu­sions I11'C impli ed also b~T t he res ul ts quoted ill sees. 2 , 3 .

] 3. Thcr e fire l11 a n~' compl emenLs Bnd gene'raliza­tions or J ense n's lheorrm due Lo J . L . " Talsh for which th e r eflder is referrcd Lo th e Ii Leratl1rr [24 25 26 27, 30]. ' , ,

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3 Compare with [51, Sec. 10, p. 6u.

55

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(Paper 66B2- 72 )