UPB Sci Bull, Series A, Vol 7, No 3, 8 ISSN 3-77 SZEGO S THEOREM STARTING FROM JENSEN S THEOREM Cǎli Alexe MUREŞAN Mai îtâi vo itroduce Teorea lui Jese şi uele coseciţe ale sale petru deteriarea uǎrului zerourilor uei fucţii aalitice î plaul coplex î iteriorul discului D (; r ) Apoi vo prezeta Teorea lui Szego şi vo deteria oi evaluǎri asupra uǎrului rǎdǎciilor reale ale uui polio cu coeficieţi coplecşi Firstly, we will itroduce Jese s theore ad soe useful cosequeces for givig the ubers of the zeros to the aalytical coplex fuctios iside the ope disc D (; r ) The, we will preset Szego s Theore ad we will get ew evaluatio about the uber of the real roots of a coplex polyoial Keywords: The uber of real roots, Jese's equality, Szego s theore Itroductio There are ay theores about the ubers of the real roots to the coplex polyoials Soe of the use, specially the Cauchy s theore i coplex plae ad the others use, specially Jese theore as we ca see i chapter 3 If P( x) = ax + + ax+ a C[ x],, a a, the legth of P is deoted by L( P ) = ai ad we deoted with t the uber of real roots of P, repeated accordig to their ultiplicity, i the first class we see for exaple Theore 33 : t ( t + ) < 4( + ) l[ ] () aa Our theore, use Jese s theore ad follow a ISchur ethod, see [] to Refereces ad our result are: Theore 3: LP ( ) LP ( ) t l + l () Assist, Matehatics Departet, Uiversity Petrol si Gaze of Ploiesti, ROMANIA
4 Alexe Căli Mureşa Jese s Equality ad its Applicatios Theore Jese s equality: Be it P(x) a aalytic fuctio i a regio which cotais the closed disk D(; r ) ; r>, i the coplex plae, if, x, x,, x C, xi < r,( ) i =,, are the zeros of P i the iterior of D(; r ) repeated accordig to their ultiplicity ad if P(), the: R i l P() θ l( ) l P( R e ) d or: x θ j j π = (3) i l R θ l P( R e ) dθ l P() l( xj ) + j = (4) Proof: see [] or [3] fro Refereces Reark This forula establishes a coectio betwee the absolute values of the zeros of the fuctio P iside the disk z <R ad the values of P(z) o the circle z = R, ad ca be see as a geeralizatio of the ea value property of haroic fuctios Corrolary Be it P(x) a aalytic fuctio i a regio which cotais the closed disk D (; ) i the coplex plae, is the uber of all zeros of P ad if, for s, x, x,, x s, xi <, ( ) i =, s, are the zeros of P i the iterior of D (; ) repeated accordig to ultiplicity, the: a) i Pz ( ) < P() (5) z= iθ b) i{ F( z) } l F( e ) dθ ax{ F( z) } z (6) = z = Proof: a) I Theore Be it R= The: s = iθ l P( e ) dθ l P() l( xi ), + j = s s iθ But xi < l( xi ) < So l Pe ( ) dθ l P() <,
Szego's theore startig fro Jese's theore 43 π l i Pz ( ) l P (), < z = i Pz ( ) < P() (7) z = b) We ca try by siillarity or see [4] to Refereces Corrolary If P( x) = ax + a x + + ax+ a, ai C; i =,, ad R =, R >, where x, x,, x are the roots repeated accordig to a ultiplicity of P(x) ad a, a, the: iθ l P[ e ] dθ l a P'( z) π a = dz π i = (8) P( z) l[ L( P)] l a LF ( ) z = a For provig see [4] to Refereces Corollary 3 Be it P(x) a aalytic fuctio i a regio which cotais the closed disk D(; r ), r> i the coplex plae, s is the uber of all zeros of P with, x, x,, x s, xi, ( ) i =, s, are the zeros of P i the iterior of D (; ) repeated accordig to ultiplicity, the: Pz ( ) l ax z = r P() s < (9) l r Proof: If s, x, x, xs, xs+, x C, xi < r,( ) i =,, are the zeros of P i the iterior of D(; r ) repeated accordig to their ultiplicity The fro Jese Theore we have: i l r = θ l P( r e ) dθ l P() l( xj ) + j = Because for, s, xi, we have s s xi l( xi ) l( xi ) < l r = ( s) l r s+
44 Alexe Căli Mureşa iθ Now we ca write: l r ( s) l r < l P( r e ) dθ l P() Ad ext P() = a, ad l ( i π θ P r e ) dθ < l[ax P( z) ] dθ = l[ax P( z) ], z = r z = r Therefore, fro the previous relatios we have: s l( r ) l[ax ( ) ] l P z π a, z = r ad ow: Pz ( ) l ax z = r P() s < l r 3 Szego s Theore Propositio 3 Be it the polyoial - P( x ) = ax + a-x + + a x + a C[ x], with a a ad - d Q ( x ) = x [ P( x) ], N, eaig Q is defied by the relatio dx - dq - ( ) ( x Q ) x = x () dx a) If we deote by α the uber of the real roots, havig absolute value bigger or equal with for the polyoial Q ad by b the uber of the real roots bigger or equal with the b + α i i b) Q( x) = aix () Proof: a) Usig Rolle s Theore Q x x P x Q x x = = ( ) = '( ); ( ) = or P x = '( ) ()
Szego's theore startig fro Jese's theore 45 Fro Rolle s Theore result that: P '( ) x has at least b - real roots havig absolute value bigger or equal with ( ) Q x has at least (b- ) + =(b- ) + = b Fro () ad ( ) result that ( ) roots It is kow the fact that the degree of Q ( x) = is the degree of P(x) ad we repeat the process It results b + α, with α the uber of roots havig absolute value bigger or equal with for Q( x) = Q ( x) b) Let see ow the expressio to Q( x) = Q ( x) First we take = the = Obtai that: Q( x) = xp'( x) = x ( ax + ( ) a x + + a (3) Q( x ) = a x + a x + + ax + ax ( ) Q( x) = x ax + a x + + ax+, ( ) i i Q( x ) = a x + a x + + a x + a x = aix (4) Now by iductio about we ca obtai: - d i i Qx ( ) = Q( x) = x [ Px ( )] = ax i (5) dx - Theore 3 For P( x ) = ax + a-x + + a x + a C[ x], with a a ; the legth of P is deoted by = ai ad let b the uber of the real roots bigger or equal with The b l (6) Deostratio: Fro previous propositio for a atural ad for the polyoial
46 Alexe Căli Mureşa i i Qx ( ) = Q( x) = ax i we have b + α where α is the uber of the real roots, havig absolute value bigger or equal with for the polyoial Q We cosider revq the reciprocal polyoials of Q( x) : i i i i (7) revq( x) = x Q( x ) = a i x = a i x If Q(x) = with x revq = ; ad ow we see that: α is the x x uber of the real roots bigger or equal with for Q(x) if ad oly if α is the uber of the real roots sall or equal with for revq ( x ) Be it r R; r > Fro Jese s iequality results that the total uber of the roots (coplex) of the polyoial revq fro z are deliitated by the quatity of the total uber of the roots (coplex) of the polyoial revq fro z r ad takig i Jese forula oly the roots with oduly at ost equally to oe we obtai fro Corollary 3: revq( z) l ax z = r revq() α < (8) l r We pick / j ax { ( ); } ( j r = e revq z z = r a j e ) j= (9) Ad if we ote g( x) = ( x) e x j the for = x, j {,,, } we will have: x j ( j ) ( x) e = e () x x g '( x) = ( x) ( ) e + ( x) e x g' ( x) = ( x) e ( x) () g ' x for x ad because of that The ( )
Szego's theore startig fro Jese's theore 47 We obtai j g g( ) ( ) j {,,, } () j ( j e ) = (3) Fro (9) ad (3) we have: j ax { ( ); ( j revq z z = r ) } a j e a j = L( P ) j= (4) j= l a L( P) Because revq() = a fro (8) relatio we obtai: α < α < l a L( P) L( P) The b + α b< + l Be it = l + a It results b < l + + l a l + l a b< l + +, l + (5) b< l + + l LP ( ) Ad because b is atural fro the last relatio we obtai: b l Theore 3 For - - P( x ) = ax + a x + + a x + a C[ x], with a a ;
48 Alexe Căli Mureşa the legth of P is deoted by = a ad let be t the uber of all the real roots of P The: t LP ( ) LP ( ) l + l (6) Deostratio: a) Be it b the uber of the real roots bigger or equal with The s=t-b is the uber of the real roots saller tha We ca observe that s is the uber of real roots bigger or equal with for the reciprocal polyoial - revp = x P = ax + ax + + a x We have deostrated i previous theore that b LP ( ) - l for P = ax + a-x + + a a (7) Usig a siilar ethod we have: s LrevP ( ) l a ( ) But L( revp) = L( P), the s l LP a The: t = b+ s LP ( ) LP ( ) l + l a a (8) ad the relatio was proved Corrolary 3 For P( x ) = a x + a - x + + a x + a C[ x], with a a with legth of P, roots of P The: i - ai = ad t the uber of all the real t 8 l ( ) L P aa (9)
Szego's theore startig fro Jese's theore 49 LP ( ) LP ( ) Proof: Fro the last Theore we have: t l + l i j N We use the relatio, ( ),, we ca write: ( ) i + j i+ j = i + j + ij i + j + ( i + j ) LP ( ) The t l + l l + l, a a a a [ LP] ( ) t 4 l, t 8 l (3) a a a - Corrolary 3 For P( x ) = ax + a-x + + a x + a C[ x] with a a, L( P ) = ai a polyoial which have at least oe real root bigger or equal to oe ad at least oe root saller tha oe, with legth of P, ad t the uber of all the real roots of P The: L( P) t 4 l (3) a Proof: Be it b the uber of the real roots bigger or equal with ad s=t-b the LP ( ) uber of the real roots saller tha, we have obtaied: b l, a s l a Fro hypotesis b,< s the b b, s< s The we have: LP ( ) LP ( ) L( P) t b+ s b + s l + l = 4 l (3) a a a Theore 33 (G Szegö) If P( x) = ax + + ax+ a C[ x],, a a, the if we oted with t the uber of real roots of P we have:
5 Alexe Căli Mureşa t ( t+ ) < 4( + ) l[ ] a a (33) For provig see [5] Corrolary 33 If Px ( ) = ax + + ax + a Zx [ ],, a a, for t, the uber of real roots of P, we have: t < 4( + )l[ L( P)] (34) Deostratio: a, a ad a, a Z The a, a ad L( P) >, a a, l > Also it is easy to prove that: > ad l[ a a a a ] > Now fro Theore 3 (G Szegö) we have: t < t ( t + ) < 4( + ) l[ ] aa (35) The because: for, ad a a so ad a a l[ ] l[ ], the relatio becoe: t < 4 ( + ) l[ L( P)] a a Reark 3 Deostratios for G Szegö s theores we ca see i [], [5], [6] The author follow a ethod fro [] ad give the ew relatios as we ca see i Theore 3 ad Theore 3 R E F E R E N C E S [] M Migotte, Itroductio to Coputatioal Algebra ad Liear Prograig, Ed Uiv Bucuresti,, p36-4 [] L V Ahlfors, Coplex Alysis, McGraw-Hill Book Copay, Secod Editio,979, p 5- ;34-37, 39-43 [3] V V Prasolov, Polyoials, Moscow Ceter for Cotiuous Math Educatio, -4, p6-7 [4] AC Mureşa The approxiatio of a Polyoial Measure with Applicatio towards Jese s Theore Geeral Matheatics vol 5 o4, 7, Ed by Departet of Matheatics of the Uiversity Lucia Blaga, Sibiu, Roaia [5] G Szego, Orthogoal Polyoials, Aer Math SocColloq Publ Volue XXIII, New York,959 [6] A Bloch, G Polya, O the roots of certai algebraic equatio, Proc Lodo Math Soc, t 33, 93, p 9-4