On The Eneström-Kakeya Theorem

Transcription

1 Applied Mahemaics,, 3, doi:436/am673 Published Olie December (hp://wwwscirporg/oural/am) O The Eesröm-Kakeya Theorem Absrac Gulsha Sigh, Wali Mohammad Shah Bharahiar Uiversiy, Coimbaore, Idia Deparme of Mahemaics, Kashmir Uiversiy, Sriagar, Idia gulshasigh@rediffmailcom, wmshah@rediffmailcom Received Augus 9, ; revised November 7, ; acceped November, I his paper, we prove some geeraliaios of resuls cocerig he Eesröm-Kakeya heorem The resuls obaied cosiderably improve he bouds by relaxig he hypohesis i some cases Keywords: Polyomial, Zeros, Eesröm-Kakeya heorem Iroducio ad Saeme of Resuls The followig resul due o Eesröm ad Kakeya [] is well kow i he heory of disribuio of he eros of polyomials Theorem A If P := a = is a polyomial of degree such ha a a a a a o, he P does o vaish i > I he lieraure, [-8], here exis exesios ad geeraliaios of Eesröm-Kakeya heorem Joyal, Labelle ad Rahma [9] exeded his heorem o a polyomial whose coefficies are moooic bu o ecessarily o egaive by provig he followig resul Theorem B Le P := a = be a polyomial of degree such ha a a a a a he all he eros of P ( ) lie i, a a a a Dewa ad Bidkham [] geeralied Theorem B ad proved he followig: Theorem C Le P := a = be a polyomial of degree such ha for some > ad <, a, a a a a a he P has all he eros i he circle a a a a a By usig Schwar's Lemma, Ai ad Mohammad [] geeralied Eesröm-Kakeya heorem i a differe way ad proved he followig: Theorem D Le P := a = be a polyomial of degree wih real posiive coefficies If > ca be foud such ha a a a, for r =,,, r r r a = a =, he all he eros of P lie i Ai ad Zargar [] also relaxed he hypohesis of Eesröm-Kakeya heorem i a differe way ad proved he followig resul Theorem E Le P := = of degree such ha for some K, Ka a a a a he all he eros of a be a polyomial >, P lie i K K While sudyig Theorem E, a aural quesio arises ha wha happes if we relax he hypohesis of Theorem D i a similar way ad oly assume ha a r arar, for r =,3,, I his paper, we sudy such a case ad prove a more geeral resul from which may kow resuls follow o a fairly uiform procedure Ifac we prove: Theorem Le P := = be a polyomial of degree such ha = a ib a ad b,,, are real umbers ad if > ca be foud such ha for r =,3,, Copyrigh SciRes

2 556 G SINGH ET AL ad for some K, a a a r r r, b b b r r r, Ka he all he eros of R, ( ) a Kb ( ) b,, P lie i K R K a b a b a b = a b a a b b a b The followig ieresig resul immediaely follows from Theorem, if we assume ha all he coefficies of he polyomial P are real Corollary Le P := a = be a polyomial of degree wih real coefficies If > ca be foud such ha r r r, a a a for r =,3,, ad for some K, he all he eros of R *, P lie i K Ka a, R Ka a a a * = a a a a Remark If we assume ha all he coefficies of P are real ad posiive, he for K =, Corollary saisfies he saeme of Theorem D ad a simple calculaio shows ha i his case also all he eros of P lie i Nex, if i he Theorem, we ake = ad assume ha coefficies o be real, we ge he followig: Corollary Le P := a = be a polyomial of degree wih real coefficies If for some > ad K, Ka a a a > a, he all he eros of P lie i a a K Ka a Remark If we pu = i Corollary, we ge he resul due o Ai ad Zargar [] ad for =, K =, Corollary reduces o Theorem B We ex prove he followig more geeral resul which is of idepede ieres Theorem Le P := = be a polyomial of degree such ha = a ib a ad b,,,,, are real umbers If > ca be foud such ha for r,3,, a a a r r r, b b b r r r, ad for some real umbers u ad v, u, v ua a, vb b, he all he eros of P () lie i ua ivb R, R = ua vba b ab ab a a b b a b } If i Theorem, we ake a b u = ad v =, a b so ha u, v, we ge he followig: Corollary 3 Le P := = be a polyomial of degree such ha = a ib a ad b,,,,, are real umbers If > ca be foud such ha a a ) a, for r =,3,, r r( r a a ( ) a, for r = + r r r b b ) b, for r =,3,, r r( r Copyrigh SciRes

3 G SINGH ET AL 557 b b ( ) b, for r = +, r r r he all he eros of P () lie i R *, R a b a b a b * = a b a a b b a b I paricular, if a a a, for r,,, he r r r r r r a a a, for r, r r r b b b, for r,,, r r r b b b, for r, a a, b b ad we ge i his case all he eros of P lie i a b a b Remark 3 A resul of Shah ad Lima [7, Theorem ] is a special case of Corollary 3, if we assume ha all he coefficies of P are real The followig resul also follows from Theorem, if we assume ha = ad = ] Corollary 4 Le P := = be a polyomial of degree such ha = a ib a ad b,,,,, are real umbers If for some u ad v, he all he eros of ua a a, vb b b, P lie i ua ivb ua vb May oher kow resuls ad geeraliaios similarly follows from Theorem wih suiable subsiuios We leave his o he readers Proofs of he Theorems Proof of Theorem Cosider he polyomial = f P This gives = () = K K () = K Ka a a a a a a a a a a i Kb b ( ( ) ) b b b b b b b b b f K Ka a a a a a a a a a a Kb b b b b b b b b b b Copyrigh SciRes

4 558 G SINGH ET AL = K Ka a Kb b a a a b b b a a a b b b a a b b a b For >, we have by usig hypohesis f K Kaa Kbb a aa b bb a a a b b b a a b b a b >, if K Ka Kb a b a b a b > a a b b a b Therefore, for, f >, if K > Ka b a b a b a b a a b b a b Hece all he eros of f whose modulus is greaer ha lie i he circle K Ka b a b a b a b a a b b a b Sice all he eros whose modulus is less ha already lie i his circle, we coclude ha all he eros of herefore P lies i K Ka b a b a b a b a a b b a b This complees he proof of he Theorem Proof of Theorem Cosider he polyomial f ad Copyrigh SciRes

5 G SINGH ET AL 559 = f P = = a a a a a a a a a a a i b b b b b b b b b b b = u a ua a a a a a a a a a a i v b vb b b b b b b b b b b = ua ivb ua a a a a ( a a( ) a) a a a i vb b b b b b b b b b b This gives ua ivb f ua a a a a a a a a a a vb b b b b b b b b b b ua ivb = ua a vb b a a a b b b a a b b a b For >, we have ua ivb f ua a vb b a a a b b b a a b b a b By usig hypohesis, his gives ua ivb f ua vb a b ab a b Copyrigh SciRes

6 56 G SINGH ET AL if a a b b a b >, ua ivb > ua vb a b a b a b Hece all he eros of a a b b a b f whose modulus is greaer ha lie i he circle ua ivb ua vb a b a b a b a a b b a b Sice all he eros whose modulus is less ha already lie i his circle, we coclude ha all he eros of herefore P lies i ua ivb R, R = ua vba b ab a b a a b b a b This proves Theorem compleely f ad 3 Ackowledgemes The auhors are graeful o he referee for useful suggesios 4 Refereces [] M Marde, Geomery of Polyomials, d Ediio, America Mahemaical Sociey, Providece, 966 [] E Egervary, O a Geeraliaio of a Theorem of Kakeya, Aca Mahemaica Scieia, Vol 5, 93, pp 78-8 [3] N K Govil ad Q I Rahma, O he Eesröm-Kakeya Theorem II, Tohoku Mahemaical Joural, Vol, 968, pp 6-36 [4] W M Shah ad A Lima, O he Zeros of a Cerai Class of Polyomials ad Relaed Aalyic Fucios, Mahemaicka Balkaicka, New Series, Vol 9, No 3-4, 5, pp [5] W M Shah, A Lima ad Shamim Ahmad Bha, O he Eesröm-Kakeya Theorem, Ieraioal Joural of Mahemaical Sciece, Vol7, No -, 8, pp - [6] G V Milovaovic, D S Miriovic ad Th M Rassias, Topics i Polyomials, Exremal Properies, Iequaliies ad Zeros, World Scieific Publishig Compay, Sigapore, 994 [7] Q I Rahma ad G Schmeisser, Aalyic Theory of Polyomials, Oxford Uiversiy Press, Oxford, [8] T Sheil-Small, Complex Polyomials, Cambridge Uiversiy Press, Cambridge, [9] Joyal, G Labelle ad Q I Rahma, O he Locaio of Zeros of Polyomials, Caadia Mahemaical Bulleio, Vol, 967, pp [] K K Dewa ad M Bidkham, O he Eesröm- Kakeya Theorem, Joural of Mahemaical Aalysis ad Applicaios, Vol 8, 993, pp 9-36 [] A Ai ad Q G Mohammad, Zero-Free Regios for Polyomials ad Some Geeraliaios of Eesröm- Kakeya Theorem, Caadia Mahemaical Bulleio, Vol 7, 984, pp 65-7 [] A Ai ad B A Zargar, Some Exesios of Eesröm- Kakeya Theorem, Glasik Maemaicki, Vol 3, 996, pp Copyrigh SciRes