Generalisation on the Zeros of a Family of Complex Polynomials

Transcription

1 Ieol Joul of hemcs esech. ISSN Volume 6 Numbe Ieol esech Publco House h:// Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd- Delh NC Id. Eml: ehvd3@gml.com Deme of hemcs Abmch Uvesy Abmch Eho. Absc I hs e we ove some esuls o he loco of zeos of ce clss of olyomls. These esuls geelze some ow esuls he heoy of he dsbuo of zeos of olyomls. Hece ou esul wll cosdebly move he bouds by elxg d weeg he hyohess some cses. Hee we ob ce geelzos d efemes of well ow Eesom Key Theoem fo olyoml ude much less escos o s coeffces. Keywods: Polyomls Zeos Eesom-Key heoem fucos Alyc INTODUCTION y esuls o he loco of zeos of olyomls e vlble he leue. Amog hem he Eesom-Key heoem [4] gve below s well ow he heoy of zeo dsbuo of olyomls. Theoem. Fo h-ode olyoml Pz = z ssume > The Pz hs ll s zeos he ds z. I he leue [-8] dvese ems hve bee mde fo geelzg he Eesom-Key heoem o olyomls d lyc fucos. ecely Choo[5] lso oved he followg heoems:

2 94 Aee sgh Neh d S.K.Shu Theoem. Cosde h-ode comlex olyoml Pz = z wh e = α d Im = β = d ssume h fo some d d fo d > The P z hs ll s zeos z whee d 3 wh 4. d 4 Theoem.Cosde h-ode comlex olyoml Pz = z wh e = α d Im = β = d ssume h fo some d fo some > The Pz hs ll s zeos z whee d 6 wh 6. d 6 Now we ove he followg heoem:-

3 le 95 Theoem. Cosde h-ode comlex olyoml Pz = z wh e = α d Im = β = d ssume h fo some d d fo δ η > d τ σ The P z hs ll s zeos z whee d 8 wh d. 8 8 Poof: Fsly we cosde he cse whee =. Fo he oue boud cosde olyoml Gz= -zpz =-α z + +{α δα +δα α - }z +α - -α - z α + -α z + +α -α - z + α - -α - z {α -τα + τα α }z+ α + [-β z + +{β ηβ + ηβ -β - }z + β - -β - z β + -β z + + β -β - z + β - -β - z {β -σβ σβ β }z + β ] 9 Now f z > < = ---- Gz z { z - } Whee = δ-α + η-β -δα + ηβ + α + β τα + σβ + τα + σβ + The Gz f z > d ll he zeos of Pz wh modulus gee he oe le he ds z.i c be show h. Coseuely he zeos of Pz wh modulus less h o eul o oe e ledy coed he ds z.

4 96 Aee sgh Neh d S.K.Shu Fo he e bouds g cosde Gz = Hz + If z < he Theefoe Hz 3 Whee = + δ-α + η-β -δα + ηβ + α +β τα + σβ + τα + σβ 4 Sce H = follows h Schwz lemm h Hz z fo z < The z < Gz - Hz - z > 5 If z < = he c be show h. Hece f = he ll he zeos of Pz le he ds z. 6 I s ow esy o fd he esul of he bove heoem follows fom he esul lcble o Pz. Hece he oof of he bove heoem s comlee. Coolly:-If he bove heoem we subsue fo ech of he bove mees τ d σ eul o uy he he bove esuls cocdes wh esuls obed by Choo[5]. Cocluso:- Hee ou heoem we showed he efeme ove Choo[5]. efeeces: [] A. Azz d B.A. Zg Some exesos of Eesom Key heoem Gls mhemc [] A. Azz d Q.G. ohmmd O zeos of ce clss of olyomls & eled lyc fuco. J. h Al. Al [3] A. Azz W.. Shh O he zeos of olyomls d eled lyc fucos Gls [4] A. Azz W.. Shh O he loco of zeos of olyomls d eled lyc fucos Nole Sudes [5] Y. Choo. Some esuls o he zeos of olyomls d eled lyc fucos I. Joul of h. Alyss [6] K. K. Dew d N. K. Govl O he Eesom Key heoem J. Aox. Theoy [7] K. K. Dew d.bdm O he Eesom Key heoem J. h.al [8] N. K. Govl d Q. I. ehm O he Eesom Key heoem Thou

5 le 97 h J [9] N. K. Govl d G. N. ctue Some exesos of Eesom Key heoem Ieol J. Aled mhemcs [].H. Gulz o he zeos of olyoml wh esced coeffces esech Joul of Pue Algeb [].H. Gulz O he Numbe of Zeos of Polyoml Pescbed ego esech ou. Of Pue Algeb [] A. Joyl G. Lbelle d Q. I. ehm O he loco of zeos of olyoml Cd hemcs Bull [3]. de Geomey of olyomls mh suveys 3; Amec hemcs Socey Povdece..I 966 [4] B.L. e. l. She Bouds fo he zeos of Polyomls Usg Eesom Key Theoem I. Joul of h Alyss V [5] B. L. e. l. She bouds fo zeos of comlex olyomls Ieol Joul of hemcl Achve [6] B. L. e. l. Some geelzo of Eesom Key heoem esech Joul of Pue Algeb- Pge: 35-3 [7] N.A. he d S.Sheel Ahmed A em o he geelzo of Eesom Key heoem. Joul of lyss & comuo [8] W Shh d A Lm. O Eesom Key heoem d eled lyc fucos Poc. Id Acd. Sc. h. Sc

6 98 Aee sgh Neh d S.K.Shu