Upper Bound of Partial Sums Determined by Matrix Theory

Transcription

1 Turish Jourl of Alysis d Nuber Theory, 5, Vol, No 6, 49-5 Avilble olie t Sciece d Eductio Publishig DOI:69/tjt--6- Upper Boud of Prtil Sus Deteried by Mtrix Theory Rbh W Ibrhi Istitute of Mtheticl Scieces, Uiversity Mly, Mlysi Correspodig uthor: rbhibrhi@yhooco Received Jue 8, 5; Revised Noveber 5, 5; Accepted Deceber, 5 Abstrct Oe of the jor probles i the geoetric fuctio theory is the coefficiets boud for fuctiol d prtil sus The iportt ethod, for this purpose, is the Hel trix Our i is to itroduce e ethod to deterie the coefficiets boud, bsed o the trix theory We utilize vrious ids of trices, such s Hilbert, Huritz d Tur We illustrte e clsses of lytic fuctio i the uit dis, depedig o the coefficiets of prticulr type of prtil sus This ethod shos the effectiveess of the e clsses Our results re pplied to the ell o clsses such s strlie d covex Oe c illustrte the se ethod o other clsses Keyords: lytic fuctio, uivlet fuctio, uit dis, prtil sus, coefficiets boud Cite This Article: Rbh W Ibrhi, Upper Boud of Prtil Sus Deteried by Mtrix Theory Turish Jourl of Alysis d Nuber Theory, vol, o 6 (5: 49-5 doi: 69/tjt--6- Itroductio The Hel deterit represets jor prt i the theory of sigulrities [,] I dditio, it utilizes i the ivestigtio of poer series ith itegrl coefficiets [] Also, it ppers i the study of eroorphic fuctios [4], d vrious properties of these deterits c be foud i [5] It is ell o tht the Feete-Szego fuctiol H ( This fuctiol is further geerlized s µ for soe µ (rel or coplex Feete d Szego itroduced shrp bouds of µ for µ rel of uivlet fuctios It is very iportt cobitio of the to coefficiets hich describes the re probles posted erlier by Groll Furtherore, reserchers cosidered the fuctiol 4 (see [6] Bblol H for soe subclsses of lytic fuctios Ibrhi [8] coputed the Hel deterit for frctiol differetil opertor i the ope uit dis Prtil sus re studied idely i the uivlet fuctio theory Szeg [9] proved tht if the fuctio f ( z z + z is strlie, the its prtil sus f z+ z re strlie for z < /4 [7] deteried the Hel deterit ( Moreover, if f ( z is covex, the its prtil sus f is covex for z < / 8 Lter O [] iposed the strlieess d covexity for specil cse of f z+ z I dditio, Drus d Ibrhi [] specified the ssuptios, hich idicted tht the prtil sus of fuctios of bouded turig re lso of bouded turig Recetly, Drus d Ibrhi [] cosidered the Cesáro prtil sus, it hs bee sho tht this type of prtil sus preserves the properties of the lytic fuctios i the ope uit dis I this or, e del ith the prtil sus of the for f z+ ( / z, > We itroduce soe clsses of lytic fuctios defied by its prtil sus The stbility of these clsses is studied by utilizig Huritz trices covolutig the ith Hilbert trix ( specil type of Hel trix Moreover, e discuss soe prtil sus forulted uder Tur deterit The upper boud s ell s the loer boud of the coefficiets This e process icludes soe ell o results Our outcoes deped o coputtiol results of differet order of the Tur deterit We sho tht soe geoetric properties, of the e clsses re estblished by coputig the Tur deterit such s strlieess d covexity Processig Let be the clss of lytic fuctios f z+ z i U { z: z } orlized by the coditios f ( f ( prtil su of the for ( < d For f z z + z + + z + z, covoluted ith the Hilbert trix eleets i the fit order, e obti the prtil sus f z + z + + z + z,,

2 5 Turish Jourl of Alysis d Nuber Theory For the bove prtil sus f ( z, e let f z + z + + z + z,, z U : b z + bz + + b z + b z+ b, b, b The iors of Huritz ( defied by ( f b b b ( f b b b b b5 ( f b b b4 b b ( trix for (, re Defiitio For z U, z is clled stble, syptoticlly stble d ustble if d oly if j >, j, j <, for ll j,,,, respectively Fro (, e defie the prtil sus the polyoil f ( g z z z ( : + We proceed to costruct e clsses bsed o g ( z A coputtio iplies zg P, g ( ( ( + z ( ( : +, z Thus for,,4,, e hve the folloig clsses: ( P( + ( P( +, ( ( We cll the bove clsses the coefficiet ( -strlie d they deoted by S ( Siilrly, e defie the coefficiet ( -covex, hich deoted by (, s follos: zg Q + + ( z g : + (, z (4 Thus for,,, e hve the folloig clsses: ( + ( ( + ( Q Q, I the se er of the bove clsses, oe c costruct -clss such s close to covex, uiforly clsses d cocve Bsed o these clsses, e c study the stbility of strlieess s ell s covexity Moreover, reltios cocerig these clsses c be forulted such s H (, H (, Outcoes (5 We hve the folloig stbility results for the clsses S ( d ( : Theore Cosider P S (, > The polyoil of degree is strlie stble, hile of degree is ot stble Proof By eployig P, i Eq (, polyoils of degree d c be expressed respectively s follos: d d p, ( + : b + b + b p, ( + + : b + b+ b + b Let >, thus e obti ( p, ( p > ( p, <, > The polyoil of degree is covex stble, hile of degree is ot stble Proof Cosider Q (, > The polyoils of degree d c be forulted respectively s follos: Theore Cosider Q ( d q, ( + : b + b + b

3 Turish Jourl of Alysis d Nuber Theory 5 d q, ( + + : b + b+ b + b Let >, thus e obti q, q > ( q 6 <, Cosider p S ( sequeces We del ith polyoil p S, (prtil sus stisfyig the recurret reltio d ( ( + ( ( ( >,, p p p p, ( ( p( + Defie the Tur deterit s follos: (6 Λ p p p+, (7 We shll prove iequlity of the for (,, c Λ C < < < c< C (8 Theore Assue p ( stisfies (6 The ( ( Λ + Λ p p, Proof By (6, e hve this yields tht p+ + p ( + p Λ p p Cosequetly, e obti ( p Λ + (9 By the defiitio of Λ ( Λ, e coclude tht ( p, ( the suig (9 d (, e rrive t the desired ssertio This copletes the proof Theore 4 Let ( be icresig sequece If > the ( { } Λ >, U \, Proof It suffices to sho tht ( Λ > By the proof of Theore d the fct tht p d p, e coclude tht ( ( Λ p p ( p > p Therefore, by the ssuptios of the theore, e hve Λ ( > Hece by iductio e obti ( Λ >, Defie fuctio g( : p+ p the g ( stisfies the folloig property : Propositio For e hve ( ( ( g + g+ g Proof A clcultio iplies tht + g+ ( g ( + p+ ( + p+ ( p+ ( p+ ( + p ( g + p p+ + p g ( ( Theore 5 For e hve ( ( ( Λ +, g g g here g ( p ( p ( +, Proof We observe tht d ( ( g p+ p p, ( g ( g ( ( p ( p ( ( p ( p ( p p + ( Subtrctig ( fro (, e coclude the desired ssertio p chieves (6 ith Theore 6 Cosider tht ( < Let (, be icresig such tht / d The (, + Λ ( c, c>, U, Proof Clerly tht ( is equivlet to ( icresig Defie the forul ( beig

4 5 Turish Jourl of Alysis d Nuber Theory ( ( A : g + g g Sice, therefore, i vie of Theore 5, e obti Λ Λ, ± (4 By Propositio, e hve the folloig expressio : ( ( + ( + g ( g ( A g g Multiplyig Eq(5 by e rrive o + (5 d replcig by, A ( A ( g ( Cosequetly, e coclude tht By itertig the qutities A A A ( A A d A, e tti i But by utilizig Eq(5 d Eq(6, e fid A ( g + g g g Therefore, (4 becoes Hece the proof A ( : c Theore 7 Cosider tht ( (6 p chieves (6 ith p < Let ( p, be decresig such tht / d The (, + Λ ( C, C >, U, (7 Proof By lettig ϒ : +, ith the folloig properties: li ϒ + ( + + ( + + ϒ+ ϒ ϒ + ϒ < The lst property is vlid by the ootoicity of ( i (7 Defie polyoil ( P by utilizig ϒ s follos: for here stisfyig ( p (, U, P Obviously, P stisfies :,,, P ϒ P+ +ϒ P (8 li P P P+ < This iplies tht ( P (, U is uiforly bouded o copct set for By the defiitio of the Tur deterit, e obti here such tht ( ( P ( P ( P ( Λ λ + λ, + li λ We coclude tht there exists costt C (, such tht (,, Λ C U Rer If i Theores 6 d 7, e obti tht the coefficiet For exple, if < (strlie clss, the /, Thus, Theore 7 iplies tht Λ < s + Moreover, the bove results c be cosidered for sequece of polyoils ( q (, 4 Applictios I this sectio, e utilize the Tur deterit to fied the coefficiets boud of the clsses S ( d ( We hve the folloig propositios Propositio 4 Cosider the clsses S ( d S ( d Proof By utilizig Λ d Λ respectively for fidig the upper boud of d A coputtio iplies tht d Λ (,

5 Turish Jourl of Alysis d Nuber Theory Λ ( I vie of Rer, e coclude tht Λ (, ± <, he < Siilrly for I the siilr er of Propositio 4, e hve the folloig result: The Propositio 4 Cosider the clsses ( Proof By utilizig q ( q ( q ( obti Λ :, e ( Λ, hich iplies tht ( 5 Coclusio Λ < he We iposed e techique for fidig the coefficiets boud This ethod bsed o severl types of trices The jor type s the Tur i the ope uit dis We proved the boudedess of this trix fro belo s ell s fro bove We defied clsses of lytic fuctios, depedig o oe coefficiets, clcultig by soe specil type of prtil sus The stbility of these clsses is cosidered by utilizig the Huritz trix We illustrted soe pplictios of this ethod for to ell defied clsses (strlie d covex The bove ethod c be eployed o other clsses such s uifor, cocve etc Coflict of Iterests The uthor declres tht there is o coflict of iterests regrdig the publictio of this rticle Refereces [] P Diees, The Tylor Series Dover, Ne Yor (957 [] A Edrei, Sur les dterits rcurrets et les sigulrits due foctio doe por so dveloppeet de Tylor Copos Mth 7, -88 (94 [] D G Ctor, Poer series ith itegrl coefficiets Bull A Mth Soc 69, 6-66 (96 [4] R Wilso, Deteritl criteri for eroorphic fuctios Proc Lod Mth Soc 4, (954 [5] R Vei, P Dle, Deterits d Their Applictios i Mtheticl Physics Applied Mtheticl Scieces, vol 4 Spriger, Ne Yor (999 [6] D Bsl, Upper boud of secod Hel deterit for e clss of lytic fuctios Appl Mth Lett 6(, -7 ( [7] K O Bblol, O H( Hel deterit for soe clsses of uivlet fuctios Iequl Theory Appl 6, -7 (7 [8] R W Ibrhi, Bouded olier fuctiol derived by the geerlized Srivstv-O frctiol differetil opertor Itertiol Jourl of Alysis, -7 ( [9] G Szego, Zur theorie der schlichte bbiluge Mth A, 88- (98 [] S O, Prtil sus of certi lytic fuctios It J Mth Mth Sci 5(, ( [] M Drus, R W Ibrhi, Prtil sus of lytic fuctios of bouded turig ith pplictios Coput Appl Mth 9(, 8-88 ( [] R W Ibrhi, M Drus, Cesáro prtil sus of certi lytic fuctios, Jourl of Iequlities d Applictios, 5, -9 (