IOSR Journal o Mahemaics IOSR-JM e-issn: 78-578 -ISSN: 39-765X. Volume Issue Ver. VI Mar - Ar. 5 PP 43-5 www.iosrjournals.org A auberian heorem or C α β- Convergence o Cesaro Means o Orer o Funcions Suash Naraan Mishra Piush Kumar riahi & Alo Agrawal Email:snmisra@lo. ami. eu riai@lo. ami. eu aagrawal@lo. ami. eu 3 Ami School o Alie Sciences Ami niversi ar Praesh Lucnow Camus Near Malhaur Railwa Saion Gomi Nagar Lucnow. P. Inia Absrac: he objecive o his aer o generalize cerain auberian resuls rove b Gehring [3] or summabili C ; α o sequences o uncions. In [] A. V. Bo generalize he auberian heorem or α convergence o Cesa ro means o sequences. In his aer we obain some auberian heorems or C α β convergence o Cesa ro means o orer o uncions an invesigae some o is roeries. Kewors: auberian heorem Absolue an Cesàro summabili Lebesgue Inegral Convergence. I. Inroucion he noaion is similar ha are in [3]wih he ollowing aiional einiions: I > hen A n B n enoe he n-h Cesa ro sums o orer or he series n= a n n= b n where b n = na n. A n B n enoe he a n b n. Summabili C ; α o a n will b C ; α o a n. Mishra an Srivasava [6] inrouce he Summabili meho C or uncions b generalizing C summabili meho. In his aer we iscuss some auberian heorems or C α β convergence o Cesa ro means o orer o uncions an invesigae some o is roeries. II. Deiniions an Some Preliminaries We woul lie o irs inrouce Summabili meho. Summabili meho is more general han ha o orinar convergence. I we are given a sequence s n we can consruc a generalize sequence σ n he arihmeic mean o s n b his sequence s n. I σ n is convergen in orinar sense or all n > hen we sa ha s n is summable C o he sum s. his C is calle Cesaro mean o irs orer. I s n s σ n = s +s +.+s n s ie i a sequence is convergen i is summable b meho o n + arihmeic mean. Also a series + + + is no convergen bu is summable o he sum. he sace o summable sequences is larger han sace o convergen sequences. I σ n s as n hen we sa ha sequence s n is summable b meho o arihmeic mean. For eamle : Consier he series n= u n = u + u +.. An le σ n = s +s +.+s n I ma haen ha whereas iverges he quaniies he arihmeic mean n + o arial sum o series converges o a einie limi as n. For eamle + + + iverges bu in his case s = s = = s = + = s 3 = s n =... Since s n = + n σ n = s +s +.+s n n + = + + + + + + n +.. + /n + = n+ + + n + erms /n + DOI:.979/578-6435 www.iosrjournals.org 43 Page
A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions = + + n I n is even hen σ 4n + n = + as n an i n is o hen σ n+ n =. So in eiher case lim n σ n = s n C bu s n ε S. hereore sace o summable sequences is larger han har o sace o convergen sequences. Le be an uncion which is Lebesgue-measurable an ha : [ + R an inegrable in or an inie an which is boune in some righ han neighbourhoo o origin. Inegrals o he orm hroughou o be aen as Le. eiss an i I or lim being a Lebesgue inegral. he inegral g g. g s as we sa ha uncion is summable D o he sum s an we s D as. We noe ha or an ie i is necessar an suicien or convergence o. ha wrie he shoul converge.. C ransorm o which we enoe b is given b.3 I his eiss or an ens o a limi s as we sa ha is summable C o s an we wrie s C. We also wrie.4 we sa ha he uncion are is summable D C i his eiss an ens o a limi s as o s. When D C an D C enoe he same meho. Here we give some Gehrings generalize auberian heorems. heorem.: Suose ha α an ha is summable A α o s hen is C α β convergen o s i an onl i he uncion αβ is C α β convergen o. heorem.: Suose ha α an ha is C α β convergen. I he uncion αβ is C α β convergen o hen is summable C α o is sum or ever >. III. Now we shall rove he ollowing heorem heorem 3.: Suose ha α an ha is summable A α o s. hen or r is summable C r α o s i an onl i he uncion α β is C α β o. Proo : Necessar Coniion: I r = he heorem immeiae ollows rom he summabili o C α. I r > hen b consisenc heorem or C r α summabili Gehring [3heorem 4..] i ollows ha boh he uncions an αβ are C α β convergen o s. B Har [ Equaion 6..6] S n n r = S r+ + DOI:.979/578-6435 www.iosrjournals.org 44 Page
r+ A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions α β an he resul ollows since a linear combinaion o uncions summable C α o isel. he suicien coniions o rove he heorem are : I r > i ma be shown as in Szasz [ 4 ] ha + αβ + n r + = u r αβ u u 3. Where αβ u = u = Case a : α = r > is obvious. Case b : α r > uing g = + αβ + n. We ge rom 3. ha g = r + αβ v v n v Where αβ u now has boune C α β- variaion over. Le N V = = r + v αβ v v N αβ r αβ r αβ v v α. α α α. hen b heorem o [5] we have V r + M v r v = M.. Where M = V α αβ :. hus αβ has boune C α β- variaion over. I is reail seen rom Minowsi s inequali ha he sum o wo C α β convergen sequences is also C α β convergen an we hereore euce ha is C α β convergen o s. Case c r=-when α = he resul reuces o auber s original heorem; when α i ollows rom above heorem. For α = he resul was rove b Hslo []. heorem 3. : Le α > γ β > γ β an suose ha a is summable C γ β o s an ha converges. hen a is summable D C α β o s. We irs rove his heorem uner unreasonable einiion.. However i he resul hols wih. hen i mus also hol uner he einiion o.3. his ollows rom he ollowing Lemmas. Lemma 3.: Le. Suose ha L or inie C accoring o he einiion.3..suose ha Deine or 3. or Le enoe he eression corresoning o bu wih relace b. hen. 3.3 hus is summable C uner he einiion.3. DOI:.979/578-6435 www.iosrjournals.org 45 Page
A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions Lemma 3.: Le he hohesis be as in Lemma 3.an eine as above. Le an.hen D C summabili o an are equivalen. Proo o Lemma 3.: We are given ha or some > 3.3 Bu since i 3.3 hols or given i hols or an greaer i mus hol or all suicienl large. Now b sanar roeries o racional inegrals an since we have u u u u 3.4 Since 3.3 hols his will ollow rom Minowsi s inequali i we rove ha 3.5 Now i ollows a once rom he einiion ha or I hen or we have so ha Cons. = b 3.4. Proo o Lemma 3.: We use noaions as in Lemma 3. an wrie urher or he eression corresoning o bu wih relace b. DOI:.979/578-6435 www.iosrjournals.org 46 Page
A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions DOI:.979/578-6435 www.iosrjournals.org 47 Page We now ha or an ie convergence o is equivalen o he convergence o.hen he conclusion will ollow rom Minowsi s inequali i we show ha 3.6 where we ae 3.6 as incluing he asserion ha he inegral eine b converges or all. For large we have 3.7 Hence he convergence o ollows a once b a resul ue o []. Now 3.6 is equivalen o c. 3.8 Le be an suicienl large consan. hen 3.8 will ollow rom Minowsi s inequali i we show ha c. 3.9 c. 3.
A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions B 3.9 we have c = O O. Hence 3.9 ollows. B 3.7 he eression on he le o 3. oes no ecee a consan. hus c o 3. B an obvious change o variables he eression 3. is equal o o o C C. he resul ollows. Proo o heorem 3. : We ivie he roo ino he ollowing cases. Case I. Case II. Case III. Here we observe ha Case I an II ollow rom case III wih he ai o heorem 3.. ' For i Choose an summabili C imlies summabili ' C b heorem 3. an i ollows rom Case III ha his imlies D C. Hence i is suicien o consier he case III onl. ' Proo o Case III : Since s C imlies ha s C or ' o here is no loss o generali in consiering he Case is a osiive ineger. We have C 3. DOI:.979/578-6435 www.iosrjournals.org 48 Page
A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions Now b einiion. Puing = an we see ha. 3.3 We also wrie. R I is clear ha whenever converges R. I ollows immeiael rom 3.3 ha is eine or > an ha R as R o an hence ha or o 3.4 Inegraing 3.4 b ars imeswe euce wih he hel o 3.3 ha C. 3.5 I is veriie ha eression in 3.6 is o. 3.6 Le R. In ac or ie we have uniorml in R. 3.7 his ma be rove b inucion on i we have DOI:.979/578-6435 www.iosrjournals.org 49 Page
A auberian heorem For C α β- Convergence O Cesa ro Means O Orer K O Funcions R = hence he resul is evien. Suose ha an assume he resul rue or. Inegraing b ars we have R. he irs erm is o require orer b 3.7 wih relace b - an he secon b inucion hohesis. Now inegraing 3.6 b ars we have = R = R. Since he inegrae erm ens o as is boune an R as. sing 3.7 an uing we see ha he eression in curl braces C C C Again using 3.8 he inner inegral C 3.8 on uing he eression on he righ o 3.9 is equal o C C Since he inegral converges. Hence he resul ollows. Reerences []. A.V. bo Some heorems on Summabili 95. []. J. M.Hslo A auberian heorem or absolue summabilij.lonon Mah.Soc.Vol. 937.76-8. [3]. F.W. Gehring A su o α variaion I rans. Amer. Mah.Soc.vol.76 954.4-443. [4]. O Szasz On roucs o summabili mehosproc.amer.mah.soc.vol.395. 57-63. [5]. G. H. Har J. E.Lilewooan Pola Inequaliies934. [6]. Mishra B. P. an Srivasava A.P. Some remars on absolue Summabili o uncions base on C mehos.o aear in Jour. Na. Aca. o Mah. Summabili DOI:.979/578-6435 www.iosrjournals.org 5 Page