P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type
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1 J N Sce & Mh Res Vol 3 No (7) -7 Alble ole h://orlwlsogocd/deh/sr P-Coey Proery Msel-Orlcz Fco Sce o Boher ye Yl Rodsr Mhecs Edco Deree Fcly o Ss d echology Uerss sl Neger Wlsogo Cerl Jdoes Absrcs Corresodg hor: yl_rodsr@wlso gocd Reced: 5 My 7 Resed : Je 7 Acceed: 3 Je 7 hs er we descrbed bo Msel-Orlcz co sces o Bocher ye hs bee obed h Msel-Orlcz co sce L o Bocher ye becoes Bch sce s descrbed lso bo P-coey o Msel-Orlcz co sce L o Bocher ye s roed h he Msel-Orlcz co sce L o Bocher ye s P-coe d oly boh sces L d re P-coe 7 JNSMR UN Wlsogo All rghs resered Key words: Msel-Orlcz Fco; Msel-Orlcz Fco; Bocher ye; P- Coe rodco he Orlcz sce ws rs rodced 93 by W Orlcz [] Orlcz sce heory hs ery or role d hs bee wdely led o ros brches o hecs oe o he o he sse o Ol Corol he deeloe d reee o he Orlcz sce sel s lso rogressg ery rdly oe he were Msel d Orlcz [] whch deelos col sce geered by odlr h hg coe roeres hs cse d he odlr coe geeres he Msel-Orlcz co sces o Bocher-ye whch s lso he Bch sce [34] he coey d relee roeres o he Bch sce lso hs bee wdely deeloed by y hecs Yg el [56] her er eled "P-coey d reley o Orlcz sces" roed h or Orlcz sces reley s eqle o P- coey he se resl or he Msel- Orlcz seqece d co sces were obed by Kolwcz d Plce [7-9] JNSMR SSN:
2 J N Sce & Mh Res Vol 3 No (7) -7 Alry Les Deo Ge : R s -esrble co Fco sd o be μ-egrble here es seqece co sch h e d or eery here es rl ber sch h d or eery Hece he e le o d l s clled Lebesge egrl o co d deoed by d or d Ge esrble se he se o ll - esrble cos ro o R deoed by M c be roed h M s ler sce For deed L ( ) M ( ) d oher word L ( ) s se o ll esrble cos M ( ) sch h -egrble Ge eeded rel led co o esrble se Sre essesl o E deed by ess s : M A A s : A M he we deed L ( ) he se o ll esrble cos by he orl L : ess s : Deo Ge ler sce No-eg co : s clled odlr o or eery y hs codos below ly (M) (M) (M3) y y wh Ler sce h coleed wh odlr s clled odlr sce d deoed by A se B Y wh Y ler sce s clled coe se or ech y B d wh s lcble y B Fco : B R s clled coes B coes d or eery y B d wh be ld y y Frherore he odlr s clled coly chrcer coes s go o coes O he e dscsso he eg odlr s he odlr h coly o chrcer coes Uless oherwse sed heore 3 () R wh he or eery or eery he Deo 4 Ge ler sce Fco : R s clled Msel Orlcz co : () or eery () ( ) ( ) (3) () co (4) () crese o ( ) (5) esrble or ech R (6) coe (7) o For he co o Msel-Orlcz s deed o co : L d wh JNSMR SSN:
3 J N Sce & Mh Res Vol 3 No (7) -7 3 or eery L So c be show h co s odlr coes he deed sce o co Msel Orlcz L wh L L : c or soe c Forherore or eery o he co Msel Orlcz s deed o co R wh s : or eery R d C be show h co s co o Msel-Orlcz heore 5 For eery co o Msel- Orlcz be ld or eery Deo 6 Fco o Msel Orlcz s clled coly codo - wres h cos d h or eery d Fherore deed co : L wh d or eery L h c be show h co s odlr For he co o Msel-Orlcz deed L : he deed : L or eery L L L be show h L R wh c s ored sce heore 7 Sce or L Bch s sce Frherore he sce co L s clled sce o co Msel Orlcz ye Bocher 3 M Resl Deo 8 Ge he sce or he se S : s clled re wh ceer O d se U : s clled dseb closed bll Deo 9 he sce Bch s clled relee or here so s y or y U wh y Deo he sce or ler s clled P coes h d N so or eery S be ld ; Le he sce Bch P coes d oly h N d so or eery \ h egers so be ld heore o eery o co Msel- Orlcz be ld he or y h so or eery R Le 3 Be dscoered d ee he codos h o co s For eery esrble h : R wh h ber d sch h or hd be ld JNSMR SSN:
4 J N Sce & Mh Res Vol 3 No (7) -7 4 h d or eery Le 4 here s deeloed seqece o esrble ses o o B so \ B d or eery N here so h or eery or e here o codo he or eery eery B d or Le 5 here s esrble co g : so g g d R d or e d heore 6 sce Bch P-coes he rele heore 7 Be dscoered co Msel Orlcz d sce Bch he he ollowg sees re eqle: () L P-coe (b) L d P-coe (c) L relec d P-coe (d) P-coe d Edece: b Be dscoered L P-coe Becse o he sce L d ebedded soerclly o L d chrcerscs P-coe s well ly o sbsce he obed P-coe b c L d Be dscoered L d P-coe Accordg o heore 9 obed L relee c d he chrcerscs o relese o he sce co o Msel-Orlcz L ele wh d d Be dscoered P-coe d C be chose N he or eery deed h g 4 4 here o co h d g s co 4 esreble wh 4 s resl 4 C be chose Becse he obed Seleced se B so lllg B d \ l here re bers Seleced l R so so l or e B whe he chose d l Obed For d or eery h d B JNSMR SSN:
5 J N Sce & Mh Res Vol 3 No (7) -7 5 JNSMR SSN: So obed l or e d he 4 he se wy obed For d h d or eery Frherore wll be show h here re bers r so or eery sce gro Bch d o M e obed r Wh : M e For ele s de so Sose here \ so Becse or M e so obed Bsed o chrcerscs coe o or e obed or M e s ssed h or ll ly he or eery C be e d ssed h O he oher hd hed So 's cordco So obed Bsed o chrcerscs coe o or e obed Wh so obed
6 J N Sce & Mh Res Vol 3 No (7) -7 6 or e M Deed r he he ber c be deered r so be ld r or eery cooe sce Bch d or h d B coly s he roed re wh r S L d e deed E : Obosly h or eery E Frherore E dded o he ollowg wo sbses: : E d E : Deed se E d E s ollows E d E \ E B E B Obed or e E E wh r r r obosly r Frherore ro he dee se E d co obed e E E \ B E \ B \ B Obed \ E he obed E E \ B d d d d \ E E \ E E becse or d he obed or s resl E E \ E E r \ E E E E E E JNSMR SSN:
7 J N Sce & Mh Res Vol 3 No (7) -7 7 r E E E E r r r E E wh So be od so h s resl becse obed q wh q obed h sce L P-coes 4 Coclso Modlr d geeres he Msel-Orlcz co sce L L : c or soe c Frherore or eery Msel - Orlcz co we dee co s whch s lso Msel - Orlcz co d s clled he coleery co he sese o Yog Modlr d geeres he Msel-Orlcz co sces o Bocher ye L L : Frherore L L s Bch sce he Msel-Orlcz co sces o L s P-coe d oly Bocher ye boh L d re P-coe Frherore s roed h reley s eqle o P- coey Acowledgee Acowledges re sbed o he Dere o MhecsUGM or sor hs reserch Reereces [] W Orlcz Ler Fcol Alyss World Scec Lodo 99 [] H Hdz A Ks d W Krc Uorly o-l Msel-Orlcz Sces Bll Acd Polo Sc Mh [3] H Hdz d S Che O Soe Coees o Orlcz d Orlcz-Bocher Sces Coe Mh U Crole 9 No [4] P Kolwcz d R Plce P-coey o Msel-Orlcz Fco Sces o Bocher ye Res Mec Colese Vol No 998 [5] Y Yg d H Yeg P-coey roery Msel-Orlcz seqece sces Collec Mh [6] DP Gesy O Coey Codo Nored Ler Sces rscos o he Aerc Mhecl Socey Vol 5 No [7] P Kolwcz d R Plce O P-coe Msel-Orlcz Sces Coe Mh U Crole [8] P Kolwcz d R Plce P-coey o Bocher-Orlcz Sces Proc Aer Mh Soc [9] P Kolwcz d R Plce P-coey o Msel-Orlcz Seqece Sces o Bocher ye Collec Mh 997 JNSMR SSN:
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