Journal of Quality Measurement and Analysis JQMA 12(1-2) 2016, Jurnal Pengukuran Kualiti dan Analisis

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1 Joural o Qualiy Measureme ad alysis JQM Jural Peguura Kualii da alisis SOME RESLTS FOR THE LSS OF LYTI FTIOS IVOLVIG SLGE IFFERETIL OPERTOR Beberapa Sia uu Kelas Fugsi alisis Melibaa Pegoperasi Pembea Salagea LWIH IBRHIM MSLI RS SEVER S RGOMIR & F- JOSEPH OLBMI 3 BSTRT Le deoe he class o ucio aalyic ad uivale i he ope ui dis which are deied ivolvig he Salagea diereial operaor such ha I his paper some properies such as a represeaio heorem ad coeicie esimaes or he class are obaied Keywords: Salagea diereial operaor; coeicie esimae; represeaio heorem BSTRK daia elas ugsi aalisis da uivale pada caera ui yag erari melibaa pegoperasi pembea Salagea sedemiia higga alam maalah ii beberapa sia ugsi di dalam elas seperi eorem perwaila da aggara peali diperoleh Kaa uci: Pegoperasi pembea Salagea aggara peali eorem perwaila Iroducio Le deoe he class o ucios o he orm a which are aalyic i he ope ui dis { : } is a complex umber We deoe he subclass o cosisig o aalyic ad uivale ucios i by S Le P be he class o aalyic ucio p o he orm p p such ha p ad Re p The ollowig classes o ucios are well ow ad have bee sudied repeaedly by may auhors such as Salagea 983 Opoola 994 bdulhalim 3 ad ohers

2 lawiah Ibrahim Maslia arus Sever S ragomir & F- Joseph Olubumi S : Re B : Re 3 : Re I 993 Salagea iroduced he ollowig operaor which is ow as he Salagea diereial operaor eiiio For a ucio he Salagea diereial operaor : {} is deied by where 4 ad The operaor has bee employed by various auhors o deie several subclasses o see Kaas 989 Obradovic 99 Opoola 994 Babalola ad Opoola 6 For isace Opoola 994 iroduced he subclass T o aalyic ucios which are deied ivolvig Salagea diereial operaor as ollows: T : Re 5 or The class T is he geeraliaio o he classes o ucios as meioed i 3 where T : B is he class o Bailevic ucios bdulhalim 3 Some properies o his class o ucios were also esablished by Opoola amely i T is a subclass o uivale ucios ii T T c iii I T c he he iegral operaor Fc d c is also i T I his paper we obai urher properies o ucios i he class T Prelimiaries We observed ha or ad o he orm ad applyig he operaor 4 he we have a 6 9

3 Some resuls or he class o aalyic ucios ivolvig Salagea diereial operaor or { } lso suppose ha he by usig biomial expasio rom we ca wrie 7 where he coeicies {34 } wih a deped o he coeicies a o ad he parameer ow i we apply he operaor 4 or he ucio 7 he we obai or 8 I order o prove our resuls we shall eed he ollowig lemmas Lemma Hayami e al 7 ucio p P give by saisies he codiio y p i ad oly i p 9 Lemma lors 966 Le be a aalyic ucio i he ui dis wih ad The ad i Sric iequaliy holds i i boh esimaes uless is a roaio o he dis e I or some he c wih a cosa c o absolue value 3 Mai Resuls I his secio we esablish some properies o ucios i he class T Firs we prove a suicie codiio or ucios belog o his class Theorem 3 ucio i is i he class T i ad oly where Q Q or some wih Proo From 5 is suggess ha here exiss a ucio p P such ha 9

4 lawiah Ibrahim Maslia arus Sever S ragomir & F- Joseph Olubumi 9 p or po seig p he rom Lemma we ge or wih I is equivale o Subsiuig 8 io yields ha ow dividig boh sides o by we obai or ay such ha Remar 3 For he case o i he resul has bee proved by Sigh e al 9 The ollowig propery o ucios i he class T is also esablished Theorem 33 Le belog o he class T The here exiss a aalyic ucio wih such ha or Proo Le us deie he ucios ad B as ollows: 3 ad B 4 or Subsiuig he equaio 3 io 4 we ge

5 Some resuls or he class o aalyic ucios ivolvig Salagea diereial operaor 93 B or ay B is a aalyic ucio or lso sice ad we have ha B ad B or Hece by Schwar s Lemma Lemma B which gives ha or equivalely where is aalyic ad or Thereore 5 Solvig or rom 5 he we ge Thus we obai he desired resul The ollowig resul is called he iegral represeaio heorem or ucios i he class T which provides urher propery o ucios i his class Theorem 34 Le belog o he class T The here exiss a aalyic ucio wih such ha d exp 6 or where 7 Proo: Le T The rom Theorem 33 we have

6 lawiah Ibrahim Maslia arus Sever S ragomir & F- Joseph Olubumi 8 Taig he logarihmic diereiaio we ge rom 8 9 ow iegraig boh sides o 9 alog he lie segme rom o we obai l d This gives us ha exp d or where or a aalyic ucio wih compleed Thus he proo o Theorem 34 is cowledgemes This wor was suppored by he research gra GGPM-6-8 rom he iversii Kebagsaa Malaysia Reereces Salagea G S 983 Subclass o ivale Fucios Lecure oe i Mah Spriger-Verlag: bdulhalim S 3 oeicies esimaes or ucios i B IJMMS Hidawi Publishig orp 59: Opoola T O 994 O a ew subclass o uivale ucios Mahemaica Tome 36 59: 95- Kaas S 989 ew subclasses o uivale ucios Folia Sciearium iv Tech Resoviesis 6: Obradovic M 99 O cerai diereial operaor ad some classes o uivale ucios Serija Mahemaica 6: 7- Babalola K O & Opoola T O 6 Iegraed iegral rasorm o aaheodory ucios ad heir applicaios o aalyic ad uivale ucios Tamag J Mah 374: Hayami T Owa S ad Srivasava H M 7 oeicie iequaliies or cerai classes o aalyic ad uivale ucios J Ieq I Pure ad ppl Mah 84: - hlors L V 966 omplex alysis d Ed ew Yor: McGraw Hill Boo omp Sigh S Gupa S & Sigh S 9 geeral coeicie iequaliy Geeral Mahs 73:99-4 School o Mahemaical Scieces Faculy o Sciece ad Techology iversii Kebagsaa Malaysia 436 Bagi Selagor MLYSI alaibra@umedumy* maslia@umedumy 94

7 Some resuls or he class o aalyic ucios ivolvig Salagea diereial operaor School o Egieerig ad Sciece Vicoria iversiy Melboure STRLI Severragomir@vueduau 3 eparme o Mahemaics iversiy o Ilori Ilori IGERI amelov@uiloriedug *orrespodig auhor 95