Integral Operator Defined by k th Hadamard Product

Transcription

1 ITB Sci Vol 4 A No Itegrl Opertor Deied by th Hdmrd Product Msli Drus & Rbh W Ibrhim School o Mthemticl Scieces Fculty o sciece d Techology Uiversiti Kebgs Mlysi Bgi 436 Selgor Drul Ehs Mlysi Emil: msli@ummy rbhibrhim@yhoocom Abstrct We itroduce itegrl opertor o the clss A o lytic uctios i the uit dis ivolvig th Hdmrd product covolutio correspodig to the dieretil opertor deied recetly by Al-Shsi d Drus New clsses cotiig this opertor re studied hrcteritio d other properties o these clsses re studied Moreover suborditio d superorditio results ivolvig this opertor re obtied Keywords: Hdmrd product; itegrl opertor; suborditio; superorditio AMS Mthemtics Subject lssiictio : 345 Itroductio Let H be the clss o uctios lytic i the uit dis U d H be the subclss o H cosistig o uctios o the orm Let A be the subclss o H cosistig o uctios o the orm U The ollowig dieretil opertor is deied i d studied i D : A A by where D N{} Received April st Revised My 4 th Accepted or publictio My 5 th

2 36 Msli Drus & Rbh W Ibrhim Remr Whe we get S l ge dieretil opertor 3 gives Ruscheweyh opertor 4 implies Al-Oboudi dieretil opertor o order 5 d whe opertor reduces to Al-shsi d Drus dieretil opertor o order 6 g A d b their covolutio or Hdmrd product g is Give two uctios g deied by g b U Ad or severl uctios m A m m U Alogous to ollows D U we deie itegrl opertor : A A s Let : F times N Ad let F be deied such tht

3 Itegrl Opertor Deied by th Hdmrd Product 37 F F The U F times N 3 Remr Whe we get the itegrl opertor 3 lso gives Noor itegrl opertor 78 Some o reltios or this itegrl opertor re discussed i the ext lemm Lemm Let A The dt t t ii i Proo i dt t dt t t ii

4 38 Msli Drus & Rbh W Ibrhim I the ollowig deiitios we itroduce ew clsses o lytic uctios cotiig the itegrl opertor 3: Deiitio Let A The S i d oly i { } > < U Deiitio Let A The i d oly i { } > < U Let F d G be lytic uctios i the uit dis U The uctio F is subordit e to G writte F G i G is uivlet F G d F U G U I geerl give two uctios F d G which re lytic i U the uctio F is sid to be suborditio to G i U i there exists uctio h lytic i U with h d h < or ll U such tht F G h or ll U Let : d let h be uivlet i U I p is lytic i U d stisies the dieretil suborditio p p h the p is clled solutio o the dieretil suborditio The uivlet uctio is clled domit o the solutios o the dieretil suborditio i p I p d p p re uivlet i U d stisy the dieretil superorditio h p p the p is clled solutio o the dieretil superorditio A lytic uctio is clled subordit o the solutio o the dieretil superorditio i p Let be lytic uctio i domi cotiig U d > The uctio A is clled lie i

5 Itegrl Opertor Deied by th Hdmrd Product 39 { } > U This cocept ws itroduced by Bricm 9 d estblished tht uctio A is uivlet i d oly i is lie or some Deiitio 3 Let be lytic uctio i domi cotiig U d or U Let be ixed lytic uctio i U The uctio A is clled lie with respect to i U The pper is orgied s ollows: Sectio discuses the chrcteritio properties or uctios belogig to the clsses S d Sectio 3 gives the suborditio d superorditio results ivolvig the itegrl opertor For this purpose we eed to the ollowig lemms i the seuel Deiitio 4 Deote by Q the set o ll uctios tht re lytic d ijective o U E where : { U : lim } d re such tht or U E E Lemm Let be uivlet i the uit dis U d d be lytic i domi D cotiig U with w whe w U Set Q : h : Q Suppose tht Q is strlie uivlet i U d h > or U Q I p p p

6 4 Msli Drus & Rbh W Ibrhim the p d is the best domit Lemm 3 Let be covex uivlet i the uit dis U d d be lytic i domi D cotiig U Suppose tht is strlie uivlet i U d { } > or U I p H Q with p U D d p p is uivlet i U d p p p the p d is the best subordit Geerl Properties o I this sectio we study the chrcteritio properties or the uctio A to belog to the clsses S d by obtiig the coeiciet bouds Theorem Let A I < 4 the S The result 4 is shrp Proo Suppose tht 4 holds Sice

7 Itegrl Opertor Deied by th Hdmrd Product 4 the this implies tht > hece } > { We lso ote tht the ssertio 4 is shrp d the extreml uctio is give by orollry Let the ssumptio o Theorem The orollry Let the ssumptio o Theorem The or d N I the sme wy we c veriy the ollowig results: Theorem Let A I

8 4 Msli Drus & Rbh W Ibrhim < 5 the The result 5 is shrp orollry 3 Let the ssumptio o Theorem The Also we hve the ollowig iclusio results Theorem 3 Let < The S S Proo By Theorem Theorem 4 Let < The Proo By Theorem Theorem 5 Let The S S Proo By Theorem Theorem 6 Let The Proo By Theorem Moreover we itroduce the ollowig distortio theorems Theorem 7 Let A d stisies 4 The or U d < d

9 Itegrl Opertor Deied by th Hdmrd Product 43 Proo By usig Theorem oe c veriy tht the Thus we obti The other ssertio c be proved s ollows This complete the proo I the sme wy we c get the ollowig results

10 44 Msli Drus & Rbh W Ibrhim Theorem 8 Let A d stisies 5 The or U d < d Also we hve the ollowig distortio results Theorem 9 Let A d d stisies 4 The or m U d < d m m Proo By usig Theorem oe c show tht the Thus we obti m m m

11 Itegrl Opertor Deied by th Hdmrd Product 45 m The other ssertio c be proved s ollows m This completes the proo I the sme wy we c get the ollowig results Theorem Let A d d stisies 5 The or U d < m d m 3 Sdwich Result By mig use o lemms d 3 we prove the ollowig suborditio d superorditio results ivolvig the itegrl opertor 3

12 46 Msli Drus & Rbh W Ibrhim Theorem 3 Let be uivlet i U such tht is strlie uivlet i U d } > { 6 I A stisies the suborditio } { the 7 d is the best domit Proo Our im is to pply Lemm Settig : p By computtio shows tht p p which yields the ollowig suborditio p p By settig : : d

13 Itegrl Opertor Deied by th Hdmrd Product 47 it c be esily observed tht re lytic i {} \ d tht whe {} \ Also by lettig Q d Q h we id tht Q is strlie uivlet i U d tht } > { } { Q h The the reltio 7 ollows by pplictio o Lemm orollry 3 Let the ssumptios o Theorem 3 hold The the suborditio implies 8 d is the best domit Proo By lettig : orollry 3 I A d ssume tht 7 holds the

14 48 Msli Drus & Rbh W Ibrhim B A B A implies < A B B A d B A is the best domit Proo By settig : d B A : where < A B orollry 33 I A d ssume tht 7 holds the implies d is the best domit Proo By settig : d : orollry 34 I A d ssume tht 7 holds the A

15 Itegrl Opertor Deied by th Hdmrd Product 49 implies d e A A e is the best domit A Proo By settig : d : e A< Theorem 3 Let be covex uivlet i the uit dis U Suppose tht d A { } > or U is strlie uivlet i U I H Q 9 where { is uivlet is U d the suborditio holds the { } } d is the best subordit Proo Our im is to pply Lemm 3 Settig

16 5 Msli Drus & Rbh W Ibrhim p : By computtio shows tht p p which yields the ollowig suborditio By settig p p : d : it c be esily observed tht re lytic i \{} d tht whe \{} Also we obti { } { } > The ollows by pplictio o Lemm 3 ombiig Theorems 3 d 3 i order to get the ollowig Sdwich theorems Theorem 33 Let be covex uivlet i the uit dis U i ' stisy 9 d 6 respectively Suppose tht d i is strlie i uivlet i U I A d H Q

17 Itegrl Opertor Deied by th Hdmrd Product 5 { is uivlet i U d the suborditio } ' { holds the d is the best subordit d is the best domit Acowledgemet ' } The wor here ws supported by UKM-ST-6-FRGS7-9 MOHE Mlysi The uthors lso would lie to th the oymous reeree or the iormtive d cretive commets give to the rticle Reereces Al-Shsi K & Drus M Dieretil suborditio with geerlied derivtive opertor to pper i AMMS Drus M & Ibrhim R Geerlitio o dieretil opertor ourl o Mthemtics d Sttistics 43 pp S l ge GS Subclsses o uivlet uctios Lecture Notes i Mth 3 Spriger-Verlg Berli pp Ruscheweyh S New criteri or uivlet uctios Proc Amer Mth Soc Al-Oboudi FM O uivlet uctios deied by geerlied S l ge opertor IMMS 7 pp Al-Shsi Drus M A opertor deied by covolusio ivolvig polylogrthms uctios ourl o Mthemtics d Sttistics 4 pp

18 5 Msli Drus & Rbh W Ibrhim 7 Noor KI O ew clsses o itegrl opertors NtGeomet 6 pp Noor KI & Noor MA O itegrl opertors o Mth Aly d Appl 38 pp Bricm L lie lytic uctios IBull Amer Mth Soc 79 pp Bulboc T lsses o irst-order dieretil superorditios DemostrMth 35 pp 87-9 Miller SS & Mocu PT Subordits o dieretil superorditios omplex Vribles 48 pp Miller SS & Mocu PT Dieretil Suborditios: Theory d Applictios Pure d Applied Mthemtics 5 Deer New Yor