Joul of Qulity Mesuemet d Alysis JQMA 7 0 4-5 Jul Peguu Kuliti d Alisis PROBLEMS AND PROPERTIES OF A NEW DIFFERENTIAL OPERATOR Mslh d Sift-sift sutu Pegopesi Peme Bhu MASLINA DARUS & IMRAN FAISAL ABSTRACT I this ppe we itoduce d study ew diffeetil opeto defied i the ope uit disc U : C. Usig this opeto we the itoduce ew suclss of lytic fuctios. Moeove we discuss coefficiet estimtes gowth d distotio theoems d iclusio popeties fo the fuctios elogig to the clss. Keywods: Alytic fuctios; covex fuctios; diffeetil opeto ABSTRAK Dlm mlh ii pegopesi peme hu dlm ce uit U : C dipeel d diji. Deg meggu pegopesi ii suels u fugsi lisis dipeel. Mlh gg peli teoem petumuh d eot d sift gum utu els tuut diicg. Kt uci: Fugsi lisis; fugsi cemug; pegopesi peme. Itoductio d Pelimiies Let A deote the clss of fuctios f of the fom f which e lytic d omlised i usul sese i the ope uit disc U : C Fo fuctio f i A we defie the followig diffeetil opeto: 0 D f f ; D f f f ; 3 D f D D f 4
Msli Dus & Im Fisl If f is give y the fom 4 we get D f 5 whee f A N 0. This geelises my opetos s follows. i Whe 0 we get D f the so-clled Al-Ooudi 004 diffeetil opeto. ii Whe 0 d we get D f the Sălăge s 983 diffeetil opeto. iii Whe 0 d we get D f diffeetil opeto give y Ulegddi d Somth 99. iv Whe 0 d eplcig y we get D f the diffeetil opeto of Cho d Sivstv 003. v Whe 0 d eplcig y we get D f 4
Polems o ew diffeetil opeto d its popeties well ow diffeetil opeto of Aouf et l. 009. Let deote the suclss of A cosistig of fuctios f which stisfy Re{ D f [ D f } 0 6 whee D f is give y 5. This implies tht it stisfies the followig iequlity D f D f D f D f whee U; 0; N0; C {0}. We ote tht i 0 7 Re{ f A: Re{ [ f f } U} 0 ii 00 D Re{ f A: Re{ [ iii 0 R Re{ f A: Re{ [ D iv 00 0 Re{ f A : Re{ [ v 0 0 R Re{ f A : Re{ [ f } U} 0 f } U} 0 f } U} 0 f } U} 0 43
Msli Dus & Im Fisl 0 vi 00 Re{ f A: Re f 0 U} 0 0 vii 0 R Re{ f A : Re f 0 U} 0 The clss R ws studied y Hlim 999 the clss y Che 974; 975 d whees the clss R y Eohi 965.. Coefficiet Iequlities I this sectio we fid the coefficiet iequlity fo the clss. Theoem. Let the fuctio f defied y stisfies the coditio [ [ 0 N 0 0 0 8 The f Poof. Suppose tht the iequlity 8 holds. The we hve fo U D f D f D f D f [ [ [ [ [ [ 44
Polems o ew diffeetil opeto d its popeties [ [ { [ [ } 0. whee D f is give y 5. This implies [ [ which shows tht f Coolly. Let the fuctio f defied y e i the clss hve The we [ [. Coolly. Let the hypotheses of Theoem. e stisfied. The fo 0 d we hve. 3. owth d Distotio Theoems A gowth d distotio popety fo fuctio f to e i the clss is give s follows: Theoem. If the fuctio f defied y is i the clss the fo we hve f [ [ f [ [ Poof. Let f the y Theoem. We hve 45
Msli Dus & Im Fisl [ [ [ [ Fom equtio we hve f Which implies f f [ [ Similly we c pove tht f [ [ Theoem 3. Let the hypotheses of Theoem e stisfied the fo f [ [ f [ [ Poof. Fom Theoem we hve f Sice [ [. d hece 46
Polems o ew diffeetil opeto d its popeties 47 [ [ [ [ we hve [ [. Fom we hve f f. Which poves tht f [ [ Similly f f shows tht f [ [
Msli Dus & Im Fisl Coolly 3. Let the hypotheses of Theoem e stisfied if 0 the fo we hve f f Theoem 4. If the fuctio f defied y is i the clss the fo we hve f [ [ f [ [ Poof. Let f the y usig Theoem we hve Also [ [ f f This shows tht f [ [ 48
Polems o ew diffeetil opeto d its popeties 49 Similly we c pove tht f [ [ 3. Iclusio popeties The iclusio popeties fo the clss e give y the followig theoem. Theoem 5. Let the hypotheses of Theoem e stisfied. The whee d Poof. Let f. The y usig Theoem we hve [ [ if implyig tht i such tht. This shows tht [ [ [ [ o
Msli Dus & Im Fisl [ [. Hece f which shows tht Similly let f the y usig Theoem we hve.this implies tht [ [ d hece. This poves tht f d filly implies tht. Employig simil pocedue we c pove tht d Fo moe detils out coefficiet ouds we efe to Joshi 007 Aouf 987 Silvem 975 Ri 997 d Ow d Aouf 989 espectively. Acowledgemets The wo peseted hee is fully suppoted y UKM-ST-06-FRS007-009. Refeeces Al-Ooudi F.M. 004. O uivlet fuctios defied y geelied Slge opeto. It. J. MtSci. 5:49-436. 50
Polems o ew diffeetil opeto d its popeties Aouf M.K. El-Ashwh R.M. & El-Dee S. M. 009. Some iequlities fo ceti p-vlet fuctios ivolvig exteded multiplie tsfomtios. Poc. Pist Acd. Sci. 46: 7-. Aouf M.K. 987. O suclsses of uivlet fuctios with egtive coefficiets III. Bull. Soc. Roy. Sci. Liege 56: 465-473. f Che M.P. 974. O fuctios stisfyig Re. Tmig J. Mth. Sci. Net. 3: 3-34. f Che M.P. 975. O the egul fuctios stisfyig Re. Bull. Is. Mth. Sci. Net. Acd. Siic 3: 65-70. Cho N.E. & Kim T.H. 003. Multiplie tsfomtios d stogly close to covex fuctios. Bull. Koe Mth. Soc. 40: 399-40. Cho N.E. & Sivstv H.M. 003. Agumet estimtes of ceti lytic fuctios defied y ceti Multiplie tsfomtios. Mth. Comput. Modelig 37: 39-49. Eohi T.. 965. Ceti estimtes i specil clsses of uivlet fuctios egul i the cicle. Dopovidi Ademiji Nu Koji RSR: 984-988. Hlim S.A. 999. O clss of fuctios of complex ode. Tmig J. Mth. 30: 47-53. Joshi S.B. 007. A uified pesettio of ceti suclss of lytic fuctio with egtive coefficiet. Mthemti 3: 3-8. Ow S. & Aouf M.K. 989. O suclsses of uivlet fuctios with the coefficiets. Pue App. Mth..Sci. 9:3-39. Ri R.K. 997. O ceti clsses of lytic fuctios d pplictios to fctiol clculus opeto. Itegl Tsfom d Specil Fuctios 5:3-9. Slge.S. 983. Suclsses of uivlet fuctios. Lectue otes i mthemtics Spige-Velg 03: 36-37. Silvem H. 975. Uivlet fuctios with egtive coefficiets. Poc. Ame. Mth. Soc. 5:09-6. Ulegddi B.A. & Somth C. 99. Ceti clsses of uivlet fuctio. I: Sivstv H.M. & Ow S. Eds.. Cuet Topics i Alytic Fuctio Theoy. Sigpoe: Wold Scietific Pulishig Compy. Cete fo Modellig d Dt Alysis DELTA School of Mthemticl Scieces Fculty of Sciece d Techology Uivesiti Kegs Mlysi 43600 UKM Bgi Selgo DE Mlysi E-mil: msli@um.my * im_fisll@yhoo.com * Coespodig utho 5