On New Bijective Convolution Operator Acting for Analytic Functions

Transcription

1 Jourl o Mthetics d Sttistics 5 (: 77-87, 9 ISSN Sciece Pulictios O New Bijective Covolutio Opertor Actig or Alytic Fuctios Oqlh Al-Rei d Msli Drus School o Mtheticl Scieces, Fculty o Sciece d Techology, Uiversiti Kegs Mlysi, Bgi 436, Selgor, Drul Ehs, Mlysi Astrct: Prole stteet: We itroduced ew ijective covolutio lier opertor deied o the clss o orlied lytic uctios. This opertor ws otivted y y reserchers ely Srivstv, Ow, Ruscheweyh d y others. The opertor ws essetil to oti ew clsses o lytic uctios. Approch: Siple techique o Ruscheweyh ws used i our preliiry pproch to crete ew ijective covolutio lier opertor. The preliiry cocept o Hdrd products ws etioed d the cocept o suorditio ws give to give shrp proos or certi suiciet coditios o the lier opertor oreetioed. I ct, the suorditig ctor sequece ws used to derive dieret types o suorditio results. Results: Hvig the lier opertor, suorditio theores were estlished y usig stdrd cocept o suorditio. The results reduced to wellkow results studied y vrious reserchers. Coeiciet ouds d iclusio properties, growth d closure theores or soe suclsses were lso otied. Coclusio: Thereore, y iterestig results could e otied d soe pplictios could e gthered. Key words: Covolutio, covex uctios, strlike uctios, prestrlike uctios, suorditios y: ( = INTRODUCTION LetA deote the clss o uctios orlied + + ( which re lytic i the ope uit disk U = { : < }, lett deote the suclss oa cosistig o uctios o the or: ( =, (, U We deote lso y K the clss o uctios A tht re covex i U : Give two uctios,g A, where = = ( = d g( = Usig the covolutio techiques, Ruscheweyh [] itroduced d studied the clss o prestrlike uctios o order β. Thus A is sid to e prestrlike uctio o order β ( β < i ( s β ( is strlike uctio o order β, where: s β( = = + C(, ( β β + ( d C( β, + = j= (j + β! Let the uctio ϕ (,; e give y: ( ϕ + (,; := (,,,...; U = ( ( their Hdrd product or covolutio g is deied y: ( g( = ( U = where (x is the Pochher syol deied y:, = ; Γ (x + (x := = x(x + (x +...(x +, N := {,,3,...} Γ(x Correspodig Auhtor: Msli Drus, School o Mtheticl Scieces, Fculty o Sciece d Techology, Uiversity Kegs Mlysi, Bgi 436, Selgor, Drul Ehs, Mlysi 77

2 J. Mth. & Stt., 5 (: 77-87, 9 Note tht (x + = (x + (x / x, ( β + (x (x = (x d = ( > β β. (! The uctio ϕ (, ; is icoplete et uctio relted to the Guss hypergeoetric uctio y ϕ (,; = F (,,,, where the hypergeoetric uctio F (,,c, is deied y: = ( + ( + F (,,c, = c! c(c +! Also, it hs lytic cotiutio to the -ple cut log the positive rel lie ro to. We ote tht ϕ(,; = / ( d ϕ (,; is the Koee uctio. Correspodig to the uctio ϕ (, ;, Crlso d Sher itroduced i [] covolutio opertor o A ivolvig icoplete et uctio s: ( L(, ( := (,; ( = + ϕ (3 = ( The Ruscheweyh derivtives o ( re L( β +, (, ( β. MATERIALS AND METHODS Usig the Hdrd product (or covolutio, we deie ew ijective covolutio opertor cts o lytic uctios i U. The we stte soe o its properties d its specil cses which will e used i this study. Correspodig to our opertor, we deie clsses o lytic uctios look like the clsses o strlike d covex uctios o order α ( α <. Suorditio priciple d kow results o suorditio ctor sequece will e used i our ivestigtio o tht clsses. Deiitio : Let the uctio φ e give y: ( φ(,,, ; := ( + ( = + Note tht: J(,,, ( = L(, (, J(,,, ( = ( L(, ( + (L(, ( = J (L(, (, J(,,, ( = J (J(,,, (, where N = {,,3,...} I =, -, -,, the J(,,,( is polyoil. I,-, -, the pplictio o the root test shows tht the iiite series or J(,,,( hs the se rdius o covergece s tht or ecuse: ( ( ( li [ + ] = li ( + ( ( ( ( + ( ( ( ( = li ( ( ( + ( li ( li ( ( = ( li ( d the lst expressio equl sice: ( li ( Hece, J(,,, psa ito itsel. So, we shll ssue, uless other wise stted, tht (, -, -, d (,-,-,. We deote y g where g A, the uctio which stisies g( g ( = / ; ( U. For > d >, i =, the: J(,,, ( = L(, ( = F (,,, (. ( = where, (,,,...; U,, Z d (x is the Pochher syol. Oe deies lier opertor J(,,, : A A y the ollowig Hdrd product (or covolutio: J(,,, ( := φ(,,,; ( ( = ( + = + ( (4 78 I >, the the opertor J(,,,( c e represeted y: J(,,, ( = F (,,,... ( + + ( ( ties F (,,/,... F (,,/, ( ties

3 i ( =,,, d J(,,, ( = F (,,,... ( ( ( F (,,+ /,... F (,,+ /, ( ( ties J. Mth. & Stt., 5 (: 77-87, 9 ( ties i ( = -,-,. Note tht J(,,, = J(,,, is the idetity opertor d i,-,-,, the J(,,, hs cotiuous iverse J(,,, d is oe-to-oe ppig o A oto itsel. Hece, J(,,, ps A oto itsel, ict it ps the clss o lytic uctios i U ito itsel. It lso provides coveiet represettio o dieretitio d itegrtio. By speciliig the preters,, d, oe c oti vrious opertors, which re specil cses o J(,,, studied erlier y y uthors, such o those opertors: J(,,, ( = J(,,, ( = L(, ( The Ruscheweyh derivtives β J(,, β +, ( = D (; β [3] J(,,, ( ; N due to Al-Ooudi [4] J(,,, (; N due to S ( l ( ge [5] The rctiol opertor due to Ow d Srivstv J(,,, ( = ( = ( D (; γ γ γ γ γ Ω Γ γ D ( is the rctiol derivtive o o order γ ; γ,3,4,.., [6]. Also, ote tht the opertors J(,,, + (; N due to Noor [7,8] d J(,,, (; >, > due to Choi l et. [9] re specil cses o the iverse opertor J(,,, (. We prove ow the ollowig two idetities which will e used i this study. Le : Let A stisies (4. The we hve the ollowig: (J(,,, ( = J(,, +, ( ( J(,,, ( For (,,,... (J(,,, ( = (,, ( ( J(,,, ( Proo: Let J(,,, ( e s i (4. The we hve: J(, 79 ( J(,, +, ( = ( + + = = J(,,, ( = + ( ( + (+ ( = ( J(,,, ( ( ( ( = ( ( J(,,, ( = ( + + = ( + ( = ( + ( ( = ( J(,,, ( + ( + ( + where (,,,,.... Hece, the proo is coplete. Now, we itroduce ew clsses o lytic uctios ivolvig our opertor J(,,,. Deiitio : A uctio A is sid to e i the clss S(,,,, α or Z,, α <, i d oly i: (J(,,, ( R > α J(,,, ( or ll U (5 Deiitio 3: A uctio A is sid to e i the clss C(,,,, α or Z,, α <, i d oly i: ((J(,,, ( R > α (J(,,, ( or ll U. (6 Rerk : S(,, ( α,, α i d oly i is prestrlike uctio o order α. Deiitio 4 [] : Let g e lytic d uivlet i U. I is lytic i U, ( = g( d ( U g( U, the oe sys tht is suordite to g i U d we write p g or ( p g(. Oe lso sys tht g is superordite to i U.

4 J. Mth. & Stt., 5 (: 77-87, 9 Deiitio 5 [] : A iiite sequece { k} k= o coplex uers is sid to e suorditig ctor sequece i or every uivlet uctio i K, oe hs: k k k p U k= ( ( ; = Le [] : The sequece { k} k= is suorditig ctor sequece i d oly i: R + k k > ( U k = RESULTS Coeiciet ouds d Iclusios: we ivestigte suiciet coditios or the uctio A to e i the clsses S(,,,, α d C(,,,, α y otiig the coeiciet ouds. Moreover, we study soe iclusio properties or soe suclsses o S(,,,, α d C(,,,, α. Theore : Let A. I ( + α + α (7 ( ( ( the S(,,,, α, where S(,,,, α is deied s i Deiitio. Proo: Suppose tht (7 holds. The y usig Le d or ll U, we hve: (J(,,, ( J(,, +, ( = J(,,, ( J(,,, ( = ( + ( (+ = ( ( + ( + = ( ( + ( (+ = ( ( (+ ( ( + ( + α ( + α ( ( d this iplies S(,,,, α. This copletes the proo o Theore. Let S (,,,, α deote the clss o uctios A whose Tylor-Mcluri coeiciets stisy the coditio (7 d deote S [,,,, α] = S (,,,, α T. We ote tht: S (,,,, α S(,,,, α. Corollry : Let A e i the clss The: α ( ( + α (+ ( Exple : The uctio give y: ( = ± elogs to the clss ( α α ε ( ( + α ( + ( S (,,,, Z, d ε C with ε =. S (,,,, α. α or α, Exple : For α <, d N, the ollowig uctios deied y: ( α ( ± ( = ( =,,...; U ( + α (+ ( re i the clss S (,,,, α. Theore : Let A. I: ( + + α + α (8 ( ( ( ( the C(,,,, α, where C(,,,, α is deied s i Deiitio 3. The lst expressio is ouded y ( α i the ollowig iequlity which is equivlet to (7 holds: Proo: Suppose tht (8 holds. The y usig Le (i, with dieretitig its oth sides d or ll U, we hve: 8

5 J. Mth. & Stt., 5 (: 77-87, 9 (J(,,, ( ( + J(,, +, ( J(,, +, ( + ( J(,,, ( = (J(,,, ( J(,, +, ( ( J(,,, ( ( + ( ( + ( + ( + + ( ( ( + ( ( (+ ( The lst expressio is ouded y ( α i the ollowig iequlity which is equivlet to (8 holds: We ext study soe iclusio properties o the clsses S (,,,, α d C (,,,, α. ( + ( + ( + α Y ( ( + + ( ( + α ( α Theore 3: Let α < α <, < < d, Z with <. The: S (,,,, α S (,,,, α where Y = ( + d this iplies C(,,,, α. This copletes the proo o Theore. Let C (,,,, α deote the clss o uctios A whose Tylor-Mcluri coeiciets stisy the coditio (8 d deote C [,,,, α] = C (,,,, α T. We ote tht C (,,,, α C(,,,, α Corollry : Let A e i the clss The: α ( ( + ( + α ( + ( Exple 3: The uctio give y: C (,,,, α. S (,,,, α S (,,,, α S (,,,, α S (,,,, α S (,, +,, α S (,,,, α S (,,, +, α S (,,,, α Proo: By usig Theore d Le. Theore 4: Let α < α <, < < d, Z with <. The: C (,,,, α C (,,,, α ( = ± elogs to the clss ( α α ε ( ( + ( + α ( + ( C (,,,, Z, d ε C with ε =. α or α, Exple 4: For α <, d N, the ollowig uctios deied y: C (,,,, α C (,,,, α C (,,,, α C (,,,, α C (,, +,, α C (,,,, α C (,,, +, α C (,,,, α Proo: By usig Theore d Le. ( = ± ( α ( ( + ( + α ( + ( ( =,,...; U re i the clss C (,,,, α. + Growth d closure theores: we prove growth theores whe S (,,,, α d whe C (,,,, α. Furtherore, we prove closure theores whe is i the suclss S [,,,, α ] d whe it is i the suclss C [,,,, α ] respectively. 8

6 J. Mth. & Stt., 5 (: 77-87, 9 Theore 5: Let S (,,,, α. The: Theore 6: Let C (,,,, α. The: α ( ( α ( + α ( ( α ( + α ( + ( α ( + α ( + ( α ( + or, {,,,...} with + + or ll N, N,, α < d U. Proo: I S (,,,, α with, {,,,...}, + + or ll N,, < α < d N {}, the i view o Theore, we hve: ( α ( + ( + This yields: Now: ( + ( + α ( ( ( α α. α α + Also: ( ( α ( + ( α ( + ( α ( + This copletes the proo o Theore 5. Corollry 3: Uder the hypothesis o Theore 5. ( is icluded i disc cetered t the origi with rdius r give y: + r = ( ( α + α By usig the se proo techique o Theore 5, we c prove the ollowig result. 8 or, {,,,...} with + + or ll N, N,, α < d U. Corollry 4: Uder the hypothesis o Theore 6. ( is icluded i disc cetered t the origi with rdius r give y: α r = + ( ( α + Correspodig to the suclsses S [,,,, α ] d C [,,,, α ], we itroduce the ollowig closure theores. Theore 7: Let ( = d: α ( + ( =, ( =,,... ( + α ( + ( The is i the clss S [,,,, α ] i d oly i it c e writte s: ( = = µ (, where µ ( d µ =. Proo: Assue tht ( = µ (. The: ( = µ ( + µ ( = = α ( = µ + µ ( + α ( + ( α ( = µ + µ µ ( + α ( + ( α ( = µ ( + α ( + (

7 J. Mth. & Stt., 5 (: 77-87, 9 Thus: α ( µ ( + α (+ ( ( ( + α ( + = ( α µ ( = = ( α( µ α Coversely, suppose is i the clss Sice: d α (, ( ( + α (+ ( We y set: The : ( = ( + α ( + ( µ = ( α ( µ = µ = = µ ( + α ( + ( = µ ( ( = = = µ + µ ( = = = µ ( α ( This copletes the proo o Theore 7. S [,,,, α ]. Corollry 5: The extree poits o the clss S [,,,, α ] re the uctios ( = d: α ( + ( =, ( =,,... ( + α ( + ( By usig the se proo techique o Theore 7, we c prove the ollowig theore. Theore 8: Let ( = d: 83 α ( + ( =, ( =,,... ( + ( + α ( + ( The is i the clss C [,,,, α ] i d oly i it c e writte s: ( = = µ ( where, µ ( d µ =. Corollry 6: The extree poits o the clss C [,,,, α ] re the uctios ( = d: + ( =,( =,,... ( + ( + α (+ ( = α ( Suorditio results: we oti shrp suorditio results ssocited with the clsses S (,,,, α d C (,,,, α y usig the se techique s i [-5]. Soe pplictios o those i results which give iportt results o lytic uctios re lso ivestigted. Theore 9: Let S (,,,, α where, {,,,...} with + + or ll N, N, d α <. The: ( α ( + ( α + ( α ( + or every g i K d: R + ( > ( ( α + ( g( p g( ( U (9 α ( The costt + (( α / ( α ( + cot e replced y lrger oe. Proo: Let S (,,,, α d let: g( = + + c

8 J. Mth. & Stt., 5 (: 77-87, 9 e y uctio i the clss K. The we redily hve: ( α ( + ( α + ( α ( + ( g( ( α ( + = c + = ( α + ( α ( + + is suorditig ctor sequece, with =. I view o Le, this is equivlet to the ollowig iequlity: ( α ( + R + > ( U ( = α + ( α ( + Now, sice: Thus, the suorditio result (9 will hold true i the sequece: ( + α ( + ( ( ( α ( + ( α + ( α ( + is icresig uctio o or N,, α < d, {,,,...} with + + or every N, we hve: ( α ( + + R + R ( α ( + ( ( α + α + = = α + ( α ( + ( α + ( + α + ( α ( + ( α ( + > r α + ( α ( + α + ( α ( + ( + ( ( r + α + = ( ( α ( + α r r > ( = r. α + ( α ( + α + ( α ( + This proves the iequlity ( d hece lso the suorditio result (9 sserted y Theore 9. The iequlity ( ollows ro (9 y tkig: Next, we cosider the uctio: g( = K ( α ( = ( α ( + which stisies the ssuptio o Theore 9. The y usig (9, we hve: 84

9 J. Mth. & Stt., 5 (: 77-87, 9 ( α ( + ( α + ( α ( + ( p ( U d 3 α R ( > α It c e esily veriied or the uctio ( tht: ( α ( + i R ( = ( U U ( α + ( α ( + which copletes the proo o Theore 9. By tkig = i Theore 9, we hve the ollowig corollry. Corollry 7: I the uctio deied y ( stisies: The costt (3 α / ( α cot e replced y lrger oe. Lettig =, = d = α ; α < i Theore 9, we hve the ollowig corollry relted to the strlikeess o the rctiol derivtive d α rctiol itegrl opertor Ω ( due to Ow d Srivstv which is etioed ove. Corollry 9: I the uctio deied y ( stisies: (! ( + α α ( < α < ( α + ( + α α ( ( where, {,,,...} with + + or ll N d α <, the or every uctio g i K, oe hs: d α ( g( p g( ( U ( α + ( α α R ( > + α The costt + ( α / ( α cot e replced y lrger oe. Puttig = d = i Theore 9, we hve the ollowig corollry. Corollry 8: I the uctio deied y ( stisies: the or every uctio g i K, oe hs: d ( g( p g( ( U 3 α 3 α R ( > The costt (3 α / cot e replced y lrger oe. It is cler ro the proo o Theore 9 tht the α uctio ( = ( α < ; U is α extrel uctio o Corollry 8. Also the ollowig exple gives o-polyoil uctio stisies the se corollry. Exple 5: Let the uctio h e deied y: h( = ( α < ; U α + (3 α ( + α α, ( α < the or every uctio g i K, oe hs: α ( g( p g( ( U (3 α 85 the ove uctio is lytic i U d it is equivlet to: The we hve: α h( = + (3 α

10 J. Mth. & Stt., 5 (: 77-87, 9 α α α ( = = (3 α (3 α 4 3α + α α Thereore, the Tylor-Mcluri coeiciets o the uctio h stisy the coditio i Corollry 8. By the se proo techique o Theore 9, we c prove the ollowig theore. Theore : Let C (,,,, α where, {,,,...} with + + or ll N, N, d α <. The: ( α ( + α + ( α ( + or every g i K d: R + ( > ( ( α + ( g( p g( ( U ( α (3 The costt + (( α / ( α ( + cot e replced y lrger oe. By tkig = i Theore, we hve the ollowig corollry. Corollry : I the uctio deied y ( stisies: ( + + α α ( ( ( Puttig = d = i Theore, we hve the ollowig corollry. Corollry : I the uctio deied y ( stisies: ( + ( + α α ( α < the or every uctio g i K, oe hs: d α ( g( p g( ( U 5 3α 5 3α R ( > 4 α The costt (5 3 α / (4 α cot e replced y lrger oe. Lettig =, = d = α, α < i Theore, we hve the ollowig corollry relted to the covexity o the rctiol derivtive d rctiol α itegrl opertor Ω ( due to Ow d Srivstv which is etioed ove. Corollry : I the uctio deied y ( stisies: (! ( + ( + α α ( α < ( α + the or every uctio g i K, oe hs: where, {,,,...} with + + or ll N d α <, the or every uctio g i K, oe hs: α ( g( p g( ( U ( α + ( α d ( g( p g( ( U 5 α R ( > ( 5 α d α R ( > + ( α The costt + ( α / ( α cot e replced y lrger oe. The 5 α cot e replced y lrger oe. α Note tht the uctio ( = ( α ( α < ; U is extrel uctio o Corollry. Also the ollowig exple gives o-polyoil uctio stisies the se corollry. Exple 6: Let the uctio h e deied y: 86

11 g( = ( α < ; U α + 4(3 α The ove uctio is lytic i U d it is equivlet to: The we hve: α g( = + = 4(3 α + α α α ( ( 4 = = 4(3 α = 4(3 α ( α + + α α Thereore, the Tylor-Mcluri coeiciets o the uctio g stisy the coditio i Corollry. DISCUSSION LetA deote the set o uctios o the or ( = , which re lytic i the ope uit disk. A ew ijective covolutio lier opertor deied oa is itroduced, which is geerlitio o Crlso-Sher opertor [] d vrious well kow opertors d clsses o lytic uctios ivolvig tht opertor re studied. Mily, severl properties o soe suclsses re ivestigted, like coeiciet ouds, iclusios, growth d closure theores. Furtherore, i suorditio results with soe pplictios re ivestigted s well. The proo thechique o those suorditio results is used erlier y y reserchers, ely Srivstv, Attyi, Ali, Rvichdr d Seeivsg. CONCLUSION We coclude this study with soe suggestios or uture reserch, oe directio is to study other clsses o lytic uctios ivolvig our opertor J(,,,. Aother directio would e studyig other properties o the clsses S(,,,, α d C(,,,, α. ACKNOWLEDGMENT The study showed here ws supported y E- ScieceFud 4---SF45, MOSTI, Mlysi. REFERENCES. Ruscheweyh, S., 977. Lier opertor etwee clsses o prestrlike uctios. Coetrii Mthetici Helvetici, 5: DOI:.7/BF56738 J. Mth. & Stt., 5 (: 77-87, Crlso, B.C. d D.B. Sher, 984. Strlike d prestrlike hypergeoetric uctios. SIAM. J. Mth. Al., 5: &idtype=cvips&gis=yes 3. Ruscheweyh, S., 975. New criteri or uivlet uctios. Proc. A. Mth. Soc., 49: Al-Ooudi, F.M., 4. O uivlet uctios deied y geerlied S ( l ( ge opertor. It. J. Mth. Mth. Sci., 7: DOI:.55/S S ( l ( ge, G.S., 983. Suclsses o uivlet uctios. Lecture Notes Mth., 3: DOI:.7/BF Ow, S. d H.M. Srivstv, 987. Uivlet d strlike geerlied hypergeoetric uctios. C. J. Mth., 39: Iyt Noor, k., 999. O ew clsses o itegrl opertors. J. Nturl Geoet., 6: Iyt Noor, k. d M.A. Noor, 999. O itegrl opertors. J. Mth. Al. Appli, 38: Choi, J.H., M. Sigo d H.M. Srivstv,. Soe iclusio properties o certi ily o itegrl opertors. J. Mth. Al. Appli, 76: DOI:.6/S-47X(5-. Good, A.W., 983. Uivlet Fuctios. Mrier Pulishig Copy, Tp, Florid, pp: 84-88, 35.. Wil, H.S., 96. Suorditig ctor sequeces or covex ps o the uit circle. Proc. A. Mth. Soc., : Srivstv, H.M. d A.A. Attiy, 4. Soe suorditio results ssocited with certi suclsses o lytic uctios. J. Iequl. Pure Applied Mth., 5: _JIPAM/3_4_www.pd 3. Attiy, A.A., 5. O soe pplictios o suorditio theore. J. Mth. Al. Appli., 3: DOI:.6/j.j Attiy, A.A., Nk Eu Cho d M.A. Kuti, 8. Suorditio properties or certi lytic uctios. It. J. Mth. Mth. Sci., 8: 8-8. DOI:.55/8/ Ali, R.M., V. Rvichdr, N. Seeivsg, 6. Suorditio y covex uctios. It. J. Mth. Mth. Sci., 4: DOI:.55/IJMMS/6/6548