Approximation by complex Durrmeyer-Stancu type operators in compact disks

Transcription

1 Re et al. Joual of Iequalities ad Applicatios 3, 3:44 R E S E A R C H Ope Access Appoximatio by complex Dumeye-Stacu type opeatos i compact disks Mei-Yig Re *, Xiao-Mig Zeg ad Liag Zeg * Coespodece: pmeiyig@63.com Depatmet of Mathematics ad Compute Sciece, Wuyi Uivesity, Wuyisha, 3543, Chia Full list of autho ifomatio is available at the ed of the aticle Abstact I this pape we itoduce a class of complex Stacu-type Dumeye opeatos ad study the appoximatio popeties of these opeatos. We obtai a Vooovskaja-type esult with quatitative estimate fo these opeatos attached to aalytic fuctios o compact disks. We also study the exact ode of appoximatio. Moe impotat, ou esults show the ovecovegece pheomeo fo these complex opeatos. MSC: 3E; 4A5; 4A36 Keywods: complex Dumeye-Stacu type opeatos; Vooovskaja-type esult; exact ode of appoximatio; simultaeous appoximatio; ovecovegece Itoductio I 986, some appoximatio popeties of complex Bestei polyomials i compact disks wee iitially studied by Loetz. Vey ecetly, the poblemof the appoximatio of complex opeatos has bee causig geat coce, which has become a hot topic of eseach. A Vooovskaja-type esult with quatitative estimate fo complex Bestei polyomials i compact disks was obtaied by Gal Also, i3 8 simila esults fo complex Bestei-Katoovich polyomials, Bestei-Stacu polyomials, Katoovich- Schue polyomials, Katoovich-Stacu polyomials, complex Favad-Szász-Miakja opeatos, complex Beta opeatos of fist kid, complex Baskajov-Stacu opeatos, complex Bestei-Dumeye polyomials, complex geuie Dumeye-Stacu polyomials ad complex Bestei-Dumeye opeatos based o Jacobi weights wee obtaied. The aim of the peset aticle is to obtai appoximatio esults fo complex Dumeye- Stacu type opeatos which ae defied fo f :, C cotiuous o, by (f ; z: f p,k (z p,k (tf ( t α dt ( α p, (z, ( whee α, β ae two give eal paametes satisfyig the coditio α β, z C,,,...,adp,k (z ( k z k ( z k. 3 Re et al.; licesee Spige. This is a Ope Access aticle distibuted ude the tems of the Ceative Commos Attibutio Licese ( which pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial wok is popely cited.

2 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page of 3 Note that, fo α β, these opeatos become the complex Dumeye-type opeatos (f ; zm (, (f ; z, this case has bee ivestigated i. Auxiliay esults I the sequel, we shall eed the followig auxiliay esults. Lemma Let e m (z z m, m N {}, z C, N, α β, the we have that (e m ; z is a polyomial of degee less tha o equal to mi(m, ad (e m ; z m j ( m j α m j j ( (e m j ; z. Poof By the defiitio give by (, the poofiseasy, heethe poofisomitted. Let m,,,accodigto,lemma,byasimplecomputatio,wehave (e ; z; z (e ; z ( ( α ; (e ; z ( z ( z ( ( αz ( ( α (. Lemma Let e m (zz m, m N {}, z C, N, α β, fo all z,, we have M (e m ; z m. Poof The poof follows diectly Lemma ad, Lemma. Lemma 3 Let e m (zz m, m, N, z C ad α β, the we have z( z ( (e m ; z M (e m ; z ((m Poof Let T T T (m z α( m (e m ; z ((m αm( α ( (m M (e m ; z. ( ( t α,k (f : p,k (tf ( t α,k (f : p,k (ttf ( t α,k (f : p,k (tt f E (f ; z: p,k (zt,k (f, dt, dt, dt,

3 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 3 of 3 the we have M (f ; ze T,k (e m T,k (e m ( α (f ; zf p,k (t p, (z, ( t α α T,k (e m α T,k (e m, ( ( t α p,k (t ( T,k (e m ( α T,k (e m. By a simple calculatio, we obtai ( t α m dt α ( t α m dt α( T,k (e m z( zp,k (z(k zp,k(z, (k ( t p,k (tt( tp,k (t. It follows that whee z( z ( E (e m ; z (k zp,k (z p,k (z ( t α p,k (z ( t α m p,k (t dt (k ( t ( t p,k (t m dt ze (e m ; z, (k ( t ( t ( t α m p,k (t dt ( t α m p,k (z t( tp,k (t dt ( p,k (z ( ( p,k (z T,k (e m ( t α m t( tp,k (t dt E α( (e m ; z p,k (zt,k (e m E (e m ; z.

4 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 4 of 3 Also, usig itegatio by pats, we have ( t α m t( tp,k (t dt ( t α m p,k (t( t dt m ( t α m p,k (tt( t dt T,k (e m T,k (e m m T,k (e m m T,k (e m (m T,k (e m αm(α ( T,k (e m. ( α m αm T,k (e m So, i coclusio, we have z( z ( E (e m ; z (m E (e m ; z α( m m z αm( α ( E (e m ; z, E (e m ; z which implies the ecuece i the statemet. Lemma 4 Let N, m,3,...,e m (zz m, S α β, we have S,m (z z( z ( M (e m ; z ((m,m (z: (m z α(m S,m ((m (z αm ((m M (e m ; z α(m ( α ( (m M (e m ; z (e m ; z z m, z C ad (m z α(m z m z m. (3 ((m Poof Usig the ecuece fomula (, by a simple calculatio, we ca easily get the ecuece (3, the poof is omitted. 3 Mai esults The fist mai esult is expessed by the followig uppe estimates.

5 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 5 of 3 Theoem Let α β, R, D R {z C : z < R}. Suppose that f : D R C is aalytic i D R, i.e., f (z m c mz m fo all z D R. (i Fo all z ad N, we have M (f ; z f (z K (f, whee K (f ( m c m m(m α β m <. (ii (Simultaeous appoximatio If < < R ae abitaily fixed, the fo all z ad, p N, we have ( (f ; z (p f (p (z K (f p! ((, p whee K (f is defied as i the above poit (i. Poof Takig e m (zz m, by the hypothesis that f (z is aalytic i D R, i.e., f (z m c mz m fo all z D R, it is easy fo us to obtai M (f ; z m c m (e m ; z. Theefoe, we get M (f ; z f (z m m c m M (e m ; z e m (z c m M (e m ; z e m (z, as (e ; ze (z. (i Fo m N, takig ito accout that (e m ; z is a polyomial of degee mi(m,, by the well-kow Bestei iequality ad Lemma,weget ( (e m ; z m max { (e m ; z : z } (m m. O the oe had, whe m,fo z, by Lemma,wehave (e ; z e (z z ( ( α z ( α β. Whe m, fo N, z, α β, iviewof (m z α(m ((m, usig the ecuece fomula (3 ad the above iequality, we have M (e m ; z e m (z S,m (z ( (m m S,m (z

6 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 6 of 3 α m α m m β ( m m ( m S,m (z α ( m m β ( m S,m (z m α β ( m. By witig the last iequality, fo m,..., we easily obtai step by step the followig: M (e m ; z e m (z ( S,m (z (m α β ( m m α β ( m S,m (z (m m(α β ( m m(m α βm. I coclusio, fo ay m, N, z, α β,wehave M (e m ; z e m (z m(m α βm, fom which it follows that M (f ; z f (z c m m(m α β m. m By assumig that f (z is aalytic i D R,wehavef ( (z m c mm(m z m ad the seies is absolutely coveget i z, soweget m c m m(m m <, which implies K (f ( m c m m(m α β m <. (ii Fo the simultaeous appoximatio, deotig by Ɣ the cicle of adius > ad cete, sice fo ay z ad υ Ɣ, wehave υ z. ByCauchy sfomula,it follows that fo all z ad N,wehave ( (f ; z (p f (p (z p! π Ɣ (f ; υ f (υ dυ (υ z p K (f p! π π ( p K (f p! (, p which poves the theoem. Theoem Let α β, R >,D R {z C : z < R}. Suppose that f : D R C is aalytic i D R, i.e., f (z k c kz k fo all z D R. Fo ay fixed, R ad all N, z, we

7 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 7 of 3 have M (f ; z f (z α (βz f (z M (f M z( z f (z, (f ( M, (f (, (4 whee M (f c k kb k, k < with B k, (k 3 3k 3k (4k 3 k 4k 6 (k 3 9k 3k 6,M, (f c k k(k α k 3 β k αβ k β k, M, (f c k k(k (α β k <. Poof Fo all z D R,wehave M (f ; z f (z α (βz f (z z( z (f ; z f (z z f (z (f ; z f (z z( zf (z zf (z : I I. M (f ; z (f ; z α βz z( z f (z f (z α βz f (z f (z By, Theoem, we have I M (f,wheem (f c k kb k, k < with B k, (k 3 3k 3k (4k 3 k 4k 6(k 3 9k 3k 6. Next, let us estimate I. By f is aalytic i D R, i.e., f (z k c kz k fo all z D R,wehave I c k c k M O the oe had, whe k, sice (e k ; z (e k ; z α βz kz k (e k ; z (e k ; z α βz kz k. k k ( k (β k j j j β k j, by Lemma,weobtai (β k (e k ; z (e k ; z α βz k ( k j α k j j ( (e k j ; z M j j kz k k ( k (e k ; z α βz kz k k ( k j α k j j ( (e k j ; z kk α ( (e k k ; z

8 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 8 of 3 k j j ( k j β k j j ( (e k k ; z α βz kz k k ( k j α k j j ( (e k j ; z kk α M (e ( k k ; z e k (z kk α k ( k j β k j ( k zk j ( (e k k ; z j kk β M (e ( k k ; z e k (z kk β ( k zk α βz j α k j k j ( k j j kz k ( k (e j ; z kk α ( k M (e k ; z e k (z k ( k j β k j j ( (e k k ; z kk β M (e ( k k ; z e k (z k ( k kαz k k ( k kβz k. By the poof of, Coollay 3, fo ay k N, z,, we have M (e k ; z k, M (e k ; z e k k k. Hece, fo ay k, z,, we ca get k ( k j α k j j ( (e k j ; z j k ( k j α k j j ( k k j k k(k (k j(k j j k(k k(k ( k j j j α k j ( k α k ( k j α k j ( k j ( k α ( k α ( k ad k k α M (e ( k k ; z e k (z k(k α k. ( Also, usig k ( k k j ( k j j β k j ( k kβ (

9 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 9 of 3 fo ay k, z,, we get M (e k ; z (e k ; z α βz kz k k(k α ( k k(k α ( k k3 β ( k k αβ ( k k β ( k k k(k α k 3 β k αβ k β ( ( k(k (α β. k k(k β ( k O the othe had, whe k, usig Lemma ad (e ; z z (see 9, by a simple calculatio, we ca get (e ; z (e ; z α βz (β (β αβ β. So, fo ay k N, z,, we have M (e k ; z (e k ; z α βz kz k k k(k α k 3 β k αβ k β ( ( k(k (α β. k Hece, we have I M, (f ( M, (f (, whee M, (f c k k(k α k 3 β k αβ k β k, M, (f c k k(k (α β k. I coclusio, we obtai M (f ; z f (z α (βz f z( z (z f (z I I M (f, (f ( M, (f (. M I the followig theoem, we obtai the exact ode of appoximatio. Theoem 3 Let α β, R >,D R {z C : z < R}. Suppose that f : D R C is aalytic i D R. If f is ot a polyomial of degee, the fo ay, R, we have M (f ; f C (f, N, whee f max{ f (z ; z } ad the costat C (f >depeds o f, adα, β, but it is idepedet of.

10 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page of 3 Poof Defie e (zz ad H (f ; z (f ; z f (z α (βz f (z Fo all z D R ad N,wehave z( z f (z. M (f ; z f (z { α (βz f (zz( zf (z H (f ; z }. I view of the popety F G F G F G, it follows M (f ; f { α (βe f e ( e f H (f ; }. Cosideig the hypothesis that f is ot a polyomial of degee i D R,wehave α (βe f e ( e f >. Ideed, supposig the cotay, it follows that α (βz f (zz( zf (z fo all z D. Defiig y(z f (z ad lookig fo the aalytic fuctio y(z udethefomy(z k a kz k, afte eplacemet i the diffeetial equatio, the coefficiets idetificatio method immediately leads to a k foallk N {}. Thisimpliesthaty(z fo all z D ad theefoe f is costat o D, a cotadictio with the hypothesis. Usig iequality (4, we get H (f ; N (f, (5 whee N (f M (f M, (f M, (f. Theefoe, thee exists a idex, depedig oly o f, ad α, β, such that fo all,wehave α (βe f e ( e f H (f ; ( α (βe f e ( e f, which implies (f ; f α (βe f e ( e f fo all.

11 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page of 3 Fo {,,..., },wehave M (f ; f W, (f, whee W, (f (f ; f >. As a coclusio, we have M (f ; f C (f fo all N, whee { C (f mi W, (f, W, (f,...,w, (f, α (βe f e ( e f }, this completes the poof. Combiig Theoem 3 with Theoem, we get the followigesult. Coollay Let α β, R >,D R {z C : z < R}. Suppose that f : D R C is aalytic i D R. If f is ot a polyomial of degee, the fo ay, R, we have M (f ; f, N, whee f max{ f (z ; z } ad the costats i the equivalece deped o f, adα, β, but they ae idepedet of. Theoem 4 Let α β, R >,D R {z C : z < R}. Suppose that f : D R C is aalytic i D R. Also, let < < Radp N be fixed. If f is ot a polyomial of degee p, the we have ( M (f ; (p f (p, N, whee f max{ f (z ; z } ad the costats i the equivalece deped o f,,, p, α ad β, but they ae idepedet of. Poof Takig ito accout the uppe estimate i Theoem, it emais to pove the lowe estimate oly. Deotig by Ɣ the cicle of adius > ad cete, by Cauchy s fomula, it follows that fo all z ad N,wehave ( M (f ; z (p f (p (z p! πi Ɣ M (f ; v f (v (v z p dv.

12 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page of 3 Keepig the otatio thee fo H (f ; z, fo all N,wehave M (f ; z f (z { α (βz f (zz( zf (z H (f ; z }. By usig Cauchy s fomula, fo all v Ɣ,weget ( M (f ; z (p f (p (z { (α (βz f (zz( zf (z (p p! H } (f ; v dv. πi Ɣ (v z p Passig ow to ad deotig e (zz, it follows ( M (f ; (p f (p ( α (βe f e ( e f (p p! πi Ɣ H (f ; v dv (v p. Sice fo ay z ad υ Ɣ we have υ z,so,byiequality(5, we get p! πi Ɣ H (f ; v dv (v p p! π π H (f ; ( p N (f p! ( p, whee N (f M (f M, (f M, (f. Takig ito accout that the fuctio f is aalytic i D R, by followig exactly the lies i Gal 5, seeig also the book Gal 6, pp (whee it is poved that (α βe f e ( e f (p >,wehave ( α (βe f e ( e f (p >. I cotiuatio, easoig exactly as i the poof of Theoem 3, we ca get the desied coclusio. Competig iteests The authos declae that they have o competig iteests. Authos cotibutios All authos cotibuted equally ad sigificatly i witig this aticle. All authos ead ad appoved the fial mauscipt. Autho details Depatmet of Mathematics ad Compute Sciece, Wuyi Uivesity, Wuyisha, 3543, Chia. Depatmet of Mathematics, Xiame Uivesity, Xiame, 365, Chia.

13 Reet al. Joual of Iequalities ad Applicatios 3, 3:44 Page 3 of 3 Ackowledgemets The authos ae most gateful to the edito ad aoymous efeee fo caeful eadig of the mauscipt ad valuable suggestios which helped i impovig a ealie vesio of this pape. This wok is suppoted by the Natioal Natual Sciece Foudatio of Chia (Gat o. 6734, the Class A Sciece ad Techology Poject of Educatio Depatmet of Fujia Povice of Chia (Gat o. JA34, ad the Natual Sciece Foudatio of Fujia Povice of Chia (Gat o. 3J7. Received: 9 Apil 3 Accepted: 3 August 3 Published: 3 Septembe 3 Refeeces. Loetz, GG: Bestei Polyomials, d ed. Chelsea, New Yok (986. Gal, SG: Vooovskaja s theoem ad iteatios fo complex Bestei polyomials i compact disks. Medite. J. Math. 5, 53-7 (8 3. Aastassiou, GA, Gal, SG: Appoximatio by complex Bestei-Schue ad Katoovich-Schue polyomials i compact disks. Comput. Math. Appl. 58, (9 4. Gal, SG: Appoximatio by complex Bestei-Katoovich ad Stacu-Katoovich polyomials ad thei iteates i compact disks. Rev. Aal. Numé. Théo. Appox. 37, (8 5. Gal, SG: Exact odes i simultaeous appoximatio by complex Bestei-Stacu polyomials. Rev. Aal. Numé. Théo. Appox. 37,47-5 (8 6. Gal, SG: Appoximatio by Complex Bestei ad Covolutio Type Opeatos. Wold Scietific, Sigapoe (9 7. Gal, SG: Exact odes i simultaeous appoximatio by complex Bestei polyomials. J. Coc. Appl. Math. 7, 5- (9 8. Gal, SG: Appoximatio by complex Bestei-Stacu polyomials i compact disks. Results Math. 53, (9 9. Gal, SG: Appoximatio by complex geuie Dumeye type polyomials i compact disks. Appl. Math. Comput. 7, 93-9 (. Gal, SG: Appoximatio by complex Bestei-Dumeye polyomials with Jacobi weights i compact disks. Math. Balk. 4, 3- (. Gal, SG, Gupta, V: Appoximatio by a Dumeye-type opeato i compact disks. A. Uiv. Feaa 57, 6-74 (. Gal, SG, Gupta, V: Appoximatio by complex beta opeatos of fist kid i stips of compact disks. Medite. J. Math., 3-39 (3 3. Gupta, V: Appoximatio popeties by Bestei-Dumeye type opeatos. Complex Aal. Ope. Theoy 7, (3 4. Gal, SG, Gupta, V, et al.: Appoximatio by complex Baskakov-Stacu opeatos i compact disks. Red. Cic. Mat. Palemo 6, ( 5. Gal, SG, Mahmudov, NI, Kaa, M: Appoximatio by complex q-szász-katoovich opeatos i compact disks, q >. Complex Aal. Ope. Theoy (. doi:.7/s Mahmudov, NI: Appoximatio popeties of complex q-szász-miakja opeatos i compact disks. Comput. Math. Appl. 6, ( 7. Mahmudov, NI, Gupta, V: Appoximatio by geuie Dumeye-Stacu polyomials i compact disks. Math. Comput. Model. 55, ( 8. Re, MY, Zeg, XM: Appoximatio by a kid of complex modified q-dumeye type opeatos i compact disks. J. Iequal. Appl., (. doi:.86/9-4x-- 9. Gupta, V, Maheshwai, P: Bezie vaiat of a ew Dumeye type opeatos. Riv. Mat. Uiv. Pama 7, 9- (3 doi:.86/9-4x-3-44 Cite this aticle as: Re et al.: Appoximatio by complex Dumeye-Stacu type opeatos i compact disks. Joual of Iequalities ad Applicatios 3 3:44.