UPPER ESTIMATE FOR GENERAL COMPLEX BASKAKOV SZÁSZ OPERATOR. 1. Introduction

Transcription

1 Joural of Classical Aalysis Volume 7, Number , doi: /jca UPPER ESTIMATE FOR GENERAL COMPLEX BASKAKOV SZÁSZ OPERATOR VIJAY GUPTA AND GANCHO TACHEV Abstract. I the preset article, we study geeral complex Baskakov-Szász operators ad establish a upper estimate for these operators attached to aalytic fuctios of expoetial growth o compact disks. 1. Itroductio It is kow that Baskakov operators are based o the egative biomial distributio. There are several itegral modificatios of the well kow Baskakov operators available i the literature. The most commo are Baskakov-Katorovich ad Baskakov Durrmeyer type operators. Apart from these there are other hybrid operators havig differet basis fuctios i summatio ad itegratio. I order to approximate itegrable fuctios o the iterval [0, Gupta ad Srivastava [12] proposed the Baskakov-Szász operators i case of real variables ad studied some approximatio properties. Very recetly Agrawal et al [2] proposed the geeralizatio of such operators based o a parameter a > 0 ad studied some approximatio properties i case of real variables. I case of a complex variable, we ca write the geeralized operators as where L a f,z = W,k a z s,k t f tdt, 1 k=0 0 z k W,k a az P k,a z=e 1+z k! 1 + z +k, s,kt=e t t k /k! k k ad P k,a= i a k i, with the risig factorial give by i = i=0 i i 1, 0 = 1. As k=0 W,k a x=1ad 0 s,ktdt = 1/, these operators reproduce costat fuctios. Also i special case a = 0, these operators iclude the well kow Baskakov-Szász operators see e.g. [12]. O the complex operators the commedable work was doe by the pioeer S. G. Gal who preseted results o overcovergece of several complex operators i his book [4] ad refereces therei. Differet forms of complex itegral operators have bee discussed i the recet years, we refer some of the papers i this directio due Mathematics subject classificatio 2010: 41A25, 41A30. Keywords ad phrases: Complex Baskakov-Szász operators, Berstei s iequality, upper estimate. c D l,zagreb Paper JCA

2 18 V. GUPTA AND G. TACHEV to Gal ad Gupta [6], [7] ad[8], Agarwal ad Gupta [1] ad Gupta [10] etc. Very recetly Gupta i [11] compiled some of the results o complex type operators. Also the recet book by Gal [5] cotais iterestig geeralizatios ad extesios of the results of [4]. For the special case a = 0thefirst author [9] established some results. But the operators L a f,z provides ratioal fuctios ad are differet from those cosidered for special case a = 0. The preset paper deals with the study of the complex geeralized Baskakov-Szász operators 1. Here, we estimate the upper boud for these operators. 2. Basic results I the sequel, we eed the followig lemmas. LEMMA 1. If we deote T a,mz=l a t m,z, the there holds the followig recurrece relatio: [ T,m+1 a + z m + 1 z=z1 T,m z+ Further by simple computatio, we have + z + az ] T,m z. 1 + z L a 1,z =1; L a t,z =z + az 1 + z + 1 ; L a t 2,z =z 2 + z2 + 4z + a2 z z 2 + 2az2 1 + z + 4az z The proof of the above lemma is similar as doe i Lemma 3 of [2] for the real case, we omit the details. We deote D R = {z C : z < R}. By H R, we mea the class of fuctios satisfyig: f : [R,+ D R C is cotiuous i R,+ D R, aalytic i D R i.e. f z= k=0 c kz k,forallz D R. LEMMA 2. Suppose that f : [R,+ D R is aalytic i D R ad there exists B,C > 0 such that f x Ce Bx, for all x [R,+. Deotig f z = k=0 c kz k,z D R, we have L a f z = k=0 c kl a e k z, for all z r with Rez 0 ad N with > B/1 h,where h= r 2 /1 + r 2 ad r < R. Proof. For ay m N, let us defie f m z= m c j z j if z r ad f m x= f x if x r,+. Sice f m z c j r j := C r,forall z r ad m N, f is cotiuous o [r,r]. Obviously from the hypothesis o f it follows that f m x C r,r e Bx,forall

3 UPPER ESTIMATE 19 x [0,+ ad ay m N. This implies that for each fixed m, N, > B ad z r with Rez 0, we have L a f m z C r,r 1 + z = C r,r 1 + z C r,r 1 + z e 1+z az P j,a z j! 1 + z e 1+z az P j,a z j j! 1 + z P j,a h j j+1 j! B j+1, j e t j 0 j! t j e Bt dt j+1 B j+1 r where h = 2 < 1, takig ito accout that for z = x + iy with x 0wehave 1+r 2 z 2 = 1 + z x 2 + y x +x 2 + y 2 x 2 + y 2 r2 1 +x 2 + y r 2. We apply the ratio test to the last series, deotig a j = P j,a j! h j j+1, we get a j+1 B j+1 a j = P j+1,a P j,a j+1 h B,where h B < 1 is equivalet to > B 1 h. Therefore, if > B 1 h the P there exists j 0, such that j+1,a P j,a j+1 h B < 1forall j j 0, therefore L a f m z is well-defied for > Deotig B 1 h. f m,k z=c k e k z if z r ad f m,k x= f x if x r,, m + 1 clearly each f m,k is of expoetial growth o [0, ad that f m z= m k=0 f m,kz. By the liearity property, we have L a f mz= m k=0 c k L a e kz, it is sufficiet to prove that lim m L a f mz =L a f z for ay fixed N with > B/1 h ad z r with Rez 0. But this is immediate from lim m f m f r = 0, from f m f B[0,+ f m f r ad from the iequality L a f m z L a f z 1 + z e z f m f B[0, M r, f m f r. Here B[0,+ deotes the uiform orm o C[0,+-the space of all complexvalued bouded fuctios o [0,+.

4 20 V. GUPTA AND G. TACHEV 3. Upper estimate Our mai result is the followig theorem for upper boud. THEOREM 1. Let f H R, r+1+ r rr + 2 < R < + ad suppose that there exist M > 0 ad A 1 R,1, with the property that c k M Ak Γk+a, for all a > 0, k= 0,1,..., which implies f z Me A z for all z D R ad f x Ce Bx, for all x [R,+. Let 1 r < r + 2 < r + 2 R+r R r < 1 A ad h = r 2 /1 + r 2. The for all z r with Rez 0 ad ad N with > B/1 h, we have where C r,a,a = M L a f z f z C r,a,a, k + a r + 2 R + r k A <. Proof. By usig the recurrece relatio of Lemma 1,wehave [ T,k+1 a + z z + k + 1 z=z1 T,k a z + + az ] T,k a 1 + z z, for all z C, k {0,1,2,...}, N. From this we immediately get the recurrece formula [ T,k a z z1 + z z + k zk = [T,k 1 a z zk 1 ] + + az ] [T,k 1 a 1 + z z zk 1 ] [ 2k 1+k 1z + + az ] z k 1, 1 + z for all z C, k, N.Nowfor1 r < R, if we deote the orm r i CD r, where D r = {z C : z r}, the by a liear trasformatio, with T,k a z = P k,z 1+z k where P k, z is a polyomial of degree k the Berstei s iequality i the closed uit disk for ratioal fuctios as give i [3] ad also i Corollary i [4], becomes T,k a z z R+r R r k r T,k a r,forall z r. Thus from the above recurrece relatio with h = z 1+z r 2 < 1, we get 1+r 2 T,k a e r1 + r k r T,k 1 a e k 1 R + r k 1 r r + r + k + a k + ar r k 1, which, by usig the otatio η = r + 2, implies T,k a e 2 + rk + a k r r + R + r T,k 1 a e k 1 r + ηk + a = r + R + r T,k 1 a e k 1 r + T a,k 1 e k 1 r k + a 2 + rr k 1 k + a η r k 1.

5 UPPER ESTIMATE 21 I what follows we prove by mathematical iductio with respect to k that for η, this recurrece implies T,k a e η Γk + a + 1 R + r k 1 k r r k 1 for all k 1. 2 Ideed for k = 1 it is trivial, as the left-had side is for k, the above recurrece relatio implies that T,k+1 a e η k + a + 1 k+1 r r + It remais to prove that r + η k + a η k + a + 1 R + r k r k. η Γk + a + 1 r k 1 + or after simplificatios, equivaletly to r + η k + a + 1 az 1+z + 1. Suppose that it is valid η Γk + a + 1 r k 1 η k + a + 1 r k R + r k η Γk + a + 2 r k, Γk + a + 1+rk + a + 1 Γk + a + 2 r, for all k N ad r 1. Sice by η,weget r+ η k+a+1 Γk+a+1+rk+a+1 r+k+a+1 Γk+a+1+rk+a+1, it is good eough if we prove that r + k + a + 1 Γk + a + 1+rk + a + 1 Γk + a + 2 r. But this last iequality is obviously valid for all k 1adfixed r 1. From the hypothesis o f, by Lemma 2 we ca write L a f z= k=0c k L a e kz= k=0 c k T a,k z, for all z D R, Rez 0, > B/1 h, which from the hypothesis o c k immediately implies for all z r with Rez 0 ad N with > B/1 h, L a f z f z M c k. T a,k z e kz M Ak + a + 1 r + 2Γk r k 1 Γk + a k + a r + 2A R + r k = C r,a,a, R + r k 1

6 22 V. GUPTA AND G. TACHEV where C r,a,a = M k + a r + 2A R + r k < for all 1 r r + 2 R+r R r < A 1, takig ito accout that the series k=1 uk is uiformly coverget i ay compact disk icluded i the ope uit disk. REMARK 1. It is observed that the geeralized Baskakov-Szász operators discussed here provides ratioal fuctios ad these operators behave differetly tha the usual Baskakov-Szász operators discussed i [9]. At this momet we are ot able to obtai asymptotic formula ad we will discuss that elsewhere. imply the i- REMARK 2. Actually the coditios 1 A equality which is true if r + 2 R + r < R, < R ad r + 2 R+r R r < 1 A R > r + 1+ r rr + 2, ad the last iequality should replace the coditio 2 < R < + cosidered for special case a = 0i[9] i the formulatio of Theorem 1. REMARK 3. I particular, if f is a etire fuctio, the uder the growth coditios i Theorem 1, the property L a f z= k=0 c kl a e kz follows by simple direct calculatio without to eed Lemma 1 ad the upper estimate i Theorem 1 holds i ay semi-disk z r, Rez 0. O the other had, if f is supposed to be aalytic oly o D R,i.e. f z = k=0 c kz k, z < R, without to require to be defied ad of expoetial growth o [0,+ too, the oe ca cosider the approximatio operator deoted by L a f z = k=0 c k L a e k z, z < R, which evidetly will satisfy the estimate i Theorem 1. Ackowledgemets. The authors are thakful to the reviewer for valuable commets. REFERENCES [1] R. P. AGARWAL, V. GUPTA, O q-aalogue of a complex summatio-itegral type operators i compact disks, J. Iequal. Appl. 2012, 2012, Art [2] P. N. AGRAWAL, V. GUPTA, A. S. KUMAR, A. KAJLA, Geeralized Baskakov Szász type operators, Appl. Math. Comput , [3] P. BORWEIN AND T. ERDÉLYI, Sharp extesios of Berstei s iequalityto ratioal spaces, Mathematika 43 2, 1996, [4] S. G. GAL, Approximatio by Complex Berstei ad Covolutio Type Operators, World Scietific Publ. Co., Sigapore, Hog Kog, Lodo, New Jersey, [5] S. G. GAL, Overcovergece i Complex Approximatio, Spriger, New York, 2013, ISBN

7 UPPER ESTIMATE 23 [6] S. G. GAL, V. GUPTA, Approximatio by a Durrmeyer-type operator i compact disks, Aali dell Uiversita di Ferrara , [7] S. G. GAL, V. GUPTA, Quatitative estimates for a ew complex Durrmeyer operator i compact disks, Appl. Math. Comput , [8] S. G. GAL, V. GUPTA, Approximatio by certai itegrated Berstei-type operator i compact disks, Lobachevskii J. Math , [9] V. GUPTA, Complex Baskakov-Szász operators i compact semi-disks, Lobachevskii J. Math , [10] V. GUPTA, Approximatio properties by Berstei-Durrmeyer type operators, Complex Aal. Operator Theory , [11] V. GUPTA, R. P. AGARWAL, Covergece Estimates i Approximatio Theory, Spriger 2014, ISBN [12] V. GUPTA, G. S. SRIVASTAVA, Simultaeous approximatio by Baskakov-Szász type operators, Bull. Math.delaSoc.Sci.deRoumaieN.S , Received July 21, 2015 Vijay Gupta Departmet of Mathematics Netaji Subhas Istitute of Techology Sector 3 Dwarka, New Delhi , Idia vijaygupta2001@hotmail.com Gacho Tachev Departmet of Mathematics Uiversity of Architecture Sofia 1046, Bulgaria gtt fte@uacg.bg Joural of Classical Aalysis jca@ele-math.com