Inegalităţi de tip Chebyshev-Grüss pentru operatorii Bernstein-Euler-Jacobi
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1 Iegalităţi de tip Chebyshev-Grüss petru operatorii Berstei-Euler-Jacobi arxiv: v1 [math.ca] 26 Ju 2015 Heier Goska, Maria-Daiela Rusu, Elea-Doria Stăilă Abstract The classical form of Grüss iequality was first published by G. Grüss ad gives a estimate of the differece betwee the itegral of the product ad the product of the itegrals of two fuctios. I the subsequet years, may variats of this iequality appeared i the literature. The aim of this paper is to cosider some Chebyshev- Grüss-type iequalities ad apply them to the Berstei-Euler-Jacobi (BEJ) operators of first ad secod kid. The first ad secod momets of the operators will be of great iterest AMS Subject Classificatio : 41A17, 41A36, 26D15. Key Words ad Phrases: Chebyshev fuctioal, Chebyshev-type iequality, Grüss-type iequality, Chebyshev-Grüss-type iequalities, Berstei-Euler-Jacobi (BEJ) operators of first ad secod kid, first ad secod momets. 1 Itroducere Î cele ce urmează vom prezeta rezultate clasice di literatură. Iegalităţile de tip Chebyshev-Grüss au fost ites studiate de-a lugul ailor, mai ales datorită umeroaselor aplicaţii. Ele reprezită legătura ître fucţioala lui Chebyshev şi iegalitatea lui G. Grüss. Fucţioala descrisă de T(f,g) := 1 b f(x)g(x)dx 1 b 1 b f(x)dx g(x)dx, b a a b a a b a a ude f, g : [a, b] R sut fucţii itegrabile, este biecuoscută î literatură drept fucţioala clasică a lui Chebyshev. Petru detalii, articolul [3] poate fi de ajutor. U prim rezultat pe care îl readucem î ateţia cititorului este dat de următoarea teoremă. Teoremă 1.1. (vezi [16]) Fie f,g : [a,b] R două fucţii mărgiite şi itegrabile, ambele crescătoaresau descrescătoare. Mai mult, fie p : [a,b] R + 0 o fucţie mărgiită şiitegrabilă. Uiversity of Duisburg-Esse, Faculty of Mathematics, Forsthausweg 2, Duisburg, Germay s: heier.goska@ui-due.de, maria.rusu@ui-due.de, elea.staila@stud.ui-due.de 1
2 Atuci avem (1.1) b a b b b p(x)dx p(x) f(x) g(x)dx p(x) f(x)dx p(x) g(x)dx. a a a Dacă ua di fucţiile f sau g este crescătoare şi cealaltă este descrescătoare, atuci iegalitatea (1.1) se iversează. Remarcă 1.1. Relaţia (1.1) a fost itrodusă petru prima oară de către P. L. Chebyshev î 1882 ( vezi [2]). Di acest motiv, este cuoscută sub umele de iegalitatea lui Chebyshev. Î cotiuare amitim uul di rezultatele eseţiale pe care se bazează cercetarea oastră, şi aume iegalitatea lui Grüss petru fucţioala lui Chebyshev. Teoremă 1.2. (vezi [11]) Fie f,g fucţii itegrabile, defiite pe itervalul [a,b] cu valori î R, astfel îcât m f(x) M, p g(x) P, petru orice x [a,b], ude m,m,p,p R. Atuci are loc iegalitatea (1.2) T(f,g) 1 (M m)(p p). 4 Următoarele iegalităţi de tip Chebyshev-Grüss vor fi folosite î cotiuare, petru a itroduce rezultatele oastre. Teoremă 1.3. (vezi [21]) Dacă f,g C[a,b] şi x [a,b] este fixat, atuci are loc iegalitatea (1.3) T(f,g;x) 4 ω 1 (f;2 H((e ) ( ) 1 x) 2 ;x) ω g;2 H((e 1 x) 2 ;x). Remarcă 1.2. Rezultatul de mai sus implică folosirea celui mai mic majorat cocav ω al primului modul de cotiuitate. O defiiţie şi detalii cu privire la acesta se găsesc, de exemplu, î [8]. Remarcă 1.3. Scopul ostru este să aplicăm iegalitatea de mai sus î cazul operatorilor BEJ de tipul I şi II, petru diferite cazuri, luâd î cosiderare diferitele momete de ordiul doi. Î cazul operatorilor liiari şi pozitivi care reproduc fucţiile costate, dar u le reproduc pe cele liiare, avem următorul rezultat. Teoremă 1.4. (vezi Corolarul 5.1. di [8]) Dacă H : C[a,b] C[a,b] este u operator liiar şi pozitiv, care reproduce fucţii costate, atuci petru f,g C[a,b] şi x [a,b] fixat, au loc următoarele iegalităţi: (1.4) T(f,g;x) 1 4 ω ( f;2 H(e 2 ;x) H(e 1 ;x) 2 ) 1 (f;2 H((e ) 4 ω 1 x) 2 ;x) (g;2 H(e ) ω 2 ;x) H(e 1 ;x) 2 ) ( ω g;2 H((e 1 x) 2 ;x) Remarcă 1.4. Petru a aplica iegalitatea de mai sus uora di cazurile operatorilor BEJ de tipul I sau II, avem î primul râd evoie de mometele de ordiul îtâi. Apoi trebuie să calculăm difereţe de tipul T(e 1,e 1 ;x) := H(e 2 ;x) H(e 1 ;x) 2. Dacă operatorul H u reproduce fucţiile liiare, se obţie o îmbuătăţire a iegalităţii (1.3). 2.
3 Operatorii Berstei-Euler-Jacobi (BEJ) sut împărţiţi î două clase: BEJ de tip I şi BEJ de tip II. Scopul acestui articol este de a aplica iegalităţile de tip Chebyshev-Grüss de mai sus acestor tipuri de operatori. Î rezultatele următoare mometele de ordiul uu şi cele de ordiul doi vor fi de mare iteres. Operatorii BEJ de tipul I, ca şi clasă de operatori pozitivi şi liiari, pot fi defiiţi după cum urmează. Petru mai multe detalii, vezi [23]. Defiiţie 1.5. Defiim R (r,a,b) m, : C[0,1] C[0,1] ca fiid compoziţia R (r,a,b) m, = B m B a,b r B, petru r > 0, a,b 1,,m > 1. Î ecuaţia de mai sus, Ba,b r este operatorul Euler-Jacobi- Beta defiit î [9] şi B,B m sut operatorii Berstei de ordi şi m. Remarcă 1.5. Operatorii BEJ de tip I reproduc costatele. Petru aumite valori ale lui a, respectiv b, şi aume petru a = b = 1, sut reproduse şi fucţiile liiare. A doua clasă de operatori liiari şi pozitivi pe care îi cosiderăm sut BEJ de tipul II. Defiiţia este dată î cotiuare. Defiiţie 1.6. Petru r,s > 0, a,b,c,d 1 şi > 1, defiim R s,c,d;r,a,b ca fiid R s,c,d;r,a,b = B c,d s B B a,b r. : C[0,1] C[0,1] B a,b r şi B c,d s sut operatori de tip Euler-Jacobi Beta şi B este operatorul Berstei de ordi. Remarcă 1.6. Operatorii BEJ de tip II reproduc costatele. Petru aumite valori ale lui a,b şi c,d, şi aume petru a = b = c = d = 1, sut reproduse şi fucţiile liiare. 2 Mometele de ordiul uu şi doi Lemă 2.1. Petru clasa de operatori BEJ de tipul I mometele de ordiul 1 şi 2 sut date de: (2.1) R m, (r,a,b) ((e 1 xe 0 ) 1 ;x) = a+1 x(a+b+2). r+a+b+2 (2.2) R m, (r,a,b) ((e 1 xe 0 ) 2 ;x) = x2 [m(a 2 +b 2 +5a+5b+2ab+6 r)+r 2 (1 m )] m(r +a+b+2)(r+a+b+3) x[m(2a2 +2ab+8a+2b+6 r)+r 2 (1 m )+mr(a b)] + m(r +a+b+2)(r+a+b+3) + [m(a+1)(a+2)+m(r(a+1)+ab+a+b+1)] m(r +a+b+2)(r +a+b+3) Petru detalii î ceea ce priveşte demostraţia, vezi [23]. 3
4 Lemă 2.2. Petru clasa de operatori BEJ de tipul II mometele de ordiul 1 şi 2 sut date de: (2.3) R (s,c,d;r,a,b) ((e 1 xe 0 ) 1 ;x) = x[r(c+d+2)+(a+b+2)(s+c+d+2)] + (r +a+b+2)(s+c+d+2) + r(c+1)+(s+c+d+2)(a+1) (r +a+b+2)(s+c+d+2). (2.4) R (s,c,d;r,a,b) ((e 1 xe 0 ) 2 ;x) = ( 1)(a2 +b 2 +2ab+5a+5b+6 r)(sx+c+1)(sx+c+2) + (r +a+b+2)(r +a+b+3)(s+c+d+2)(s+c+d+3) + (a2 +b 2 +2ab+5a+5b+6 r (2ab+2a 2 +8a+2b+6 r))(sx+c+1) + (r +a+b+2)(r+a+b+3)(s+c+d+2) a 2 +3a+2 + (r +a+b+2)(r+a+b+3) + (sx+c+1)(sx+c+2) (s+c+d+2)(s+c+d+3) sx+c+1 (s+c+d+2) c2 +d 2 +2cd+5c+5d+6 s (s+c+d+2)(s+c+d+3) x cd+2c2 +8c+2d+6 s (s+c+d+2)(s+c+d+3) x c 2 +3c+2 (s+c+d+2)(s+c+d+3) + 2r(sx+c+1) + (r +a+b+2)(s+c+d+2) 2r(sx+c+1)(sx+c+2) (r +a+b+2)(s+c+d+2)(s+c+d+3) + 2r(sx+c+1)(sx+c+2) + (r +a+b+2)(s+c+d+2)(s+c+d+3) 2(a+1)x r +a+b+2 +2x2. 2(a+1 x(2r +a+b+2))(sx+c+1) + (r+a+b+2)(s+c+d+2) Petru detalii î ceea ce priveşte demostraţia, vezi [23]. Mulţi operatori uzuali pot fi regăsiţi dacă particularizăm valorile idicilor. Astfel î primele cici tabele prezetate i Aexă regăsim mometele de ordiul doi, petru cazurile particulare pe care am reuşit să le localizăm î literatură, determiate pe baza ecuaţiilor (2.2) şi (2.4), folosid coveţia B = B a,b = Id. 3 Iegalităţi Chebyshev-Grüss petru operatorii BEJ de tip I şi II Î cotiuare aplicăm iegalitatea de tip Chebyshev-Grüss, dată de (1.3), operatorilor BEJ de tipul I şi II şi obţiem următoarele teoreme. Teoremă 3.1. Dacă f,g C[a,b] şi x [a,b] este fixat, atuci iegalitatea T(f,g;x) 1 ) ( ) (f;2 4 ω R m, (r,a,b) ((e 1 x) 2 ;x) ω g;2 R m, (r,a,b) ((e 1 x) 2 ;x) are loc, petru r > 0, a,b 1,,m > 1. 4
5 Teoremă 3.2. Dacă f,g C[a,b] şi x [a,b] este fixat, atuci iegalitatea T(f,g;x) 1 ) ( ) (f;2 4 ω ((e 1 x) 2 ;x) ω g;2 ((e 1 x) 2 ;x) R (s,c,d;r,a,b) are loc, petru r,s > 0, a,b,c,d 1, > 1. R (s,c,d;r,a,b) Remarcă 3.1. Particularizâd valorile idicilor î ecuaţiile (2.2) şi (2.4), obţiem iegalităţile Chebyshev-Grüss corespuzătoare operatorilor biecuoscuţi î literatură. Mometele de ordiul doi î cazurile particulare sut prezetate î Tabelele 1 5 di Aexă. Dacă aplicăm iegalitatea (1.4) operatorilor BEJ de tipul I şi II care u reproduc fucţiile liiare, obţiem următorul rezultat. Teoremă 3.3. Petru f,g C[a,b] şi x [a,b] fixat, următoarea iegalitate are loc: T(f,g;x) 1 (f;2 T(e ) ( ) 4 ω 1,e 1 ;x) ω g;2 T(e 1,e 1 ;x). Difereţele de mai sus sut date de ecuaţiile: respectiv T(e 1,e 1 ;x) := R (r,a,b) m, ((e 1 xe 0 ) 2 ;x) [R (r,a,b) m, ((e 1 xe 0 ) 1 ;x)] 2, T(e 1,e 1 ;x) := R (s,c,d;r,a,b) ((e 1 xe 0 ) 2 ;x) [R (s,c,d;r,a,b) ((e 1 xe 0 ) 1 ;x)] 2. Remarcă 3.2. Mometele de ordiul îtâi şi difereţele de tip T(e 1,e 1 ;x) petru cazurile particulare se regăsesc î Tabelele 6 7 di Aexă. 5
6 4 Aexă 6 Notaţie Deumire Mometul de ordiul doi Referiţe U = operatorul origial U Berstei-Durrmeyer ((e 1 xe 0 ) 2 ;x) := 2X [4] [10] +1 R, (, 1, 1) = R,, ;, 1, 1 M = R, (,0,0) = R,, ;,0,0 D <α> = R, (,α,α) = R,, ;,α,α M ab = R, (,a,b) = R,, ;,a,b U = R (, 1, 1), = R,, ;, 1, 1 P = R, (c,a,b) = R,, ;c,a,b L = R, (, 1, 1) = R, 1, 1;,, V 0,0 = R, (,0,0) = R,0,0;,, operatorul lui Durrmeyer operatorul Berstei-Durrmeyer cu poderi simetrice operatorul lui Durrmeyer cu poderi Jacobi -operatorul lui Păltăea operatorul lui Mache-Zhou operator de tip Stacu M ((e 1 xe 0 ) 2 ;x) := 2X( 3)+2 (+2)(+3) D <α> ((e 1 xe 0 ) 2 ;x) := X(2 4a2 10a 6)+(a+1)(a+2) (+2a+2)(+2a+3) M ab ((e 1 xe 0 ) 2 ;x) := a2 +b 2 +2ab+5a+5b+6 2 x 2 (+a+b+2)(+a+b+3) 2a2 +2ab+8a+2b+6 2 (+a+b+2)(+a+b+3) x+ (a+1)(a+2) (+a+b+2)(+a+b+3) U ((e 1 xe 0 ) 2 ;x) := X(1+ ) +1 P ((e 1 xe 0 ) 2 ;x) := a2 +b 2 +2ab+5a+5b+6 c c 2 (c+a+b+2)(c+a+b+3) (2a2 +2ab+8a+2b+6 c c 2 ) x+(a+1)(a+2) (c+a+b+2)(c+a+b+3) L ((e 1 xe 0 ) 2 ;x) := 2X 2+1 operatorul lui Lupaş V 0,0 ((e 1 xe 0 ) 2 ;x) := X(22 6)+3+1 (+2)(+3) x 2 [5] [13] [18] [19] [15] [14] [13] Tabelul 1: Mometele de ordiul doi - partea I
7 7 Notaţie Deumire Mometul de ordiul doi Referiţe V α,β = R, (,α,β) = R,α,β;,, S α = R ( 1 α, 1, 1), = R 1 α, 1, 1;,, Q,c,d = R, (,c,d) = R,c,d;,, B, B 1, 1 = R, (, 1, 1) = R,, ;, 1, 1 = R, 1, 1;,, B λ, T λ = R ( 1 λ, 1, 1), = R,, ;1 λ, 1, 1 = R 1 λ, 1, 1;,, B, B 0,0 = R(,0,0), = R,, ;,0,0 = R,0,0;,, operatorul lui Lupaş cu poderi Jacobi V α,β ((e 1 xe 0 ) 2 ;x) := α2 +β 2 +2αβ +5α+5β +6 2 x 2 (+α+β +2)(+α+β +3) 2α2 +2αβ +9α+β α+αβ +3+α+β +1 (+α+β +2)(+α+β +3) x+α2 (+α+β +2)(+α+β +3) operatorul lui Stacu S α ((e 1 xe 0 ) 2 ;x) := X(2 +1) (+1) operator de tip Stacu cu parametri c şi d operatorul Beta de a doua speţa al lui Lupaş operatorul Beta al lui Mühlbach operatorul Beta de prima speţa al lui Lupaş Q,c,d ((e 1 xe 0 ) 2 ;x) := c2 +d 2 +2cd+5c+5d+6 2 x 2 ( +c+d+2)( +c+d+3) 2c2 +2cd+8c+2d+6 2 x+ ( +c+d+2)( +c+d+3) c 2 +3c+c + +2+c+d+cd+1 ( +c+d+2)( +c+d+3) B ((e 1 xe 0 ) 2 ;x) := X +1 B λ ((e 1 xe 0 ) 2 ;x) := λx 1+λ B ((e 1 xe 0 ) 2 ;x) := X( 6)+2 (+2)(+3) I. Raşa hadwritte otes, 19 August [22] H. Goska hadwritte otes, 18 March [12] [17] [12] Tabelul 2: Mometele de ordiul doi - partea a II-a
8 B 1,β Notaţie Deumire Mometul de ordiul doi Referiţe = R (, 1,β) B 1,β ((e 1 xe 0 ) 2 X +(β +1)(β +2)x2 ;x) := [9] (+β +1)(+β +2), = R,, ;, 1,β = R, 1,β;,, B α, 1 = R (,α, 1), = R,, ;,α, 1 = R,α, 1;,, B α,β, = R,, ;,α,β = R,α,β;,, = R (,α,β) B = R (,, ), = R, (,, ) = R,, ;,, operatorul Beta cu poderi Jacobi operatorul lui Berstei B α, 1 ((e 1 xe 0 ) 2 X +(α+1)(α+2)(x 1)2 ;x) := (+α+1)(+α+2) B α,β((e 1 xe 0 ) 2 ;x) := α2 +β 2 +2αβ +5α+5β +6 x 2 (+α+β +2)(+α+β +3) 2α 2 +2αβ +8α+2β +6 (+α+β +2)(+α+β +3) x+ (α+1)(α+2) (+α+β +2)(+α+β +3) B ((e 1 xe 0 ) 2 ;x) := X [9] [9] [1] 8 Tabelul 3: Mometele de ordiul doi - partea a III-a Notaţie Deumire Mometul de ordiul doi Referiţe D = R (,, ),+1 R m, = R (,, ) m, R m, = R(, 1, 1) m, D ((e 1 xe 0 ) 2 ;x) := 2X +1 Rm, ((e 1 xe 0 ) 2 ;x) := X(+m 1) m R m,((e 1 xe 0 ) 2 ;x) := X( +m +m) m( +1) [7] [23] [23] Tabelul 4: Mometele de ordiul doi - petru operatori BEJ de primul tip
9 Notaţie Deumire Mometul de ordiul doi Referiţe operatori de tip Beta B α,λ ((e 1 xe 0 ) 2 ;x) := X(α+λ+αλ) [20] (1+α)(1+λ) geeralizaţi B (α,λ) = R 1 α, 1, 1; 1 λ, 1, 1 F α = R 1 α, 1, 1; 1, 1 B (α,λ) = R 1 α, 1, 1; 1 λ, 1, 1 operatorul lui Fita operatorul de tip Beta al lui Piţul F α ((e 1 xe 0 ) 2 ;x) := X(α+α+2) (α+1)(+1) B (α,λ) ((e 1 xe 0 ) 2 ;x) := X(α+λ+αλ+1) (1+α)(1+λ) Tabelul 5: Mometele de ordiul doi - petru operatori BEJ de tipul al doilea [6] [20] 9
10 10 Mometul de ordiul Notaţie uu M ((e 1 xe 0 ) 1 ;x) := M 1 2 x +2 D<α> ((e 1 D <α> xe 0 ) 1 ;x) := 2 x (α+1)+α+1 +2 α+2 M ab M ab ((e 1 xe 0 ) 1 ;x) := a+1 x (a+b+2) +a+b+2 P ((e 1 xe 0 ) 1 ;x) := P a+1 x(a+b+2) c+a+b+2 V 0,0 V α,β V 0,0 ((e 1 xe 0 ) 1 ;x) := 1 2x +2 V α,β ((e 1 xe 0 ) 1 ;x) := α+1 x (α+β +2) +α+β +2 Difereţe de tipul T(e 1,e 1 ;x) T(e 1,e 1 ;x) := (2 X +1) (+1) (+2) 2 (+3) T(e 1,e 1 ;x) := X ( α )+1++2α+α+α 2 (+2α+2) 2 (+2α+3) T(e 1,e 1 ;x) := 2X(+b+1)+ x2 (b a)+1++a+b+a+ab (+a+b+2) 2 (+a+b+3) T(e 1,e 1 ;x) := X c2 ( c+a+b++2)+x c(a+b) (c+a+b+2) 2 (c+a+b+3) 10a2 +5ac+15a+3b+2a 2 c+3c+2a 2 b+2a 3 +5ab+7 (c+a+b+2) 2 (c+a+b+3) T(e 1,e 1 ;x) := 2(+1)(2 X ++1) (+2) 2 (+3) T(e 1,e 1 ;x) := β + 2 α+2 2 (+α+β +2) 2 (+α+β +3) x2 + α α+2α β 2 2β β x+ (+α+β +2) 2 (+α+β +3) 2 4 3α 3β α 2 4αβ 3αβ α 2 β 2 5α β 2 α 3β 2 2 α 2 β 2 2 α (+α+β +2) 2 (+α+β +3) Tabelul 6: Difereţe de tipul T(e 1,e 1 ;x)- Partea I
11 11 Notaţie Mometul de ordiul uu Difereţe de tipul T(e 1,e 1 ;x) Q ρ,c,d Q ρ,c,d ((e 1 xe 0 ) 1 ;x) := c+1 x (c+d+2) ρ+c+d+2 T(e 1,e 1 ;x) := 22 ρ ρ ρ 2 d+ 2 cρ ρ 3 (ρ+c+d+2) 2 (ρ+c+d+3) x2 2 cρ+ 2 dρ+2 2 ρ ρ ρ 2 d+ 2 cρ ρ 3 x+ B, B 0,0 B 1,β B α, 1 B α,β B ((e 1 xe 0 ) 1 ;x) := 1 2x +2 B 1,β ((e 1 xe 0 ) 1 ;x) := x(β +1) +β +1 B α, 1 ((e 1 xe 0 ) 1 ;x) := (α+1)(1 x) +α+1 B α,β((e 1 xe 0 ) 1 ;x) := α+1 x(α+β +2) +α+β +2 (ρ+c+d+2) 2 (ρ+c+d+3) 2 4cρ 2cdρ 2 ρ d 3ρ 3c c 4cd c 2 d 2 c 2 ρ (ρ+c+d+2) 2 (ρ+c+d+3) + 2 cρ 2 cρ 2 cd 3d d 2 c 2 ρ 2 c 2 d 2dρ (ρ+c+d+2) 2 (ρ+c+d+3) T(e 1,e 1 ;x) := X( )++1 (+2) 2 (+3) T(e 1,e 1 ;x) := 2 X +β + (+β +1) 2 (+β +2) T(e 1,e 1 ;x) := 2 X x(1+α)++ α (+α+1) 2 (+α+2) T(e 1,e 1 ;x) := X 2 +x(β α)+α+β +αβ +α++1 (+α+β +2) 2 (+α+β +3) Tabelul 7: Difereţe de tipul T(e 1,e 1 ;x)- Partea a II-a
12 Bibliografie [1] S.N. Berstei: Démostratio du théorème de Weierstrass fodée sur le calcul des probabilités, Comm. Soc. Math. Kharkow 13 (1912/13), 1-2. [Also appears i Russia traslatio i Berstei s Collected Works]. [2] P. L. Chebyshev: O priblizheyh vyrazheijah odih itegralov cherez drugie, Soobscheija i Protokoly Zasedaij Mathematicheskogo Obschestva pri Imperatorskom Khar kovskom Uiversitete 2 (1882), 93 98; Poloe Sobraie Sochieii P. L. Chebysheva. Moskva-Leigrad, 3(1978), [3] P.L. Chebysev: Sur les expressios approximatives des itégrales défiies par les autres prises etre les mêmes limites, Proc. Math. Soc. Kharkov 2 (1882), 93-98(Russia), traslated i Oeuvres, 2 (1907), [4] W. Che: O the modified Berstei-Durrmeyer operator. I: Report of the Fifth Chiese Coferece o Approximatio Theory, Zhe Zhou, Chia, [5] J.L. Durrmeyer: Ue formule d iversio de la trasformée de Laplace: Applicatio à la théorie des momets, Thèse de 3e cycle, Faculté des Scieces de l Uiversité de Paris, [6] Z. Fita: Quatitative estimates for some liear ad positive operators, Studia Uiv. Babeş-Bolyai, Mathematica 47 (2002), [7] H. Goska, P. Piţul, I. Raşa: O differeces of positive liear operators, Carpathia J. Math. 22 (2006), [8] H. Goska, I. Raşa, M.-D. Rusu: Čebyšev-Grüss-type iequalities revisited, Math. Slovaca 63, o. 5, (2013), [9] H. Goska, I. Raşa, E.D. Stăilă: Beta operators with Jacobi weights, arxiv: (2014). [10] T.N.T. Goodma, A. Sharma: A Berstei-type operator o the Simplex, Math. Balkaica 5 (1991), b [11] G. Grüss: Über das Maximum des absolute Betrages vo f(x)g(x)dx b a a 1 b (b a) 2 f(x)dx b g(x)dx, Math. Z. 39 (1935), a a [12] A. Lupaş: Die Folge der Betaoperatore, Ph.D. Thesis, Stuttgart: Uiversität Stuttgart [13] A. Lupaş: The approximatio by meas of some liear positive operators. I Approximatio Theory (M.W. Müller et al., eds.), , Berli: Akademie-Verlag [14] A. Lupaş, L. Lupaş: Polyomials of biomial type ad approximatio operators, Studia Uiv. Babeş-Bolyai, Mathematica XXXII 4,
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