The Exact Order of Approximation for Bivariate Complex Bernstein-Schurer Polynomials

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Vol. 9, No. 3, 2016, 322-332

ISSN 1307-5543 – www.ejpam.com

The Exact Order of Approximation for Bivariate Complex

Bernstein-Schurer Polynomials

Nurhayat ˙Ispir, ¸

Sule Yüksel Güngör

∗

Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey

Abstract. In this paper we study the approximation properties of the tensor product kind bivariate complex Bernstein-Schurer polynomials. We obtain the order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for bivariate complex Bernstein-Schurer polyno-mials attached to analytic functions on compact polydisks.

2010 Mathematics Subject Classifications: 30E10, 41A25, 41A28.

Key Words and Phrases: Bivariate complex polynomials, Berstein-Schurer polynomials, Rate of con-vergence, Voronovskaja’s theorem, Exact order of approximation.

1. Introduction

The Bernstein polynomial attached to f :[0, 1]→Rwas introduced by Bernstein to give a proof of the Weierstrass Theorem. If f is continuous on[0, 1]then lim_nB_n(f)(x) = f(x) whereBn(f)(x) =

Pn k=0

n k

xk(1−x)n−kf k_n,x ∈[0, 1].

If f :G→Cis an analytic function in the open setG⊂C, with D₁⊂G (where

D1={z∈C:|z|<1}), then S. N. Bernstein[6]proved that the complex Bernstein polynomials defined by

B_n(f)(z) =

n

X

k=0

n k

zk(1−z)n−kf

k n

uniformly converge to f in D₁. But Bernstein obtained this convergence result without any quantitative estimate. Recently, Voronovskaja- type results with quantitative estimates for the complex Bernstein, complex q-Bernstein, complex Bernstein- Kantorovich, complex Kan-torovich - Stancu polynomials attached to analytic functions on compact disks and the exact order of simultaneous approximation by these complex operators were obtained by S. G. Gal [5].

∗_{Corresponding author.}

Email addresses:[email protected](N. ˙Ispir),[email protected](¸S. Güngör)

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The complex Bernstein-Schurer polynomials (introduced and studied in the case of real variable in[7]) are defined for any fixedp=0, 1, 2, . . . by

B_n_,_p(f)(z) =

n+p

X

k=0

n+p k

zk(1−z)n+p−kf (k/n),z∈C_.

The approximation properties of these polynomials are investigated and the exact order of approximation with quantitative estimates were gave in [1]. It is clear that for p = 0 these polynomials become the classical complex Bernstein polynomials studied in[5]. In real case the approximation properties of bivariate Berstein-Schurer polynomials is studied by D. Barbasu[2–4].

In this note we would like to extend the approximation results from the univariate case, obtained for the complex Bernstein-Schurer polynomials, to the bivariate case.

First we present a few concepts in the bivariate case which are natural extensions of the usual concepts in the univariate case. LetD_R

j :=

z_j∈C:z_j

<R_j,j=1, 2 and

P(0;R) = DR1×DR2 denotes an open polydisk (of center 0 and radiusR) whereR= (R1,R2)

andz₁ ≤r₁,

z₂

≤r₂,r₁<R₁ withr₂<R₂. Let also

P_R:=P(0,R) = z₁,z₂∈C2:z_j

≤R_j, j=1, 2

denotes the closed polydisk. For f(z1,z2) is an analytic function of two complex variables (z1,z2)in the polydiskP(0;R)we can define the tensor product kind Bernstein-Schurer poly-nomials as follows

Bn,m,p,q(f) z1,z2

=

n+p

X

k=0

m+q

X

j=0

bn,k z1

bm,j z2

f k/n,j/m (1)

where b_n_,_k z₁= n+_kpz₁k(1−z₁)n+p−k, b_m_,_j z₂= m+_jqz₂j(1−z₂)m+q−j,n,m∈Nand

p,q∈N∪ {0}.

The goal of this paper is to obtain the exact order of approximation for the polynomi-als given by (1) on compact polydisks. First we give the order of approximation and the Voronovskaja- type theorems with quantitative estimate for the polynomials Bn,m f

(z) de-fined by (1). These results allow us to obtain the exact order in approximation by the polyno-mialsB_n_,_m f(z).

2. The Convergence Results with Quantitative Estimates

Theorem 1. For fixed p,q∈N∪ {₀}_{and R}₁_>_p₊₁_{, R}₂_>_q₊₁_{suppose that f} _:_P_(0;_R₎→C_is

analytic in P(0;R) =D_R

1×DR2, that is f(z1,z2) = P∞

k=0

P∞

j=0ck,jz

k

1z

j

2 for all(z1,z2)∈P(0;R),

R= (R1;R2). Then we have

(i) For allz₁ ≤r₁,

z₂

≤r₂with1<r₁,(p+1)r₁<R₁,1<r₂,(q+1)r₂<R₂and n,m∈N

B_n_,_m_,_p_,_q(f) z₁,z₂

−f(z₁;z₂)≤Mp,q

(3)

where

M_rp,q

1,r2,n,m(f) = ∞ X

k=1

∞ X

j=1

c_k_,_j

rj

2

3k(k−1)

n+p

p+1r₁k+1

n

p+1r₁k− r

k

1

n

+

∞ X

k=1

∞ X

j=1

c_k_,_j

rk

1





3j j−1

m+q

q+1r₂j+ 1

m

q+1r₂j− r

j

2

m





and Mrp1,,qr2,n,m(f)<∞.

(ii) Let k₁,k₂∈Nbe with k₁+k₂≥1,1≤r₁<r₁∗≤(1+p)r₁<R₁,

1≤r₂<r₂∗≤(1+q)r₂<R₂. Then for allz₁ ≤r₁,

z₂

≤r₂and n,m∈N, p,q∈N∪{0}

we have

∂k1+k2_B

n,m,p,q(f)

∂k1z₁∂k2z₂ z1,z2

− ∂

k1+k2_f

∂k1z₁∂k2z₂ z1,z2

≤M_rp∗,q

1,r2,n,m∗

(f). k1!

r₁∗−r₁k1+1

k₂!

r₂∗−r₂k2+1

where M_rp∗,q

1,r2,n,m∗

(f)is given as at the above point (i).

Proof. (i)Denotee_k_,_j(z₁,z₂) =e_k(z₁)e_j(z₂)where e_k(u) =uk. Since

f(z₁,z₂) =P∞_k₌₀P∞_j

=0ck,jek,j(z1,z2)and by the definition of the operator (1) we get

B_n_,_m_,_p_,_q(f) z₁,z₂

−f(z1,z2)

≤

∞ X

k=0

∞ X

j=0

c_k_,_j

B_n_,_m_,_p_,_q(e_k_,_j) z₁,z₂

−e_k,j(z1,z2)

.

By the simple calculation we can write

B_n_,_m_,_p_,_q(e_k_,_j) z₁,z₂

−e_k_,_j(z₁,z₂)=

Bn,p(ek)(z1).Bm,q(ej)(z2)−z

k

1z

j

2

≤ z

j

2

B_n_,_p(e_k)(z₁)−zk

1

+

B_n_,_p(e_k)(z₁)

Bm,q(ej)(z2)−z

j

2

≤r₂jA+B_n_,_p(e_k)(z₁)

B, (2)

say. By the Stirling numbers of second kindS k,j, we can write

Bn,p(ek)(z1) =

k

X

j=1

S k,j n+p

. . . n+p− j−1

n+pj

zj

(for complex Bernstein polynomials in one variable see[5, page 27]). SinceS k,j¾0, for alln,k∈N, p∈N∪ {0}andPk_j₌₁S k,j n+p. . . n+p− j−1= n+pkit follows

B_n_,_p(e_k)(z₁) ¶

k

X

j=1

S k,j n+p

. . . n+p− j−1

n+pk

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for allz₁

≤r₁, with 1<r₁,(p+1)r₁<R₁. To estimateAandBfor fixedn,m∈N, we should

consider two possible cases:

(1) 0≤k≤n+p, 0≤ j≤m+qand (2)k>n+p, j>m+q.

We start with case (1). Ifk=0, j=0 then obviously we getB_n_,_p(e_k)(z₁)−(e_k)(z₁) =0

andB_m_,_q(e_j)(z₂)−(e_j)(z₂)

=0. Therefore, let us suppose that 1≤k≤n+p,

1≤ j≤m+q. Denoting by∆k the finite difference of orderk, as in the case of the classical Bernstein polynomials we easily can write the representation formulas

B_n_,_p(f)(z₁) =

n+p

X

v=0

∆₁v_/_nf(0)e_v(z₁), B_m_,_q(f)(z₂) =

m+q

X

w=0

∆w₁_/_mf(0)e_w(z₂).

For simplicity, we use the following notations

C_np_,_v_,_k=

_n₊_p

v

∆₁v_/_nek(0) =

_n₊_p

v

0,1

n. . . , j n;ek

v!/nv,

C_nq_,_w_,_j=

m+q

w

∆w₁_/_mej(0) =

m+q

w

0, 1

m, . . . , j m;ej

w!/mw.

Since e_k,e_j are convex of any order, it follows that all C_np_,_v_,_k ≥ 0, C_nq_,_w_,_j ≥0 and taking into account thatBn,p(f)(1) = f (n+p)/p

, Bm,q(f)(1) = f m+q

/mwe get

n+p

X

v=0

C_np_,_v_,_k=B_n_,_p(e₁)(1) =

n+p n

k

,

m+q

X

w=0

C_nq_,_w_,_j=B_n_,_p(e₂)(1) =

m+q

m

j

. (3)

Using the result in the proof of Theorem 2.1 in[1]for anyz₁ ≤r₁,

z₂

≤r₂ with

1≤r1≤(p+1)r1<R1, 1≤r2≤(q+1)r2<R2, directly we can write

A=

Bm,q(ej)(z2)−z

j

2

≤

j j−1

m+q

q+1r2

j

+ 1

m

q+1r2

j − r

j

2

m

and

B=B_n_,_p(e_k)(z₁)−e_k(z₁) ≤

k(k−1)

n+p

p+1r₁k+ 1

n

p+1r₁k− r

k

1

n.

We now consider second case. Fork>n+p,z₁

≤ r₁ with 1≤ r₁≤(p+1)r₁ <R₁ and

for j>m+q,z₂

≤r₂with 1≤r₂≤(q+1)r₂<R₂, and considering (3) we get

B=B_n_,_p(e_k)(z₁)−e_k(z₁) ≤

B_n_,_p(e_k)(z₁) +rk

1

≤

_n₊_p

n

k

r₁n+p+r₁k

≤2 p+1r₁k≤ 2(k−1) n+p

p+1r₁k,

and in similar way,A=B_m_,_q(e_j)(z₂)−(e_j)(z₂) ≤

2(j−1)

m+q

q+1r2

j

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Combining all of the results obtained forAandB, we have the desired inequality. (ii) Now we give the rate of convergence in simultaneous approximation. Let 1≤ r1 < r1∗ <

R1

2, 1 ≤ r2 < r2∗ <

R2

2 andγ1 =

u₁−z₁

= r∗

1, γ2 =

u₂−z₂

= r∗

2. By the Cauchy’s formula

∂k1+k2_B

n,m,p,q(f)

∂k1z₁∂k2z₂ z1,z2

− ∂

k1+k2_f

∂k1z₁∂k2z₂ z1,z2

= k1!k2! (2πi)2

Z

γ2 Z

γ1

B_n_,_m_,_p_,_q(u₁,u₂)− f(u₁,u₂)du₁du₂ u₁−z₁k1+1

u₂−z₂k2+1

passing to absolute value withz₁ ≤r₁,

z₂

≤r₂and taking into account that

u₁−z₁ ≥r∗

1−r1,

u₂−z₂

≥r∗

2−r2, by applying the estimate in (i) we easily obtain

∂k1+k2_B

n,m,p,q(f)

∂k1z 1∂k2z2

z₁,z₂− ∂

k1+k2_f ∂k1z

1∂k2z2

z₁,z₂

≤M_rp∗,q

1,r2,n,m∗

(f). k1!

r₁∗−r1

k1+1

k2!

r₂∗−r2

k2+1

which proves the theorem.

In what follows a Voronovskaja’s result forB_n_,_m_,_p_,_q(f)is presented. It will be the product of the parametric extensions generated by the Voronovskaja’s formula in univariate case in The-orem 2.2 in[1]. Indeed, for f(z₁,z₂)defining the parametric extensions of the Voronovskaja’s formula by

z1Ln(f)(z1,z2):=Bn,p f(.,z2)

(z1)−f(z1,z2)−

p nz1

∂f

∂z₁ z1,z2

−z1(1−z1) 2n

∂2f

∂z₁2(z1,z2),

z2Lm(f)(z1,z2):=Bm,q f(z1, .)

(z₂)−f(z₁,z₂)− q

mz2

∂f

∂z₂ z1,z2

−z2(1−z2) 2m

∂2f

∂z₂2(z1,z2),

their product (composition) gives

z2Lm(f)(z1,z2)oz1Ln(f)(z1,z2)

=B_m_,_q

B_n_,_p f(., .)(z₁)− f(z₁, .)− p

nz1

∂f

∂z1

z₁, .−z1(1−z1) 2n

∂2f

∂z₁2(z1, .)

z₂

−

B_n_,_p f(.,z₂)(z₁)−f(z₁,z₂)− p

nz1

∂f

∂z1

z₁,z₂−z1(1−z1)

2n

∂2_f

∂z₁2(z1,z2)

− q

mz2

B_n,p

∂f

∂z2 (.,z2)

(z1)− ∂f

∂z2 (z1,z2)

−p

nz1

∂f

∂z₁

∂f

∂z₂

z1,z2

−z1(1−z1) 2n

∂2f

∂z₁2

∂f

∂z₂

z1,z2

− z2(1−z2) 2m .

Bn,p

∂2f

∂z₂2(.,z2)

(z1)− ∂2f

∂z2₂(z1,z2)−

−p

nz1

∂f

∂z1

∂2f

∂z2₂

z₁,z₂−z1(1−z1)

2n

∂2 ∂z₁2

∂2f

∂z₂2

(z₁,z₂)

(6)

:=E₁−E₂−E₃−E₄.

After simple calculation, we can write

z2Lm(f)(z1,z2)oz1Ln(f)(z1,z2) =Bn,m,p,q f

z₁,z₂−B_m_,_q f z₁, . z₂

− p

nz1.Bm,q

∂f

∂z₁ z1, .

z2

−z1(1−z1) 2n .Bm,q

∂2f

∂z₁2(z1, .)

(z2)

−B_n_,_p f(.,z₂)(z₁)−f(z₁,z₂)− p

nz1

∂f

∂z1

z₁,z₂−z1(1−z1)

2n

∂2_f

∂z2₁(z1,z2)

− q

mz2 Bn,p

∂f

∂z2 (.,z₂)

(z₁) + q

mz2

∂f

∂z2

(z₁,z₂) + p

nz1 q mz2

∂2f

∂z1∂z2

z₁,z₂

+ q

mz2

z₁(1−z₁) 2n

∂2f

∂z₁2

∂f

∂z₂

z1,z2

−z2(1−z2) 2m .Bn,p

∂2_f ∂z2₂(.,z2)

(z₁) +z2(1−z2) 2m

∂2_f

∂z₂2(z1,z2)

+z2(1−z2) 2m

p nz1

∂f

∂z1

∂2_f ∂z₂2

z₁,z₂+z1(1−z1) 2n

z2(1−z2) 2m

∂4_f

∂z2₁∂z₂2(z1,z2)

from which immediately can be derived the commutativity property

z2Lm(f)(z1,z2)oz1Ln(f)(z1,z2) =z1 Ln(f)(z1,z2)oz2Lm(f)(z1,z2).

Now we can give the Voronovskaja- type theorem.

Theorem 2. For fixed p,q∈N∪ {₀}_{and R}₁_>_p₊₁_{, R}₂_>_q₊₁_{suppose that}

f :P(0;R)→Cis analytic in P(0;R) =D_R

1×DR2, that is f(z1;z2) = P∞

k=0

P∞

j=0ck,jz

k

1z

j

2 for all

z_1,z₂∈P(0;R), R= (R₁,R₂). For allz₁ ≤r₁,

z₂

≤ r₂ with1<r₁,(p+1)r₁ <R₁,1<r₂,

(q+1)r₂<R₂and n,m∈Nwe have

_z

2Lm(f)(z1,z2)oz1Ln(f)(z1,z2)

≤Mp,q

r1,r2(f)

1

n2 + 1

m2

,

where

M_rp,q

1,r2(f) =max (_∞

X

k=2

∞ X

j=0

c_k_,_j q+1r₂jD_k_,_p_,_r₁,

∞ X

k=1

∞ X

j=1

c_k_,_j

j

q+1r₂j−1D_k_,_p_,_r₁,

∞ X

k=2

∞ X

j=2

c_k_,_jj(j−1) q+1r₂j−2D_k_,_p_,_r

1 )

,

D_k_,_p_,_r

1= (k−1)Ak

p+1r₁k−1+C_k_,_p p+1r₁k−2,A_k= (k−1) [4(k−1) (k−2) +2]

and

Ck,p= (k−1)

(7)

Proof. Since f(z1;z2) =

P∞

k=0

P∞

j=0ck,jz

k

1z

j

2 for all z1,z2

∈P(0;R), we can write

f(z₁;z₂) =P∞_k₌₀f_k(z₂)z₁kwithf_k(z₂) =P∞_j

=0ck,jz

j

2. It follows ∂f

∂z1(z1,z2) = P∞

k=1 fk z2

kz₁k−1,

∂2_f ∂z2 1

(z1,z2) =

P∞

k=2fk z2

k(k−1)z₁k−2 and ∂_∂2_z2f 2

(z1,z2) =

P∞

k=0 ∂2 ∂z2

2

fk z2

z₁kwhere

∂2 ∂z2 2

fk z2

=P∞_j

=2ck,jj(j−1)z

j−2

2 . HenceBn,p f(.,z2)

(z1) =

P∞

k=0 fk z2

Bn,p e1k

z1

and

B_n_,_p f(.,z₂)(z₁)−f(z₁,z₂)− p

nz1

∂f

∂z1

z₁,z₂−z1(1−z1)

2n

∂2f

∂z₁2(z1,z2)

=

∞ X

k=2

f_k z₂

B_n_,_p e₁k z₁− e₁k z₁− p

nkz

k

1−

z₁k−1(1−z₁)k(k−1) 2n

.

ApplyingB_m,qto the last expression with respect toz2, we obtain

E1=

∞ X

k=2

Bm,q fk

z2

Bn,p e1k

z1

− ek₁ z1

− p

nkz

k

1−

z₁k−1(1−z₁)k(k−1) 2n

=

∞ X

k=2

∞ X

j=0

c_k,jBm,q

e₁j z2

!

B_n,p e1k

z1

− e₁k z1

− p

nkz

k

1−

z₁k−1(1−z₁)k(k−1) 2n

.

Passing now to absolute value withz₁ ≤ r₁,

z₂

≤ r₂ and considering the estimates in the

proof of Theorem 2.1 and Theorem 2.2 in[1], we can write

E₁ ≤ ∞ X

k=2

∞ X

j=0 c_k_,_j

1+qr₂j





(k−1)A_k p+1r₁k−1+C_k_,_p p+1r₁k−2 n2





where A_k = (k−1) [4(k−1) (k−2) +2] and C_k_,_p = (k−1)p(5k−4) +p2+k(4k−7). Similarly, E₂ ≤ ∞ X

k=2

f_k z₂





(k−1)Ak

p+1r1

k−1

+Ck,p

p+1r1

k−2

n2   ≤ ∞ X

k=2

∞ X

j=0 c_k_,_j

1+qr₂j





(k−1)A_k p+1r₁k−1+C_k_,_p p+1r₁k−2 n2



.

Then

B_n_,_p

∂f

∂z2 (.,z₂)

(z₁) =

∞ X

k=0 ∂fk

∂z2

z₂B_n_,_p e₁k z₁

=

∞ X

k=0

∞ X

j=1

(8)

and

Bn,p

∂f

∂z₂(.,z2)

(z1)− ∂f

∂z₂(z1,z2)− p nz1

∂f

∂z₁

∂f

∂z₂

z1,z2

−z1(1−z1) 2n

∂2_f ∂z₁2

∂f

∂z2

z₁,z₂

=

∞ X

k=1

∞ X

j=1

ck,jjz j−1 2

Bn,p e1k

z1

− e₁k z1

− p

nkz

k

1−

z₁k−1(1−z₁)k(k−1) 2n

,

in the same way, we get

E₃

≤

∞ X

k=1

∞ X

j=1

c_k_,_j

j

q+1r₂j−1





(k−1)A_k p+1r₁k−1+C_k_,_p p+1r₁k−2 n2



.

Similarly

B_n_,_p

∂2f

∂z₂2(.,z2)

(z₁) =

∞ X

k=0 ∂2f_k

∂z₂2 z2

B_n_,_p ek₁ z₁

=

∞ X

k=0

∞ X

j=2

ck,jj j−1

z₂j−2Bn,p e1k

z1

and

B_n_,_p

∂2f

∂z₂2(.,z2)

(z₁)−∂ 2_f

∂z₂2(z1,z2)− p nz1

∂f

∂z₁

∂2f

∂z₂2

z₁,z₂−z1(1−z1)

2n

∂2 ∂z₁2

∂2f

∂z₂2

(z₁,z₂)

=

∞ X

k=2

∞ X

j=2

ck,jj j−1

z₂j−2

Bn,p e1k

z1

− e₁k z1

− p

nkz

k

1−

z₁k−1(1−z₁)k(k−1) 2n

hence by similar opinion we have

E₄

≤

∞ X

k=2

∞ X

j=2

c_k_,_j

j j−1 q+1

r₂j−2





(k−1)A_k p+1r₁k−1+C_k_,_p p+1r₁k−2 n2



.

Interchanging above the places of n and mwe obtain a similar order of approximation for

_z

1Lm(f)(z1,z2)oz2Ln(f)(z1,z2)

therefore

_z

2Lm(f)(z1,z2)oz1Ln(f)(z1,z2) ≤

E₁

+

E₂

+

E₃

+

E₄

≤M_rp,q

1,r2(f) ₁

n2 + 1

m2

withMrp1,,qr2(f)given by the statement.

The Voronovskaja- type theorem will be used to find the exact order in approximation by

Bn,n,p,p f

(9)

Theorem 3. For fixed p,q∈N∪ {0}and R₁>p+1, R₂>q+1suppose that f :P(0;R)→Cis analytic in P(0;R) =D_R

k=0

P∞

j=0ck,jz

k

1z

j

2 for all (z₁,z₂)∈P(0;R), R= (R₁;R₂). Denotingf

r1,r2 =sup

f(z₁;z₂) :

z₁

≤r₁,

z₂

≤r₂ , if f

is not a solution of the complex partial differential equation

pz1 ∂f

∂z₁ +

z₁(1−z₁) 2

∂2f

∂z₁2(z1,z2) +qz2

∂f

∂z₂ +

z₂(1−z₂) 2

∂2f

∂z₂2(z1,z2) =0, (4) for any(z1,z2)∈P(0;R), then we have

B_n_,_n_,_p_,_p−f

r_1,r₂≥

Kr1,r2,f

n , for all n∈N (5)

where Kr1,r2,f is independent on n.

Proof. We can write

Bn,n,p,p f

z1,z2

−f z1,z2

=2

n

¨

p

2z1 ∂f

∂z1

+z1(1−z1) 4

∂2f

∂z₁2(z1,z2) + p

2z2 ∂f

∂z2

+z2(1−z2) 4

∂2f

∂z₂2(z1,z2)

+2

n

n2

4 z2Ln(f)oz1Ln(f)

(z1,z2)

+Rn f

(z1,z2)

where

Rn f

(z1,z2) =n

2

Bn,p f(z1, .)

(z2)−f(z1,z2)−

p nz2

∂f

∂z₂ z1,z2

−z2(1−z2) 2n

∂2f

∂z₂2(z1,z2)

+n 2

B_n_,_p f(.,z₂)(z₁)−f(z₁,z₂)− p

nz1

∂f

∂z1

z₁,z₂−z1(1−z1)

2n

∂2f

∂z₁2(z1,z2)

+ p 2n

z1

Bn,p

_∂_f ∂z1 z1, .

z2

−_∂∂_zf

1 z1,z2

+z₂ B_n_,_p_∂∂_zf

2(.,z2)

(z₁)−_∂∂_zf

2 z1,z2

!

+z2(1−z2) 4

B_n,p

∂2f

∂z₂2(.,z2)

(z1)− ∂2f

∂z2₂(z1,z2)− p nz1

∂f

∂z₁

∂2f

∂z2₂

z1,z2

+z1(1−z1) 4

B_n_,_p

∂2f

∂z₁2(z1, .)

(z₂)−∂ 2_f

∂z2₁(z1,z2)− p nz2

∂2f

∂z₁2

∂f

∂z2

z₁,z₂

+z1(1−z1)z2(1−z2) 8n

∂4f

∂z₁2∂z₂2(z1,z2)

From Theorem 2.2 in[1]and by the reasonings in the above Theorem 2, it is immediate that

R_n f

r1,r2→0 asn→ ∞. Also, by Theorem 2 we obtain

n2

4

_z

2Lm(f)oz1Ln(f)

r1,r2≤

Mrp1,,qr2(f)

(10)

which implies

2

n

n2

4 z2Lm(f)oz1Ln(f)

+R_n f

r1,r2

→0, asn→ ∞.

Denoting

H z₁,z₂= p 2z1

∂f

∂z1

+z1(1−z1) 4

∂2f

∂z₁2(z1,z2) + q

2z2 ∂f

∂z2

+ z2(1−z2) 4

∂2f

∂z₂2(z1,z2)

and from the inequalities

kF+Gk_r₁_,_r₂≥kFk_r

1,r2− kGkr1,r2

≥ kFk_r

1,r2− kGkr1,r2

it follows

B_n_,_n_,_p_,_p− f

r1,r2 ≥2

n

¨

kHk_r₁_,_r₂−

2

n

n2

4 z2Lm(f)oz1Ln(f)

+R_n f

r1,r2 «

≥2

n

1

2kHkr1,r2=

1

nkHkr1,r2,

for all n≥n₀, with n₀ depending only on f,r₁ and r₂. We used here that by hypothesis we havekHkr1,r2>0.

Forn∈1, 2, . . .n₀−1 it is easily seen thatB_n_,_n_,_p_,_p−f

r1,r2

≥Ar_1,r_2,n,p(f)

n with

Ar1,r2,n,p f

=nB_n_,_n_,_p_,_p− f

r_1,r₂ which finally implies (5) where

K_r₁_,_r₂_,_f =max

§

A_r₁_,_r₂_,1,_p f, . . . ,A_r₁_,_r₂_,_n₀₋_1,_p f,1

2kHkr1,r2 ª

.

This completes the proof.

Combining now Theorem 1 with Theorem 3 we immediately obtain the following exact order.

Corollary 1. For fixed p,q∈N∪ {₀}_{and R}₁_>_p+1, R2>q+1suppose that f :P(0;R)→Cis

analytic in P(0;R) = D_R

k=0

P∞

j=0ck,jz

k

1z

j

2 for all(z1,z2)∈P(0;R),

R= (R1;R2). Assume that1< r1,(p+1)r1<R1,1<r2,(q+1)r2<R2. If f is not a solution

of the equation(4)then we have

B_n_,_n_,_p_,_p−f

r1,r2 ∼ 1

n

(11)

References

[1] G. A. Anastassiou and S. G. Gal. Approximation by complex Bernstein Schurer and Kan-torovich Schurer polynomials in compact disks, Computers & Mathematics with Applica-tions, 58, 734-743. 2009.

[2] D. Barbosu.Bivariate operators of Schurer-Stancu type, Analele Stiintifice ale Universitatii Ovidius Constanta, 11(1), 1-8. 2003.

[3] D. Barbosu.On the Schurer-Stancu approximation formula, Carpathian Journal of Math-ematics, 21(1 - 2), 7 - 12. 2005.

[4] D. Barbosu and T. O. Pop.Bivariate Schurer-Stancu operators revisited, Carpathian Journal of Mathematics, 26(1), 24 - 35. 2010.

[5] S. G. Gal.Approximation by complex Bernstein and convolution type operators, Series on Concrete and Applicable Mathematics, 8. World Scientific Publishing Private Limited Company, Hackensack, NJ, xii+337. 2009.

[6] G. G. Lorentz.Bernstein Polynomials, 2nd edition, Chelsea Publication, New York, 1986. [7] F. Schurer.On linear positive operators in approximation theory, Mathematical Institute of