Return to Article Details Approximation by complex Bernstein-Kantorovich and Stancu-Kantorovich polynomials and their iterates in compact disks

APPROXIMATION BY COMPLEX BERNSTEIN-KANTOROVICH AND STANCU-KANTOROVICH POLYNOMIALS AND THEIR ITERATES IN COMPACT DISKS*

SORIN G. GAL $^{†}$

Abstract

In this paper, Voronovskaja-type results with quantitative upper estimates and the exact orders in simultaneous approximation by some complex Kantorovich-type polynomials and their iterates in compact disks in $C$ are obtained.

MSC 2000. Primary: 30E10; Secondary: 41A25, 41A28.
Keywords. Complex Bernstein-Kantorovich polynomials, complex StancuKantorovich polynomials, Voronovskaja's theorem, exact orders in simultaneous approximation.

1. INTRODUCTION AND AUXILIARY RESULTS

The complex Bernstein polynomials, the complex Bernstein-Stancu polynomials depending on two parameters

0 \leq α \leq β

and the complex BernsteinStancu polynomials depending on one parameter

0 \leq γ

, are defined by the same formulas as in the case of real variable, by

\begin{matrix} B_{n} (f) (z) = \sum_{k = 0}^{n} p_{n, k} (z) f (k / n), see e.g. [9], \\ S_{n}^{(α, β)} (f) (z) = \sum_{k = 0}^{n} p_{n, k} (z) f [(k + α) / (n + β)], see [14], \\ S_{n}^{< γ >} (f) (z) = \sum_{k = 0}^{n} p_{n, k}^{< γ >} (z) f (k / n), see [13], \end{matrix}

respectively, where

z \in C, p_{n, k} (z) = (\binom{n}{k}) z^{k} (1 - z)^{n - k}

and

\begin{matrix} p_{n, k}^{< γ >} (z) = \\ (\binom{n}{k}) \frac{z (z + γ) \dots (z + (k - 1) γ) (1 - z) (1 - z + γ) \dots (1 - z + (n - k - 1) γ)}{(1 + γ) (1 + 2 γ) \dots (1 + (n - 1) γ)} \end{matrix}

In the very recent book [2] and the papers [3], [4], [5], results on simultaneous approximation and of Voronovskaja-type, with quantitative estimates in compact disks, for the above defined complex Bernstein-type polynomials and their iterates were obtained.

The main aim of this paper is to extend these kind of results to the following Kantorovich variants of these polynomials, defined by

K_{n} (f) (z) = (n + 1) \sum_{k = 0}^{n} p_{n, k} (z) \int_{k / (n + 1)}^{(k + 1) / (n + 1)} f (t) d t, see [8]

and

K_{n}^{(α, β)} (f) (z) = (n + 1 + β) \sum_{k = 0}^{n} p_{n, k} (z) \int_{(k + α) / (n + 1 + β)}^{(k + 1 + α) / (n + 1 + β)} f (t) d t, see [1]

For our purpose, we need the following known results.
Theorem 1.1. Let

f : D_{R} \to C

be analytic in

D_{R} = {z \in C; | z | < R}

with

R > 1

, i.e.

f (z) = \sum_{k = 0}^{\infty} c_{k} z^{k}

, for all

z \in D_{R}

. Suppose

1 \leq r < r_{1} < R

. Then for all

| z | \leq r

and

n, p \in N

, we have:
(i) (a) (see [2, pp. 264, Theorem 3.4.1 (v)] or [4, Theorem 2.1, the case

α = β = 0]

)

| B_{n}^{(p)} (f) (z) - f^{(p)} (z) | \leq \frac{M_{2, r_{1}} (f) p! r_{1}}{n {(r_{1} - r)}^{p + 1}}

where

0 < M_{2, r_{1}} (f) = 2 \sum_{j = 2}^{\infty} j (j - 1) | c_{j} | r_{1}^{j} < \infty;

(b) (see [3, Theorem 2.1 (ii)])

| B_{n} (f) (z) - f (z) - \frac{z (1 - z)}{2 n} f^{''} (z) | \leq \frac{5 (1 + r)^{2}}{2 n} \cdot \frac{M_{r} (f)}{n}

where

M_{r} (f) = \sum_{k = 3}^{\infty} | c_{k} | k (k - 1) (k - 2)^{2} r^{k - 2} < \infty;

(ii) (a) (see [4, Theorem 2.1])

\begin{matrix} | {[S_{n}^{(α, β)} (f)]}^{(p)} (z) - f^{(p)} (z) | \leq \frac{M_{2, r_{1}}^{(β)} (f) p! r_{1}}{(n + β) {(r_{1} - r)}^{p + 1}} \\ where 0 < M_{2, r_{1}}^{(β)} (f) = 2 \sum_{j = 2}^{\infty} j (j - 1) | c_{j} | r_{1}^{j} + 2 β r \sum_{j = 1}^{\infty} j | c_{j} | r^{j - 1} < \infty \end{matrix}

(b) (see [4, proof of the Theorem 2.2])

| S_{n}^{(α, β)} (f) (z) - f (z) + \frac{β z - α}{n + β} f^{'} (z) - \frac{n z (1 - z)}{2 (n + β)^{2}} f^{''} (z) | = O [\frac{1}{(n + β)^{2}}]

where the positive constant in

O (1 / (n + β)^{2})

depends on

f, r, α

and

β

, but is independent of

n

and

z

;
(c) (see [4, Theorem 3.2]) Denoting the

m

th iterate by

^{m} S_{n}^{(α, β)} (f) (z)

, we have

|^{m} S_{n}^{(α, β)} (f) (z) - f (z) | \leq \frac{2 m}{n + β} \sum_{k = 1}^{\infty} | c_{k} | \cdot | β k + k (k - 1) | r^{k}

(iii) (see [5, Theorem 2.1])

| {[S_{n}^{< γ >} (f)]}^{(p)} (z) - f^{(p)} (z) | \leq \frac{M_{2, r_{1}, n}^{< γ >} (f) p! r_{1}}{{(r_{1} - r)}^{p + 1}}

where

0 < M_{2, r_{1}, n}^{< γ >} (f) = \frac{2}{n} \sum_{j = 2}^{\infty} j (j - 1) | c_{j} | r_{1}^{j} + \frac{γ (r_{1} + 1)}{6 r_{1}} \sum_{j = 2}^{\infty} j (j - 1) (2 j - 1) | c_{j} | r_{1}^{j} < \infty

.
(iv) (see [6, Theorem 3.1]) If

f

is not a polynomial of degree

\leq max {1, p - 1}

, then we have

{‖ B_{n}^{(p)} (f) - f^{(p)} ‖}_{r} \sim \frac{1}{n},

where

‖ f ‖_{r} = sup {| f (z) |; | z | \leq r}

and the constants in the equivalence depend only on

f, r

and

p

.
(v) (see [7, Theorem 3.1]) Let

0 \leq α \leq β

with

α + β > 0

. If

f

is not a polynomial of degree

\leq p - 1

then we have

{‖ {[S_{n}^{(α, β)} (f)]}^{(p)} - f^{(p)} ‖}_{r} \sim \frac{1}{n + β},

where the constants in the equivalence depend only on

f, α, β, r

and

p

Remark 1.2. The Voronovskaja-type result in [4, Theorem 2.2] holds for

| z | \leq 1

. The proof of the above point (ii) (b), is immediate by replacing in the proof of Theorem 2.2 in [4] the condition

| z | \leq 1

| z | \leq r

2. COMPLEX BERNSTEIN-KANTOROVICH POLYNOMIALS

For our purpose also will be useful the next classical result.
Theorem 2.1. (see e.g. [9, pp. 30]) Denoting

F (z) = \int_{0}^{z} f (t) d t

, we have the relationship

K_{n} (f) (z) = B_{n + 1}^{'} (F) (z), z \in C

Now, as a consequence of Theorem 2.1 and Theorem 1.1, (iv), we immediately get the following.

Corollary 2.2. Let

f : D_{R} \to C

be analytic in

D_{R}

with

R > 1

and

1 \leq r < R

.
(i) If

f

is not a polynomial of degree

\leq 0

then for all

n \in N

we have

{‖ K_{n} (f) - f ‖}_{r} \sim \frac{1}{n}

where the constants in the equivalence depend only on

f

and

r

.
(ii) If

f

is not a polynomial of degree

\leq max {1, p - 1}

then for all

p, n \in N

we have

{‖ K_{n}^{(p)} (f) - f^{(p)} ‖}_{r} \sim \frac{1}{n}

with the constants in the equivalence depending only on

f, r

and

p

.
Proof. We combine Theorem 2.1, (i) with Theorem 1.1, (iv).
(i) We get

{‖ K_{n} (f) - f ‖}_{r} = {‖ B_{n + 1}^{'} (F) - F^{'} ‖}_{r} \sim \frac{1}{n + 1}

F

is not a polynomial of degree

\leq max {1, 1} = 1

, which ends the proof.
(ii) We obtain

{‖ K_{n}^{(p)} (f) - f^{(p)} ‖}_{r} = {‖ B_{n + 1}^{(p + 1)} (F) - F^{(p + 1)} ‖}_{r} \sim \frac{1}{n + 1}

F

is not a polynomial of degree

\leq max {1, p} = p

, which ends the proof.

Upper estimates with explicit constants in Voronovskaja's theorem and in approximation by

K_{n} (f)

can be derived as follows.

Theorem 2.3. Let

f : D_{R} \to C

be analytic in

D_{R} = {z \in C; | z | < R}

with

R > 1

, i.e.

f (z) = \sum_{k = 0}^{\infty} c_{k} z^{k}

, for all

z \in D_{R}

. Suppose

1 \leq r < r_{1} < R

. Then for all

| z | \leq r

and

n, p \in N

, we have:
(i)
(ii)

\begin{array}{r} | K_{n}^{(p)} (f) (z) - f^{(p)} (z) | \leq \frac{C_{2, r_{1}} (f) (p + 1)! r_{1}}{(n + 1) {(r_{1} - r)}^{p + 2}}, \\ where 0 < C_{2, r_{1}} (f) = 2 \sum_{j = 2}^{\infty} (j - 1) | c_{j - 1} | r_{1}^{j} < \infty; \end{array}

(ii)

| K_{n} (f) (z) - f (z) - \frac{1 - 2 z}{2 (n + 1)} \cdot f^{'} (z) - \frac{z (1 - z)}{2 (n + 1)} \cdot f^{''} (z) | \leq \frac{r_{1} C_{r_{1}, n + 1} (f)}{{(r_{1} - r)}^{2}}

where

C_{r_{1}, n} (f) = \frac{5 {(1 + r_{1})}^{2}}{2 n} \cdot \frac{\sum_{k = 3}^{\infty} | c_{k - 1} | (k - 1) (k - 2)^{2} r_{1}^{k - 2}}{n} .

Proof. (i) Combining Theorem 2.1 with Theorem 1.1, (i) (a), we obtain

| K_{n}^{(p)} (f) (z) - f^{(p)} (z) | = | B_{n + 1}^{(p + 1)} (F) (z) - F^{(p + 1)} (z) | \leq \frac{M_{2, r_{1}} (F) (p + 1)! r_{1}}{(n + 1) {(r_{1} - r)}^{p + 2}}

where

0 < M_{2, r_{1}} (F) = 2 \sum_{j = 2}^{\infty} j (j - 1) | C_{j} | r_{1}^{j} < \infty

and

F (z) = \sum_{k = 0}^{\infty} C_{k} z^{k}, z \in D_{R}

.
But we also get

F (z) = \int_{0}^{z} [\sum_{k = 0}^{\infty} c_{k} t^{k}] d t = \sum_{k = 0}^{\infty} \frac{c_{k}}{k + 1} z^{k + 1} = \sum_{k = 1}^{\infty} \frac{c_{k - 1}}{k} z^{k}

which implies

C_{k} = \frac{c_{k - 1}}{k}

and

C_{2, r_{1}} (f) = 2 \sum_{j = 2}^{\infty} (j - 1) | c_{j - 1} | r_{1}^{j}

.
(ii) Replacing in Theorem 1.1, (i) (b),

n

n + 1, r

r_{1}

and

f

F

, for all

| z | \leq r_{1}

and

n \in N

, we obtain

| B_{n + 1} (F) (z) - F (z) - \frac{z (1 - z)}{2 (n + 1)} F^{''} (z) | \leq \frac{5 {(1 + r_{1})}^{2}}{2 (n + 1)} \cdot \frac{M_{r_{1}} (F)}{n + 1}

where

\begin{aligned} M_{r_{1}} (F) & = \sum_{k = 3}^{\infty} | C_{k} | k (k - 1) (k - 2)^{2} r_{1}^{k - 2} = \\ = \sum_{k = 3}^{\infty} | c_{k - 1} | (k - 1) (k - 2)^{2} r_{1}^{k - 2} := A_{r_{1}} (f) \end{aligned}

Here again we wrote

F (z) = \sum_{k = 0}^{\infty} C_{k} z^{k}

, for all

z \in D_{R}

.
Now, denoting

C_{r_{1}, n} (f) = \frac{5 {(1 + r_{1})}^{2}}{2 n} \cdot \frac{A_{r_{1}} (f)}{n}

, by

Γ

the circle of radius

r_{1} > r

and center 0 , and

E_{n} (F) (z) = B_{n + 1} (F) (z) - F (z) - \frac{z (1 - z)}{2 (n + 1)} F^{''} (z)

, since for any

| z | \leq r

and

v \in Γ

, we have

| v - z | \geq r_{1} - r

, by the Cauchy's formula it follows that for all

| z | \leq r

and

n \in N

, we obtain

| E_{n}^{'} (F) (z) | = \frac{1}{2 π} | \int_{Γ} \frac{E_{n} (f) (z)}{(v - z)^{2}} d v | \leq C_{r_{1}, n + 1} (f) \frac{1}{2 π} \frac{2 π r_{1}}{{(r_{1} - 1)}^{2}} = C_{r_{1}, n + 1} (f) \cdot \frac{r_{1}}{{(r_{1} - r)}^{2}}

But by Theorem 2.1 we obtain

E_{n}^{'} (F) (z) = K_{n} (f) (z) - f (z) - \frac{1 - 2 z}{2 (n + 1)} \cdot f^{'} (z) - \frac{z (1 - z)}{2 (n + 1)} \cdot f^{''} (z)

which proves the theorem.

3. COMPLEX STANCU-KANTOROVICH POLYNOMIALS DEPENDING ON TWO PARAMETERS

For our purpose will be useful the next result.
Theorem 3.1. Denoting

F (z) = \int_{0}^{z} f (t) d t

, we have the relationship

K_{n}^{(α, β)} (f) (z) = \frac{n + 1 + β}{n + 1} {[S_{n + 1}^{(α, β)} (F)]}^{'} (z), z \in C

Proof. The theorem is immediate by the following formula

\begin{aligned} {[S_{n + 1}^{(α, β)} (F)]}^{'} (z) = & (n + 1 + β) \sum_{k = 0}^{n} p_{n, k} (z) [F (\frac{k + α + 1}{n + β + 1}) - F (\frac{k + α}{n + 1 + β})] \\ - β \sum_{k = 0}^{n} p_{n, k} (z) [F (\frac{k + α + 1}{n + β + 1}) - F (\frac{k + α}{n + 1 + β})] \\ = & K_{n}^{(α, β)} (f) (z) - \frac{β}{n + 1 + β} K_{n}^{(α, β)} (f) (z) \end{aligned}

As a consequence of Theorem 3.1 and Theorem 1.1, (v), we also get the following.

Corollary 3.2. Let

f : D_{R} \to C

be analytic in

D_{R}

with

R > 1, 1 \leq r < R

and

0 \leq α \leq β, α + β > 0

.
(i) If

f

is not identical 0 , then for all

n \in N

we have

{‖ K_{n}^{(α, β)} (f) - f ‖}_{r} \sim \frac{1}{n + β}

where the constants in the equivalence depend only on

f, r, α

and

β

.
(ii) If

f

is not a polynomial of degree

\leq p - 1

then for all

p, n \in N

we have

{‖ {[K_{n}^{(α, β)} (f)]}^{(p)} - f^{(p)} ‖}_{r} \sim \frac{1}{n + β},

with the constants in the equivalence depending only on

f, r, α, β

and

p

Proof. We combine Theorem 3.1 with Theorem 1.1, (v).
(i) We get

{‖ K_{n}^{(α, β)} (f) - f ‖}_{r} = {‖ {[S_{n + 1}^{(α, β)} (F)]}^{'} - F^{'} ‖}_{r} \sim \frac{1}{n + β},

F

is not a polynomial of degree

\leq 0

, which ends the proof.
(ii) We obtain

{‖ {[K_{n}^{(α, β)} (f)]}^{(p)} - f^{(p)} ‖}_{r} = {‖ {[S_{n + 1}^{(α, β)} (F)]}^{(p + 1)} - F^{(p + 1)} ‖}_{r} \sim \frac{1}{n + β},

F

is not a polynomial of degree

\leq p

, which ends the proof.
Upper estimates with explicit constants in Voronovskaja's theorem and in approximation by

K_{n}^{(α, β)} (f) (z)

polynomials can be derived as follows.

Theorem 3.3. Let

f : D_{R} \to C

be analytic in

D_{R} = {z \in C; | z | < R}

with

R > 1

, i.e.

f (z) = \sum_{k = 0}^{\infty} c_{k} z^{k}

, for all

z \in D_{R}

. Suppose

1 \leq r < r_{1} < R

. Then for all

| z | \leq r

and

n, p \in N

, we have:
(i)
(ii)

\begin{matrix} | {[K_{n}^{(α, β)} (f)]}^{(p)} (z) - f^{(p)} (z) | \leq \frac{C_{2, r_{1}}^{(β)} (f) (p + 1)! r_{1}}{(n + 1) {(r_{1} - r)}^{p + 2}} + \frac{β}{n + 1} | | f | ‖_{r}, \\ where 0 < C_{2, r_{1}}^{(β)} (f) = 2 \sum_{j = 2}^{\infty} (j - 1) | c_{j - 1} | r_{1}^{j} + 2 β \sum_{j = 1}^{\infty} | c_{j - 1} | r_{1}^{j} < \infty; \end{matrix}

(ii)

\begin{aligned} | K_{n}^{(α, β)} (f) (z) - f (z) + (\frac{β z - α}{n + 1} - \frac{1 - 2 z}{2 (n + β + 1)}) f^{'} (z) - \frac{z (1 - z)}{2 (n + β + 1)} f^{''} (z) | \leq \\ \leq \frac{C (f, r_{1}, α, β)}{(n + 1) (n + β + 1)} \cdot \frac{r_{1}}{{(r_{1} - r)}^{2}} \end{aligned}

where

C (f, r_{1}, α, β)

is a positive constant depending only on

f, r_{1}, α

and

β

Proof. (i) Combining Theorem 3.1 with Theorem 1.1, (ii) (a), for all

| z | \leq r

we obtain

\begin{aligned} | {[K_{n}^{(α, β)} (f)]}^{(p)} (z) - f^{(p)} (z) | = \\ = | \frac{n + 1 + β}{n + 1} {[S_{n + 1}^{(α, β)} (F)]}^{(p + 1)} (z) - F^{(p + 1)} (z) | \\ \leq \frac{n + 1 + β}{n + 1} | {[S_{n + 1}^{(α, β)} (F)]}^{(p + 1)} (z) - F^{(p + 1)} (z) | + \frac{β}{n + 1} | F^{(p + 1)} (z) | \\ \leq \frac{n + 1 + β}{n + 1} \cdot \frac{M_{2, r_{1}}^{(β)} (F) (p + 1)! r_{1}}{(n + β + 1) {(r_{1} - r)}^{p + 2}} + \frac{β}{n + 1} \cdot | f^{(p)} (z) | \\ \leq \frac{M_{2, r_{1}}^{(β)} (F) (p + 1)! r_{1}}{(n + 1) {(r_{1} - r)}^{p + 2}} + \frac{β}{n + 1} \cdot {‖ f^{(p)} ‖}_{r} \end{aligned}

and reasoning exactly as in the proof of Theorem 2.3, (i), we get

\begin{aligned} M_{2, r_{1}}^{(β)} (F) & = 2 \sum_{j = 2}^{\infty} j (j - 1) | C_{j} | r_{1}^{j} + 2 β \sum_{j = 1}^{\infty} j | C_{j} | r_{1}^{j} \\ = 2 \sum_{j = 2}^{\infty} (j - 1) | c_{j - 1} | r_{1}^{j} + 2 β \sum_{j = 1}^{\infty} | c_{j - 1} | r_{1}^{j} := C_{2, r_{1}}^{(β)} (f) \end{aligned}

(ii) Replacing in Theorem 1.1, (ii) (b),

n

n + 1, r

r_{1}

and

f

F

, for all

| z | \leq r_{1}

and

n \in N

, we obtain

| S_{n + 1}^{(α, β)} (F) (z) - F (z) + \frac{β z - α}{n + β + 1} F^{'} (z) - \frac{(n + 1) z (1 - z)}{2 (n + β + 1)^{2}} F^{''} (z) | \leq \frac{C (f, r_{1}, α, β)}{(n + β + 1)^{2}}

where the positive constant

C (f, r_{1}, α, β)

depends only on

f, r, α

and

β

. Let us denote

E_{n} (F) (z) = S_{n + 1}^{(α, β)} (F) (z) - F (z) + \frac{β z - α}{n + β + 1} F^{'} (z) - \frac{(n + 1) z (1 - z)}{2 (n + β + 1)^{2}} F^{''} (z)

Γ

is the circle of radius

r_{1} > r

and center 0 , and since for any

| z | \leq r

and

v \in Γ

, we have

| v - z | \geq r_{1} - r

, by the Cauchy's formula it follows that for all

| z | \leq r

and

n \in N

, we obtain as in the proof of Theorem 2.3, (ii)

| E_{n}^{'} (F) (z) | \leq C (f, r_{1}, α, β) \cdot \frac{r_{1}}{{(r_{1} - r)}^{2}} \cdot \frac{1}{(n + β + 1)^{2}}

But by Theorem 3.1 we obtain

\begin{aligned} E_{n}^{'} (F) (z) & = \frac{n + 1}{n + 1 + β} K_{n}^{(α, β)} (f) (z) - f (z) + \frac{1}{n + β + 1} [(β z - α) f (z)]^{'} \\ - \frac{n + 1}{2 (n + β + 1)^{2}} {[(z - z^{2}) f^{'} (z)]}^{'} \\ = \frac{n + 1}{n + β + 1} \cdot A \end{aligned}

where

A = K_{n}^{(α, β)} (f) (z) - f (z) + f^{'} (z) (\frac{β z - α}{n + 1} - \frac{1 - 2 z}{2 (n + β + 1)}) - \frac{z (1 - z)}{2 (n + β + 1)} f^{''} (z)

which immediately proves the theorem.
Concerning the

m

th iterates

^{m} K_{n}^{(α, β)} (f) (z)

, we obtain the following result.

Theorem 3.4. Let

f : D_{R} \to C

be analytic in

D_{R} = {z \in C; | z | < R}

with

R > 1

, i.e.

f (z) = \sum_{k = 0}^{\infty} c_{k} z^{k}

, for all

z \in D_{R}

. Suppose

1 \leq r < r_{1} < R

. Then for all

| z | \leq r

and

n, p \in N

, we have

| {[^{m} K_{n}^{(α, β)} (f)]}^{(p)} (z) - f^{(p)} (z) | \leq \frac{2 m}{n + 1 + β} \sum_{k = 1}^{\infty} | c_{k - 1} | \cdot | β + (k - 1) | r^{k} \cdot \frac{(p + 1)! r_{1}}{{(r_{1} - r)}^{p + 1}}

Proof. First we easily observe that

^{m} K_{n}^{(α, β)} (f) (z) = \frac{d}{d z} [^{m} S_{n + 1}^{(α, β)} (F)] (z),

where

F (z) = \int_{0}^{z} f (t) d t = \sum_{k = 0}^{\infty} C_{k} z^{k}

. Taking into account Theorem 1.1, (ii) (c), the Cauchy's theorem and reasoning exactly as in the proofs of Theorem 2.3, (i) and 3.3, (i), it follows

\begin{aligned} | {[^{m} K_{n}^{(α, β)} (f)]}^{(p)} (z) - f^{(p)} (z) | & = | {[^{m} S_{n + 1}^{(α, β)} (F)]}^{(p + 1)} (z) - F^{(p + 1)} (z) | \\ \leq \frac{2 m}{n + 1 + β} \sum_{k = 1}^{\infty} | C_{k} | \cdot | β k + k (k - 1) | r^{k} \cdot \frac{(p + 1)! r_{1}}{{(r_{1} - r)}^{p + 1}} \\ = \frac{2 m}{n + 1 + β} \sum_{k = 1}^{\infty} | c_{k - 1} | \cdot | β + (k - 1) | r^{k} \cdot \frac{(p + 1)! r_{1}}{{(r_{1} - r)}^{p + 1}} \end{aligned}

which proves the theorem.
Remark 3.5. For

β = 0

in Theorem 3.4 we get corresponding results for the iterates of classical complex Kantorovich polynomials. Note that in the real case, some asymptotic results for the iterates of Kantorovich polynomials were obtained in [10].

REMARK 3.6. If

\frac{m_{n}}{n} \to 0

when

n \to \infty

, then by Theorem 3.4 it is immediate that

{[^{m_{n}} K_{n}^{(α, β)} (f)]}^{(p)} (z) \to f^{(p)} (z),

uniformly with respect to

| z | \leq 1

, for any

1 \leq r < R

.
Remark 3.7. The Stancu-Kantorovich polynomials depending on the parameter

0 \leq γ

were introduced in [12] by

K_{n}^{< γ >} (f) (z) = (n + 1) \sum_{k = 0}^{n} p_{n, k}^{< γ >} (z) \int_{k / (n + 1)}^{(k + 1) / (n + 1)} f (t) d t

where

p_{n, k}^{< γ >} (z) = (\binom{n}{k}) \frac{z (z + γ) \dots (z + (k - 1) γ) (1 - z) (1 - z + γ) \dots (1 - z + (n - k - 1) γ)}{(1 + γ) (1 + 2 γ) \dots (1 + (n - 1) γ)} .

To prove analogous results for these polynomials too, we would need a similar connection between

{[S_{n + 1}^{< γ >} (F)]}^{'} (z)

and

K_{n}^{< γ >} (f) (z)

, with those in Theorems 2.1 and 3.1. But this study is left as an open question.

Remark 3.8. The complex Kantorovich polynomials of second order can be defined as in the case of real variable ([11]) by

Q_{n} (f) (z) = {[B_{n + 2} (H) (z)]}^{''}, z \in C

where

H (z) = \int_{0}^{z} F (u) d u, F (u) = \int_{0}^{u} f (t) d t

and

B_{n + 2}

is the (

n + 2

)-th Bernstein polynomial.

It is easy to see that similar approximation results with those for

K_{n} (f) (z)

in Section 2 can be obtained for

Q_{n} (f) (z)

too.

REFERENCES

[1] Bărbosu, D., Kantorovich-Stancu type operators, J. Ineq. Pure Appl. Math., 5, No. 3, Article 53 (electronic), 2004.
[2] Gal, S.G., Shape Preserving Approximation by Real and Complex Polynomials, Birkhauser Publ., Boston, Basel, Berlin, 2008.
[3] Gal, S.G., Voronovskaja's theorem and iterations for complex Bernstein polynomials in compact disks, Mediterr. J. Math., 5, no. 3, pp. 253-272, 2008.
[4] Gal, S.G., Approximation by complex Bernstein-Stancu polynomials in compact disks, Results in Math., accepted for publication.
[5] Gal, S.G., Approximation and geometric properties of some complex Bernstein-Stancu polynomials in compact disks, Rev. Anal. Numér. Théor. Approx. (Cluj), 36, No. 1, pp. 67-77, 2007. ㄸ
[6] Gal, S.G., Exact orders in simultaneous approximation by complex Bernstein polynomials, J. Concr. Applic. Math., 2009, accepted for publication.
[7] Gal, S.G., Exact orders in simultaneous approximation by complex Bernstein-Stancu polynomials, Rev. Anal. Numér. Théor. Approx. (Cluj), 37, No. 1, 2008, in press. ©
[8] Kantorovich, L.V., Sur certains developpments suivant les polynômes de la forme de S. Bernstein, I, II, C.R. Acad. Sci. URSS, pp. 563-568, pp. 595-600, 1930.
[9] Lorentz, G.G., Bernstein Polynomials, 2nd edition, Chelsea Publ., New York, 1986.
[10] Nagel, J., Asymptotic properties of powers of Kantorović operators, J. Approx. Theory, 36, pp. 268-275, 1982.
[11] Nagel, J., Kantorovic̃ operators of second order, Monatsh. Math., 95, pp. 33-44, 1983.
[12] Razi, Q., Approximation of a function by Kantorovich type operators, Mat. Vest., 41, pp. 183-192, 1989.
[13] Stancu, D.D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine. Math. Pures Appl., 13, pp. 1173-1194, 1968.
[14] Stancu, D.D., On a generalization of Bernstein polynomials (in Romanian), Stud. Univ. "Babeş-Bolyai", ser. Math., 14, No. 2, pp. 31-44, 1969.

Received by the editors: May 8, 2008.

*This work has been supported by the Romanian Ministry of Education and Research, under CEEX grant, 2-CEx 06-11-96.
$^{†}$ Department of Mathematics and Computer Science, University of Oradea, Universităţii str., no. 1, 410087 Oradea, Romania, e-mail: galso@uoradea.ro.