APPROXIMATION AND GEOMETRIC PROPERTIES OF SOME COMPLEX BERNSTEIN-STANCU POLYNOMIALS IN COMPACT DISKS

. In this paper, the order of simultaneous approximation, convergence results of the iterates and shape preserving properties, for complex Bernstein-Stancu polynomials (depending on one parameter) attached to analytic functions on compact disks are obtained.


INTRODUCTION
Concerning the convergence of Bernstein polynomials in the complex plane, Bernstein proved (see e.g.[6, p.  Estimates of order O( 1 n ) of this uniform convergence and, in addition, of the simultaneous approximation, were found in [2].Also, in [2] it was proved that the complex Bernstein polynomials preserve (beginning with an index), the univalence, starlikeness, convexity and spirallikeness.
In [3], quantitative and qualitative approximation results for iterates of complex Bernstein polynomials were obtained.
The goal of this paper is to extend the above mentioned approximation results, to the following kind of complex Bernstein-Stancu polynomials: where γ may to depend on n and .
For γ = 0, these polynomials become the classical complex Bernstein polynomials.

APPROXIMATION PROPERTIES
Concerning the approximation orders by the Bernstein-Stancu polynomials defined in Introduction, the main results are expressed by the following.
Let 0 ≤ γ which can be dependent on n and 1 ≤ r < R.Then, for all |z| ≤ r and n ∈ N, we have Also, if 1 ≤ r < r 1 < R, then for all |z| ≤ r and n, p ∈ N, we have To estimate |S <γ> n (e k )(z) − e k (z)| for any fixed n ∈ N, we will consider two possible cases: 1) 0 ≤ k ≤ n; 2) k > n.
It follows
Collecting all the above estimates, we get for all |z| ≤ r Case 2).We have Reasoning as in the above Case 1), we get Collecting all the results in the Cases 1) and 2), we immediately obtain, for all |z| < r and k = 0, 1, 2, ...,

6
, which implies the corresponding estimate in statement.
For the simultaneous approximation, denoting by Γ the circle of radius r 1 > r and center 0, since for any |z| ≤ r and v ∈ Γ, we have |v − z| ≥ r 1 − r, by the Cauchy's formulas it follows that for all |z| ≤ r and n ∈ N, we have which proves the theorem.

This implies that
for all m, n ∈ N, which proves the theorem.
We also have the following quantitative result.
. Then, for all |z| ≤ r we have Proof.From the proof of Theorem 3.1, it follows that for all n, m, k ∈ N and |z| ≤ r, we have Case 1).With the notations for g j (α)(z) in the proof of Theorem 2.1 and for h j (z), c (j) i in the proof of Theorem 3.1, we can write Reasoning exactly as in the proof of Theorem 3.1, we easily get for all j, p and |z| ≤ r that Taking into account the formula for 1 − D n,k,k in the proof of Theorem 2.1, we get 2n . Also, Finally, But, taking into account the inequalities D n,k,k ≤ 1 and Collecting all these inequalities, we obtain Case 2).We get As a conclusion, from both Cases 1) and 2), we obtain we can write f (z) = zg(z), with g(z) = 0, for all z ∈ D 1 , where g is analytic in D 1 and continuous in D 1 .
Writing P n (f )(z) in the form which by the uniform convergence in D 1 of P n (f ) to f and by the maximum modulus principle, implies the uniform convergence in D 1 of Q n (f )(z) to g(z).
Since g is continuous in D 1 and |g(z)| > 0 for all z ∈ D 1 , there exist an index n 1 ∈ N and a > 0 depending on g, such that |Q n (f )(z)| > a > 0, for all z ∈ D 1 and all n ≥ n 0 .Also, for all |z| = 1, we have which from the maximum modulus principle, the uniform convergence of P n (f ) to f and of Q n (f ) to g, evidently implies the uniform convergence of Q n (f ) to g .
Since P n (f )(z) differs from S <γ(n)> n (f )(z) only by a constant, this proves the starlikeness of S <γ(n)> n (f )(z), for sufficiently large n.If f is supposed to be starlike only in D 1 , the proof is identical, with the only difference that instead of D 1 , we reason for D r .
The proofs in the cases when f is convex or spirallike of order η are similar and follow from the following uniform convergences (on D 1 or on D r ) Re zP n (f )(z) P n (f )(z) + 1 → Re zf (z) f (z) + 1, Re e iη zP n (f )(z) Pn(f )(z) → Re e iη zf (z) f (z) , as n → ∞, which proves the theorem.Remark 4.2.If f is univalent in D 1 , then from the uniform convergence in Theorem 2.1 and a well-known result in complex analysis, concerning sequences of analytic functions converging locally uniformly to an univalent function, it is immediate that for sufficiently large n, the complex polynomials S <γ(n)> n (f )(z) (where γ(n) → 0, for n → ∞), must be univalent in D 1 .
88]) that if f : G → C is analytic in the open set G ⊂ C, with D 1 ⊂ G (where D 1 = {z ∈ C : |z| < 1}), then the complex Bernstein polynomials B n (f )(z) = n k=0 n k z k (1 − z) n−k f ( k n ), uniformly converge to f in D 1 .