Geometric properties of the complex Baskakov-Stancu operators in the unit disk

abstract: In this article, we determine certain conditions under which the partial sums involving the complex Baskakov-Stancu operators of analytic univalent functions of bounded turning are also of bounded turning. Moreover, we consider some geometric properties such as starlikeness and convexity for these partial sums. The lower bound of the partial sums of univalent functions is computed using the lower bound of the complex Baskakov-Stancu operators of analytic functions.


Introduction
A central concept in Geometric Function Theory is that of univalence.Many sufficient conditions of geometric type that impose univalence are important, like : starlikeness, convexity, close-to-convexity, α-convexity, spirallikeness and bounded turning.All these geometric sufficient conditions for univalence are mainly studied for analytic functions, because in this case they can obviously be expressed by nice (and simple) differential inequalities.Also, because of the Riemann Mapping Theorem, in general it suffices to study these properties on the open unit disk.
Concerning these properties, it is normal to ask how well can be approximated an analytic function having a given property in Geometric Function Theory, by polynomials having the same property.The history of this problem contains three main directions of research, depending on the methods used : approximation preserving geometric properties by the partial sums (for resent work see [1], [2]); approximation preserving geometric properties by Cesàro means (for recent work see [3]); 2000 Mathematics Subject Classification: 30C45 Rabha W. Ibrahim approximation of univalent functions by subordinate polynomials, by using the concept of maximal polynomial range (for recent work see [4], [5]).
It was shown that the partial sums of the Libera integral operator of univalent functions is starlike in |z| < 3  8 .The number 3 8 is sharp ( [6]).In [7], it was also shown that the partial sums of the Libera integral operator of functions of bounded turning are also of bounded turning.Moreover, in [1], the authors determined conditions under which the partial sums of some multiplier integral operators of analytic univalent functions of bounded turning are also of bounded turning.In [8], Owa considered the starlikeness and convexity of special classes of partial sums of certain analytic functions in the open unit disk.In [9], Silverman studied the radii properties for the sequence of partial sums of subfamilies of univalent functions.In [10], Latha and Shivarudrappa gave some results concerning partial sums of certain meromorphic functions.In [11], Goyal et.all., concerned on partial sums of certain meromorphic multivalent functions.
In this paper, we use the partial sums method in order to obtain new results concerning the preservation of geometric properties, such as bounded turning, by approximating and interpolating polynomials.These partial sums involve the complex Baskakov-Stancu operators of analytic univalent functions.

Concepts in geometric function theory
One of the major branches of complex analysis is univalent function theory: the study of one-to-one analytic functions.A domain E of the complex plane is said to be convex if and only if the line segment joining any two points of E lies entirely in E : An analytic, univalent function f in the unit disk U mapping the unit disk onto some convex domain is called a convex function.Moreover, A set D ⊂ C is said to be starlike with respect to the point z 0 ∈ D if the line segment joining z 0 to all points z ∈ D lies in D. A function f (z) which is analytic and univalent in the unit disk U, f (0) = 0 and maps U onto a starlike domain with respect to the origin.

Let H be the class of functions analytic in the open unit disk
Let A be the subclass of H consisting of functions of the form (2.1) A function f ∈ A is called starlike of order µ if it satisfies the following inequality for some 0 ≤ µ < 1.We denoted this class S * (µ).
A function f ∈ A is called convex of order µ if it satisfies the following inequality For 0 ≤ µ < 1, let B(µ) denote the class of functions f of the form (2.1) so that We need the following results in the sequel.
Lemma 2.1.[7] For z ∈ U we have The operator ( * ) stands for the Hadamard product or convolution of two power series in A,

The complex Baskakov-Stancu operators
In the present paper we concern about the following complex Baskakov-Stancu operator: where [x 0 , ..., x m ; f ] denotes the divided difference of the function f on the distinct points x 0 , ..., x m and for ν = 0, we put n(n + 1)...(n + ν − 1) = 1.Note that the operator V α,β n (f, z) is well defined for all z ∈ C. In [13], Gal et.all.studied the rate of the approximation of analytic functions for the Baskakov-Stancu operator V α,β n (f, z).
Now by employing the Hadamard product of analytic functions f ∈ A, we may define a linear operator using (2.6) as follows:

Main results
By making use Lemma 2.1 and Lemma 2.2, we illustrate the conditions under which the k−th partial sums of the operator (2.6) of analytic univalent functions of bounded turning are also of bounded turning.
Proof: Let f be of the form (2.1) and f (z Applying the convolution properties of power series to P ′ k (z) we may write Now in virtue of Lemma 2.1, we receive Furthermore, for sufficient large n, b j satisfies the inequality where ψ k,n := e(n)k!α k and α ≥ 1.Thus (3.3) and (3.4) yield A computation gives On the other hand, the power series satisfies: P (0) = 1 and Therefore, by Lemma 2.2, we have This completes the proof of Theorem 3.1.✷ We define the function S k which is a partial sum of f ∈ A by Now we proceed to compute the radii of starlikness and convexity of S k (z).
Theorem 3.2.The function S k (z) satisfies and that for cos θ → 1, we obtain Moreover, we also observe that This completes the proof.✷ Next we drive the radiuses of convexity.
Therefor, for cos θ → 1, we obtain Moreover, we impose Consider that This completes the proof.✷ where C is a sufficient large positive constant.

Conclusion
We defined new partial sums in the unit disk using the modified complex Baskakov-Stancu operator.We studied these partial sums geometrically by employing some concepts of the geometric functions theory.Partial sums play important roles in the univalent function theory.It has been shown that the proposal partial sums preserved most of the geometric properties such as bounded turning, starlikeness and convexity.

Lemma 2 . 2 .
[12,  Vol.I] Let P (z) be analytic in U, such that P (0) = 1, and ℜ(P (z)) > 1 2 in U .For functions Q analytic in U the convolution function P * Q takes values in the convex hull of the image on U under Q.

. 9 ) 3 . 4 .
Geometric properties of the complex Baskakov-Stancu operators 31 Finally we compute the lower bound of the partial sumsf m (z) = z + m j=2 z j , z ∈ U. (3Theorem Let f m be the partial sums defined in (3.9).Then|f m