ON UNIVALENT FUNCTIONS DEFINED BY A GENERALIZED SĂLĂGEAN OPERATOR

We introduce a class of univalent functions 
 R n ( λ , α ) defined by a new differential operator 
 D n f ( z ) , n ∈ ℕ 0 = { 0 , 1 , 2 , … } , where 
 D 0 f ( z ) = f ( z ) , D 1 f ( z ) = ( 1 − λ ) f ( z ) + λ z f ′ ( z ) = D λ f ( z ) , λ ≥ 0 , and 
 D n f ( z ) = D λ ( D n − 1 f ( z ) ) . Inclusion relations, 
extreme points of R n ( λ , α ) , some convolution 
properties of functions belonging to R n ( λ , α ) , and 
other results are given.


Introduction. Let A denote the class of functions of the form
analytic in the unit disc ∆ = {z : |z| < 1}.We denote by R(α) the subclass of A for which Ref (z) > α in ∆.For a function f in A, we define the following differential operator: (1.2) If f is given by (1.1), then from (1.3) and (1.4) we see that When λ = 1, we get Sȃlȃgean's differential operator [8].
The Hadamard product or convolution of two power series The object of this paper is to derive several interesting properties of the class R n (λ, α) such as inclusion relations, extreme points, some convolution properties, and other results.
2. Inclusion relations.Theorem 2.3 shows that the functions in R n (λ, α) belong to R(α) and hence are univalent.We need the following lemmas.The assertion of Lemma 2.1 follows by using the Herglotz representation for p.The next lemma is due to Fejér [3].
A sequence a 0 ,a 1 ,...,a n ,... of nonnegative numbers is called a convex null sequence if a n → 0 as n → ∞ and Now we prove the following theorem.
Proof.Let f belong to R n+1 (λ, α) and let it be given by (1.1).Then from (1.5), we have Applying Lemma 2.1 to (D n f (z)) , we get the required result.
We also have a better result than Theorem 2.3.
It is shown in [9], as an example, that if λ 0 and Now an application of Lemma 2.1 to (D n f (z)) in the previous theorem completes the proof.
which is an improvement of the result of Saitoh [7] for λ ≥ 1, where he shows that, for λ > 0, Using Theorem 2.4 ((n − m) times ) we get, after some calculations, the following theorem.
(2.12) If we put m = 0 in Theorem 2.6, we obtain the following interesting result.
The following theorem deals with the partial sum of the functions in R n (λ, α).For the proof we need the following result, due to Ahuja and Jahangiri [2].
) and let it be given by (1.1).Then from (1.5) we have Re 1 , λ>0. (2.18) From Lemma 2.9, we see that, for λ ≥ 1/s = 0.21892, Re and the result follows by application of Lemma 2.1.Now we prove the following theorem.
This result is sharp as can be seen by the function f x given by (3.1).

Extreme points.
The extreme points of the closed convex hull of R(α) were determined by Hallenbeck [4].We denote the closed convex hull of a family F by clco F , and we make use of some results in [4] to determine the extreme points of R n (λ, α).Theorem 3.1.The extreme points of R n (λ, α) are Proof.Since D n : f → D n f is an isomorphism from R n (λ, α) to R(α), it preserves the extreme points and, in [4], it is shown that the extreme points of R(α) are Hence from (1.5), we see that the extreme points of clcoR n (λ, α) are given by (3.1).Since the family R n (λ, α) is convex (Theorem 2.6) and therefore equal to its convex hull, we get the required result.
As consequences of Theorem 3.1, we have the following corollary.Corollary 3.2.Let f belong to R n (λ, α) and let it be given by (1.1).Then This result is sharp as shown by the function f x (z) given by (3.1).
This result is sharp as shown by the function f x (z) given by (3.1) at z = xr .

Convolution properties.
Ruscheweyh and Sheil-Small [6] verified the Polya-Schoenberg conjecture and its analogous results, namely, C * C ⊂ C, C * S ⊂ S , and C * K ⊂ K, where C, S , and K denote the classes of convex, starlike, and close-toconvex univalent functions, respectively.In the following, we prove the analogue of the Polya-Schoenberg conjecture for the class R n (λ, α).
Using convolution properties, we have and the result follows by application of Lemma 2.1.

Remark 2 . 5 .
If we put n = 1 in Theorem 2.4, then we have Re