On New Bijective Convolution Operator Acting for Analytic Functions

Abstract: Problem statement: We introduced a new bijective convolution linear op erator defined on the class of normalized analytic functions. This op erator was motivated by many researchers namely Srivastava, Owa, Ruscheweyh and many others. The op erator was essential to obtain new classes of analytic functions. Approach: Simple technique of Ruscheweyh was used in our prel iminary approach to create new bijective convolution linear operator . The preliminary concept of Hadamard products was mentioned and the concept of subordination was given to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. I n fact, the subordinating factor sequence was used to derive different types of subordination results. Results: Having the linear operator, subordination theorems were established by using standard concept of subordination. The results reduced to wellknown results studied by various researchers. Coeff icient bounds and inclusion properties, growth and closure theorems for some subclasses were also obta ined. Conclusion: Therefore, many interesting results could be obtained and some applications cou ld be gathered.


INTRODUCTION
Let A denote the class of functions f normalized by: Using the convolution techniques, Ruscheweyh [1] introduced and studied the class of prestarlike functions of order β. Thus f∈ A is said to be prestarlike function of order β (0 ϕ is an incomplete beta function related to the Gauss hypergeometric function by 2 1 (a, b;z) = z F (1,a,b, z) ϕ , where the hypergeometric function 2 1 F (a, b,c, z) is defined by: Corresponding to the function (a, b;z) ϕ , Carlson and Shaffer introduced in [2] a convolution operator on A involving an incomplete beta function as: The Ruscheweyh derivatives of f (z) are β ≥ − .

MATERIALS AND METHODS
Using the Hadamard product (or convolution), we define a new bijective convolution operator acts on analytic functions in U . Then we state some of its properties and its special cases which will be used in this study. Corresponding to our operator, we define classes of analytic functions look like the classes of starlike and convex functions of order α (0 < 1). ≤ α Subordination principle and known results of subordination factor sequence will be used in our investigation of that classes.
Note that: If a = 0, -1, -2,…, then J(m,λ,a,b)f(z) is a polynomial. If a ≠ 0,-1, -2, then application of the root test shows that the infinite series for J(m,λ,a,b)f(z) has the same radius of convergence as that for f because: and the last expression equal 1 since: n n n (a) = 1 lim (b) →∞ Hence, J(m,λ,a,b) maps A into itself. So, we shall assume, unless other wise stated, that (a ≠ 0, -1, -2,…) and (b ≠ 0,-1,-2,…). We denote by g −1 where g∈ A , the function which satisfies For a > 0 and b > 0, if λ = 0, then: If λ > 0, then the operator J(m, λ ,a,b)f(z) can be represented by: and is a one-to-one mapping of A onto itself. Hence, J(m, ,a,b) λ maps A onto itself, infact it maps the class of analytic functions in U into itself. It also provides a convenient representation of differentiation and integration. By specializing the parameters m,λ,a and b, one can obtain various operators, which are special cases of J(m, ,a,b) λ studied earlier by many authors, such of those operators: ; m ∈ N due to Al-Oboudi [4] • J(m,1,1,1)f (z); m ∈ N due to S a ( l a ( gean [5] • The fractional operator due to Owa and Srivastava z J(0,0, 2, 2 )f (z) = f (z) = (2 )z D f (z); is the fractional derivative of f of order γ ; 2,3, 4,.. γ ≠ , [6] . Also, note that the operators J(0,0, 2, n 1)f (z); + n ∈ N due to Noor [7,8] ) and J(0,0, b,a)f (z); a > 0, b > 0 due to Choi al et. [9] are special cases of the inverse We prove now the following two identities which will be used in this study.
Lemma 1: Let f ∈ A satisfies (4). Then we have the following: be as in (4). Then we have: Hence, the proof is complete. Now, we introduce new classes of analytic functions involving our operator J(m, ,a,b) λ .

Definition 2:
, if and only if: for all z ∈ U Definition 3: A function f ∈ A is said to be in the class , if and only if: for all z ∈ U . Definition 4 [10] : Let g be analytic and univalent in U . If f is analytic in U , f (0) = g(0) and f ( ) g( ) ⊂ U U , then one says that f is subordinate to g in U and we write f g p or f (z) g(z) p . One also says that g is superordinate to f in U .
Definition 5 [10] : An infinite sequence k k =1 {b } ∞ of complex numbers is said to be a subordinating factor sequence if for every univalent function f in K, one has: Lemma 2 [11] : The sequence k k =1 {b } ∞ is a subordinating factor sequence if and only if: Proof: Suppose that (7) holds. Then by using Lemma 1 and for all z ∈ U , we have:

Coefficient bounds and Inclusions
The last expression is bounded by (1 ) − α if the following inequality which is equivalent to (7) holds: Then: Theorem 2: The last expression is bounded by (1 ) − α if the following inequality which is equivalent to (8) Proof: By using Theorem 1 and Lemma 1.
Proof: By using Theorem 2 and Lemma 1.
This completes the proof of Theorem 5.

Corollary 3:
Under the hypothesis of Theorem 5. f (z) is included in a disc centered at the origin with radius r given by: (2 )(1 ) a − α + − α + λ By using the same proof technique of Theorem 5, we can prove the following result.   technique as in [12][13][14][15] . Some applications of those main results which give important results of analytic functions are also investigated.
for every g in K and: Thus, the subordination result (9) will hold true if the sequence: is subordinating factor sequence, with a 0 = 1. In view of Lemma 2, this is equivalent to the following inequality: m n 1 n m n = 0 Now, since: This proves the inequality (11) and hence also the subordination result (9) asserted by Theorem 9. The inequality (10) follows from (9) by taking: z g(z) = K 1 z ∈ − Next, we consider the function: which satisfies the assumption of Theorem 9. Then by using (9), we have: It can be easily verified for the function f 1 (z) that: which completes the proof of Theorem 9. By taking = 0 λ in Theorem 9, we have the following corollary.

Corollary 7:
If the function f defined by (1) satisfies: with | a n | | b n | + ≥ + for all n ∈ N and 0 < 1 ≤ α , then for every function g in K, one has: replaced by a larger one. Putting = 0 λ and a = b in Theorem 9, we have the following corollary.
The constant ( α Ω due to Owa and Srivastava which is mentioned above.

Corollary 9:
If the function f defined by (1) satisfies: then for every function g in K, one has: The constant (3 ) / 2 − α cannot be replaced by a larger one.
It is clear from the proof of Theorem 9 that the is an extremal function of Corollary 8. Also the following example gives a non-polynomial function satisfies the same corollary.
Therefore, the Taylor-Maclaurin coefficients of the function h satisfy the condition in Corollary 8.
By the same proof technique of Theorem 9, we can prove the following theorem.
for every g in K and: The constant ( ) be replaced by a larger one. By taking = 0 λ in Theorem 10, we have the following corollary. with | a n | | b n | + ≥ + for all n ∈ N and 0 < 1 ≤ α , then for every function g in K, one has: replaced by a larger one. Putting = 0 λ and a = b in Theorem 10, we have the following corollary.

DISCUSSION
Let A denote the set of functions f of the form f(z) = z+a 1 z 2 +a 2 z 3 +…, which are analytic in the open unit disk. A new bijective convolution linear operator defined on A is introduced, which is a generalization of Carlson-Shaffer operator [2] and various well known operators and classes of analytic functions involving that operator are studied. Mainly, several properties of some subclasses are investigated, like coefficient bounds, inclusions, growth and closure theorems. Furthermore, main subordination results with some applications are investigated as well. The proof thechnique of those subordination results is used earlier by many researchers, namely Srivastava, Attyia, Ali, Ravichandran and Seenivasagan.

CONCLUSION
We conclude this study with some suggestions for future research, one direction is to study other classes of analytic functions involving our operator J(m, ,a,b) λ .
Another direction would be studying other properties of the classes S(m, ,a, b, ) λ α and C(m, ,a, b, ) λ α .