An Application of a Poisson Distribution Series on Certain Analytic Functions

f (z) ∈ C (λ, α) ⇐⇒ zf 󸀠 (z) ∈ T (λ, α) . (5) The classes T(λ, α) and C(λ, α) were extensively studied by Altintas and Owa [1] and certain conditions for hypergeometric functions and generalized Bessel functions for these classes were studied by Mostafa [2] and Porwal and Dixit [3]. It is worthy to note that T(0, α) ≡ T(α), the class of starlike functions of order α (0 ≤ α < 1) and C(0, α) ≡ C(α), the class of convex functions of order α (0 ≤ α < 1) (see [4]). A variable x is said to have Poisson distribution if it takes the values 0, 1, 2, 3, . . . with probabilities e,


Introduction
Let denote the class of functions of the form which are analytic in the open unit disk = { : ∈ and | | < 1} and satisfy the normalization condition (0) = (0) − 1 = 0. Further, we denote by the subclass of consisting of functions of the form (1) which are also univalent in and let be the subclass of consisting of functions of the form ( Let ( , ) be the subclass of consisting of functions which satisfy the condition for some (0 ≤ < 1), (0 ≤ < 1) and for all ∈ . Also, we let ( , ) denote the subclass of consisting of functions which satisfy the condition for some (0 ≤ < 1), (0 ≤ < 1) and for all ∈ .
A variable is said to have Poisson distribution if it takes the values 0, 1, 2, 3, . . . with probabilities − , . . , respectively, where is called the parameter. Thus Now, we introduce a power series whose coefficients are probabilities of the Poisson distribution: We note that, by ratio test, the radius of convergence of the above series is infinity. Now, we introduce the series Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see [5][6][7][8][9][10]) and generalized Bessel functions (see [3,[11][12][13]), we obtain necessary and sufficient conditions for function ( , ) belonging to the classes ( , ) and ( , ). Finally, we give conditions for an integral operator ( , ) belonging to the classes ( , ) and ( , ).

Main Results
To establish our main results, we will require the following Lemmas according to Altintas and Owa [1].
Lemma 2 (see [1]). A function ( ) defined by (2) Proof. Since according to Lemma 1, we must show that But this last expression is bounded previously by 1 − if and only if (11) holds. Thus the proof of Theorem 3 is established. Proof. Since according to Lemma 2, we must show that But this last expression is bounded above by 1 − if and only if (15) holds. This completes the proof of Theorem 4.

An Integral Operator
In the following theorem, we obtain similar results in connection with a particular integral operator ( , ) as follows: Proof. Since Journal of Complex Analysis 3 by Lemma 2, we need only to show that Proof. The proof of this theorem is similar to that of Theorem 5. Therefore we omit the details involved.