Application of Pascal distribution series to Rønning type starlike and convex functions

In this article we investigate the connections between the Pascal distribution series and the class of analytic functions f normalized by f(0) = f ′(0) − 1 = 0 in the open unit disk U = {z ∈ C : |z| < 1} and its coe cients are probabilities of the Pascal distribution.More precisely ,we determine such connection with parabolic starlike and uniformly convex functions in the open unit disk U .


Introduction
Let U represent the unit disk {z ∈ C : |z| < 1} and H represent the set of analytic functions in U. We suppose A denote the subset of H comprising of functions f (z) = z + ∞ n=2 a n z n z ∈ U, normalized by f (0) = 0 = f (0) − 1 and univalent in U. Denote by T the subclass of A whose members are f (z) = z − ∞ n=2 a n z n , a n ≥ 0.
For functions f 1 (z) = z + ∞ n=2 a n,1 z n and f 2 (z) = z + ∞ n=2 a n,2 z n , in A then the Hadamard product (or convolution) of f 1 and f 2 by (f 1 * f 2 )(z) = z + ∞ n=2 a n,1 a n,2 z n , z ∈ U.
Special functions (series) play a vital role in geometric function theory, exclusively in the proof by de Branges of the famous Bieberbach conjecture. The astonishing use of special functions (hypergeometric functions) has provoked renewed attention in function theory in the last few decades (see [4,6,12,16,17]) and lately by probability distribution series [2,5,8,10,11].
A variable χ is said to be Pascal distribution if it takes the values 0, 1, 2, 3, . . . with probabilities , . . . respectively, where q and κ are called the parameter,and thus Lately, for κ ≥ 1; 0 ≤ q ≤ 1, El-Deeb et al. [5] gave a power series whose coecients are probabilities of Pascal distribution We note by the familiar Ratio Test that the radius of convergence of the above series is innity. More recently, Bulboac and Murugusundaramoorthy [2] introduced a linear operator by the convolution (or Hadamard) product I κ q : A → A which is dened as follows: Motivated by the aforementioned works on hypergeometric functions [4,6,12,16,17], and distribution function [2,5,8,10,11] we give the connections between Pascal distribution series with the classes M * µ (ϑ, ν) and N * µ (ϑ, ν) by applying the convolution operator given by (6).
Making use of the Lemma 2.1,in the following theorem we will establish the connection between Pascal distribution series with the class N µ (α, ν).
Proof. Let f be given by (1) and a member of R τ (υ, δ). By asset of Lemma 1.1 and (4) it suits to show that Since f ∈ R τ (µ, δ) then by Lemma 12 we have Since 1 + υ(n − 1) ≥ nυ, we get Proceeding as in Theorem 1.1, we get But the expression L 3 (κ, µ, ϑ, ν) is bounded above by 1 − ϑ if (15) holds. Thus the proof is complete.
That is, Now by expressing n = (n − 1) + 1 and following the lines of Theorem 1.1, we get Concluding Remark: By specializing µ = 0 or µ = 1 and xing ϑ = 0 in Theorems proved in present paper ,one can deduce for the classes studied in [14] and similar manner by taking ν = 0 we can easily deduce for the function classes studied in [13].The details involved may be port as an exercise for the attracted reader.