Analytic Univalent Functions Defined by Generalized Discrete Probability Distribution

The close-to-convex analogue of a starlike functions by means of generalized discrete probability distribution and Poisson distribution was considered. Some coefficient inequalities and their connection to classical Fekete-Szego theorem are obtained. Our results provide strong connection between Geometric Function Theory and Statistics.


Introduction
Let A denote the class of analytic functions f in the unit disk E = {z : |z| < 1}. We denote by S the subclass of A of analytic univalent functions f with normalization condition f (0) = f (0) − 1 = 0 in E.
We recall here that for any f ∈ A the power series representation is given by (1.1) Let S * (0) ⊂ S be the family of starlike functions of order zero defined by the geometric condition A function f ∈ A is said to be convex if it satisfies the condition We denote by C(0) the class of convex functions of order zero.
For easy handling, we also use the notations S * = S * (0) and C = C(0).
A function f ∈ A is said to be close-to-convex if there exists ρ ∈ (− π 2 , π 2 ) and a function g ∈ S * such that Whenever ρ = 0 the class of close to convex function is denoted by K and S * ⊂ K ⊂ A.
Here we recall the class of Caratheodory function P which are analytic in E and satisfy p(0) = 1 and Rep(z) > 0 for z ∈ E with series representation It is clear that S * µ ⊂ S * . The above class was introduced and investigated by Singh [9] (see also [10] where a k ≥ 0, for all k ∈ N . The generalized discrete probability distribution whose probability mass functions given as p(k) = a k S , k = 0, 1, 2, ..., p(k) is the probability mass function because p(k) ≥ 0 and k p k = 1.
Furthermore, we let φ(x) = ∞ k=0 a k z k , then from S = ∞ k=0 a k series φ is convergent for |x| < 1 and x = 1.
Of particular interest to the present investigation is the power series whose coefficient are probabilities of the generalized distribution of the form which was recently studied in [ [1], [2], [7]]. The purpose of the present investigation is to derive a robust geometric properties that establish a more stronger connections between Geometric Function Theory and Statistics; and to further point out the consequential effects when the quantity a ns is replaced by Poisson probability distribution. Our result is robust because obtaining sharp estimates for the coefficients represents a much more difficult problem most especially when to extend from starlikeness to close-to-convexity. To achieve this our method shall follow the pattern of that of Allu et al. [10].
Next we define the close to convex analogue of the class S * µ associated with generalized distribution as follows.
where K µ represents the natural close-to-convex analogue of S * µ .

Preliminaries and Lemmas
Let Ω denote the class of analytic function of the form which satisfy w(0) = 0 and |w| < 1 for z ∈ E. Class of such functions is called Scharwz function if w ∈ A. For p ∈ P given by (1.2), let p(z) = 1+w(z) 1−w(z) for some w ∈ Ω, where A is given by (1.1); on equating coefficients we obtain The following lemmas shall be tools for the present investigation.

Lemma 3. [10]
Let p ∈ P be given by (1.2). Then for µ ∈ C, The inequality is sharp for each complex µ.

4)
Then for any µ ∈ C, both inequalities are sharp.
Recently Allu et al. [10] provided means by which the result in S * µ can be extended to K µ . Following their method we give the following results due to generalized discrete probability distribution. given by (1.3). Then

Coefficient bounds for class
Proof. We let for some g ∈ S * and some w ∈ Ω, on equating coefficients in (3.5) and using both (2.1) and (3.5), we obtain where |b k | ≤ k and |w k | ≤ 1 for k ≥ 1. Therefore (3.6) yields It is obvious that the maximum over H occurs on the boundary δH which we now consider. Suppose Solving the above equation using Wolfram Alpha .
Hence | a 3 S | ≤ 1 4 × 7.3731 = 1.8443. Applying the same method, result for a 4 S can be obtained. There is a backward shift between our result and that of [10]. Furthermore, our results established more connections between statistics and univalent functions and geometric function theory and it converges faster.
The results in Theorem 1 and Theorem 2, further strengthen the connections between statistics and geometric function theory.
For special values of a k−1 various well known discrete probability distribution such as Yule-Simeon distribution, logarithmic distribution, Poisson distribution, Binomial distribution, Beta-Binomial distribution, Zeta distribution, geometric distribution and Bernoulli distribution can be obtained.
In the next section, we shall consider the Poisson distribution.

Coefficients of Poisson Distribution
Recently, Porwal in [8] introduced a power series whose coefficients are probability of Poisson distribution as follows By ratio test the radius of convergence of the series (4.1) is infinity, m is called the parameter and m > 0. Using the above the author obtained some interesting results on classes of analytic functions (see also [3] and the relevance references therein. The Poisson distribution is a tool used in probability theory statistics to predict the amount of variation from a known average of occurrence, within a given time. That is to determine how much variation there will likely be from that average number of occurrence and also determine the probable maximum and minimum number of times the event will occur within the specified time frame. Company can also use the Poisson distribution to examine how they may be able to take steps to improve their operational efficiency.