A New Subclass of Analytic Functions Related to Mittag-Leffler Type Poisson Distribution Series

Nazek Alessa , B. Venkateswarlu, P. Thirupathi Reddy, K. Loganathan , and K. Tamilvanan 5 Department of Mathematical Sciences, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia Department of Mathematics, GSS, GITAM University, Bengaluru Rural, Doddaballapur 562 163, Karnataka, India Department of Mathematics, Kakatiya University, Warangal, 506 009 Telangana, India Research and Development Wing, Live4Research, Tiruppur, 638 106 Tamilnadu, India Department of Mathematics, Government Arts College for Men, Krishnagiri, 635 001 Tamilnadu, India


Introduction
Let Y = fω : |ω|<1g be an open unit disc in C. Consider the analytic class function A that indicates j specified on the unit disk along with normalization and has the form indicated by S, the subclass of A lying of functions that are univalent in Y: A function j ∈ A is stated in k − USTðιÞ, and k − UCVðιÞ, "the class of k-uniformly starlike functions and convex functions of order ι, 0 ≤ ι < 1," if and only if The classes UCV and UST were introduced by Goodman [1] and studied by Ronning [2]. Due to Sakaguchi [3], the class ST s of starlike functions w.r.t. symmetric points are defined as follows.
In [14], Porwal, Poisson distribution series, gives a gracious application on analytic functions; it exposed a new way of research in GFT. Subsequently, the authors turned on the distribution series of confluent hypergeometric, hypergeometric, binomial, and Pascal and prevail necessary and sufficient stipulation for certain classes of univalent functions.
Lately, Porwal and Dixit [15] innovate Mittag-Leffler type Poisson distribution and prevailed moments, mgf, which is an abstraction of Poisson distribution using the definition of this distribution. Bajpai [16] innovated Mittag-Leffler type Poisson distribution series and discussed about necessary and sufficient conditions.
The probability mass function for this is where The series (7) converges for all finite values of ω if RðτÞ > 0, RðυÞ > 0: This suggest that the series E τ,υ ðℏÞ is convergent for τ, υ, ℏ > 0: For further details of the study, see [17]. It is easy to see that the series (7) are reduced to exponential series for τ = υ = 1: A variable x is said to have Poisson distribution if it takes the values 0, 1, 2, 3, ⋯ with probabilities e −ℏ , ℏe −ℏ /1!, ℏ 2 e −ℏ /2 !, ℏ 3 e −ℏ /3!, ⋯ , respectively, where ℏ is called the parameter. Thus, This motivates researchers (see [15,17,18], etc.) to introduce a new probability distribution if it assumes nonnegative values and its probability mass function is given by (6). It is easy to see that Pðℏ, τ, υ ; nÞðωÞ given by (6) is the probability mass function because It is worthy to note that for α = β = 1, it reduces to the Poisson distribution.
Also note that In [18], Chakrabortya and Ong introduced and discussed about the Mittag-Leffler function distribution-a new generalization of hyper-Poisson distribution. The Mittag-Leffler type Poisson distribution series was innovated by Porwal and Dixit [15] and given as Equation (11) is a normalization function in S, since Kðℏ, τ, υÞð0Þ = 0 and K ′ ðℏ, τ, υÞð0Þ = 1: After that, in [19], Porwal et al. discussed about the geometric properties of (11).

Neighborhood Properties
The notion of β-neighbourhood was innovated and studied by Goodman [21] and Ruscheweyh [22].

Partial Sums
Theorem 8. If the function j is of the form (2) fulfill (18) then The estimate (38) is sharp, for every m, with Proof. Now, we define ℘; we can define Then, from (42), we attain Journal of Function Spaces It is enough to prove that the LHS of (44) is bounded above by ∑ ∞ n=2 χ n o n , which implies To show that the mapping disposed by (41) gives the exact result, we notice that for ω = re iπ/n , Taking limit ω tends to 1 − , we have Hence, the proof is completed.

Theorem 9.
If j of the form (2) which fulfill (18) then The result is sharp with (41).
Proof. Define where This last inequality is It is enough to prove that the LHS of (51) is bounded above by ∑ ∞ n=2 χ n o n , which implies This completes the proof.