Applications of Mittag-Leffer Type Poisson Distribution to a Subclass of Analytic Functions Involving Conic-Type Regions

In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Le ﬄ er type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and su ﬃ cient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly de ﬁ ned class.


Introduction, Definitions, and Motivation
Let A represent the collections of holomorphic (analytic) functions f defined in the open unit disc: such that the Taylor series expansion of f is given by a n z n z ∈ D ð Þ: By convention, S stands for a subclass of class A comprising of univalent functions of the form (2) in the open unit disc D. Let P represent the class of all functions p that are holomorphic in D with the condition and has the series representation p z ð Þ = 1 + 〠 ∞ n=1 c n z n z ∈ D ð Þ: Next, we recall the definition of subordination, for two functions h 1 , h 2 ∈ A, we say h 1 is subordinated to h 2 and is symbolically written as if there exists an analytic function wðzÞ with the properties such that Further if h 2 ∈ S, then the above condition becomes Now, recall the definition of convolution, let f ∈ A given by (2) and hðzÞ given by then their convolution denoted by ð f * hÞðzÞ is given by The most important and well-known family of analytic functions is the class of starlike functions denoted by S * and is defined as Next, for −1 ≤ B < A ≤ 1, Janowski [1] generalized the class S * as follows.
Janowski also proved that for a function p ∈ P , a function hðzÞ belongs to P ½A, B if the following relation holds Also, function f of form (2) belongs to the class S * ½A, B if Kanas et al. (see [2,3]; see also [4,5]) were the first to define the conic domain Ω k ðk ≧ 0Þ as follows: Moreover, for fixed k, Ω k represents the conic region bounded successively by the imaginary axis ðk = 0Þ. For k = 1, it is a parabola, and for 0 < k < 1, it is the right-hand branch of the hyperbola, and for k > 1, it represents an ellipse.
For these conic regions, the following functions play the role of extremal functions: where and κ ∈ ð0, 1Þ is chosen such that λ = cosh ðπK ′ ðκÞ/ð4KðκÞÞÞ.
Here KðκÞ is Legendre's complete elliptic integral of first kind and K ′ ðκÞ = Kð ffiffiffiffiffiffiffiffiffiffiffi ffi 1 − κ 2 p Þ, that is, K ′ ðκÞ is the complementary integral of KðκÞ. Assume that Then, in [6], it has been shown that, for (16), one can have Noor and Malik [7] combined the concepts of the Janowski functions and the conic regions and gave the following definition.
Geometrically, hðzÞ ∈ k − P ½A, B takes all values in domain Δ k ½A, B, which is defined as follows the domain Δ k ½A, B represents conic-type regions, which was introduced and studied by Noor and Malik [7] and is further generalized by the many authors, see for example [8] and the references cited therein.
The generalized exponential series: is one special-type function with single parameter α, was introduced by Mittag-Leffer (see [9]), and is therefore known as the Mittag-Leffler function. Another function E α,β ðzÞ with two parameters α and β having similar properties to those of Mittag-Leffler function is given by and was introduced by Wiman [10,11] Agrawal [12], and by the many other (see for example [13][14][15][16]). It can be seen that the series E α;β ðzÞ converges for all finite values of z if During the last years, the interest in Mittag-Leffler type functions has considerably increased due to their vast potential of applications in applied problems such as fluid flow, electric networks, probability, and statistical distribution theory. For a detailed account of properties, generalizations and applications of functions (27) and (28), one may refer to [17][18][19] and [20].
Geometric properties including starlikeness, convexity, and close-to-convexity for the Mittag-Leffler function E α,β ð zÞ were recently investigated by Bansal and Prajapat in [21]. Differential subordination results associated with generalized Mittag-Leffler function were also obtained in [22].
It is easy to see that (30) is a mass probability function because The power series Yðψ, zÞ given by which coefficients are probabilities of Poisson distribution is introduced by Porwal [23]. We can see that by ratio test the radius of convergence of Yðψ, zÞ is infinity. Porwal [23] also defined and introduced the following series: The works of Porwal [23] motivate researchers to introduced a new probability distribution if it assumes the positive 3 Journal of Function Spaces values and its mass function is given by (30) (see for example [24][25][26]).
It was Porwal and Dixit [24] who studied and connected the Poisson distribution and the well-known Mittag-Leffer function systematically. They called it the Mittag-Leffer type Poisson distribution and prevailed moments. The Mittag-Leffer type Poisson distribution is given by (see [24]) where Yðψ, α, βÞðzÞ is a normalized function of class S, since The probability mass function for the Mittag-Leffer type Poisson distribution series is given by where E α;β ðψÞ is given by (28). It is worthy to note that the Mittag-Leffer type Poisson distribution is a generalization of Poisson distribution. Furtheremore, Bajpai [27] also studied and obtain some necessary and sufficient conditions for this distribution series. Very recently, using the Mittag-Leffer type Poisson distribution series, Alessa et al. [28] defined the convolution operator as where Using this convolution operator, they defined and studied a new subclass of analytic function systematically. They obtained certain coefficient estimates, neighborhood results, partial sums, and convexity and compactness properties for their defined functions class.
In recent years, binomial distribution series, Pascal distribution series, Poisson distribution series, and Mittag-Leffer type Poisson distribution series play important role in the geometric function theory of complex analysis. The sufficient ways were innovated for certain subclasses of starlike and convex functions involving these special functions (see for example [26,[29][30][31][32]). Motivated by the abovementioned works and from the work of Alessa et al. [28], in this article, by mean of certain convolution operator for Mittag-Leffer type Poisson distribution, we shall define a new subclass of starlike functions involving both the conic-type regions and the Janowski functions. We then obtain some interesting properties for this newly defined function class including for example necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums. We now define a subclass of Janowski-type starlike functions involving the conic domains by mean of certain convolution operator for Mittag-Leffer type Poisson distribution as follows. where For the proofs of our key findings, we need the following lemma.
Lemma 6 [33]. Let p ∈ P have the series expansion of form (4), then

Main Results
Theorem 7. Let f ∈ k − ΩS * ðα, β, A, BÞ and is of the form (2), then The result is sharp for the function given in (51).
Proof. Suppose that inequality (42) holds true, then it is enough to show that For this, consider Journal of Function Spaces As we have set therefore, after some straightforward simplifications, we have By using (42), the above inequality is bounded above by 1, and hence, the proof is completed. ☐

Example 8.
For the function such that we have Hence, f ∈ k − ΩS * ðα, β, A, BÞ and the result is sharp.

Corollary 9.
Let the function f of the form (2) be in the class k − ΩS * ðα, β, A, BÞ: Then, The result is sharp for the function f t ðzÞ given by Proof. Let f k ðzÞ ∈ k − ΩS * ðα, β, A, BÞ such that It is enough to show that As Now, by Theorem 7, we have Hence, which completes the proof. ☐

Journal of Function Spaces
After comparing the (69) and (70), we get Now, by making use of (71) and (72), in conjunction with 6 Journal of Function Spaces Lemma, we have which is the required result. ☐

Partial Sums
In this section, we will examine the ratio of a function of form (2) to its sequence of partial sums when the coefficients of f are sufficiently small to satisfy condition (42). We will determine sharp lower bounds for Theorem 14. If f of form (2) satisfies condition (42), then where The result is sharp for the function given in (51).
Proof. It is easy to verify that ρ n+1 > ρ n > 1 for n > 2: Thus, in order to prove the inequality (76), we set n=2 a n z n−1 + ρ j+1 ∑ ∞ n=j+1 a n z n−1 1 + ∑ j n=2 a n z n−1 We now set Then, we find after some suitable simplification that Thus, clearly, we find that w z ð Þ = ρ j+1 ∑ ∞ n=j+1 a n z n−1 2 + 2∑ j n=2 a n z n−1 + ρ j+1 ∑ ∞ n=j+1 a n z n−1 : ð83Þ By applying the trigonometric inequalities with jzj < 1, we arrived at the following inequality: w z ð Þ j j≤ ρ j+1 ∑ ∞ n=j+1 a n j j 2 − 2∑ j n=2 a n j j − ρ j+1 ∑ ∞ n=j+1 a n j j : We can now see that if and only if 2ρ j+1 〠 ∞ n=j+1 a n j j ≤ 2 − 2 〠 j n=2 a n j j, ð86Þ which implies that 〠 j n=2 a n j j + ρ j+1 〠 ∞ n=j+1 a n j j ≤ 1: Finally, to prove the inequality in (76), it suffices to show that the left-hand side of (87) is bounded above by the following sum: ρ n a n j j, ð88Þ