Subclass of analytic functions associated with Poisson distribution series
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Subclass of analytic functions associated with Poisson distribution series B. A. Frasin1 · Mohammed M. Gharaibeh1 Received: 19 April 2019 / Accepted: 25 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we find the necessary and sufficient conditions, inclusion relations for Poisson m n−1 −m n distribution series K(m, z) = z + ∞ z belonging to a subclass T S (λ, α, β) n=2 (n−1)! e of analyticfunctions with negative coefficients. Further, we consider the integral operator z G (m, z) = 0 F (m,t) dt belonging to the above class. t Keywords Analytic functions · Hadamard product · Poisson distribution series Mathematics Subject Classification 30C45
1 Introduction and definitions Let A denote the class of the normalized functions of the form f (z) = z +
∞
an z n ,
(1.1)
n=2
which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Further, let T be a subclass of A consisting of functions of the form, f (z) = z −
∞
|an | z n ,
z ∈ U.
(1.2)
n=2
A function f ∈ A is said to be in the class Rτ (A, B),τ ∈ C\{0}, −1 ≤ B < A ≤ 1, if it satisfies the inequality f (z) − 1 (A − B)τ − B[ f (z) − 1] < 1, z ∈ U.
B
B. A. Frasin [email protected] Mohammed M. Gharaibeh [email protected]
1
Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan
123
B.A. Frasin, M. M. Gharaibeh
This class was introduced by Dixit and Pal [4]. In [8], Magesh and Prameela defined the following class: Definition 1.1 A function f of the form (1.2) is in S (λ, α, β) if it satisfies the condition z f (z) + λ(2λ − 1)z 2 f (z) R −α 4λ(1 − λ)z + λ(2λ − 1)z f (z) + (2λ2 − 3λ + 1) f (z) z f (z) + λ(2λ − 1)z 2 f (z) >β − 1 4λ(1 − λ)z + λ(2λ − 1)z f (z) + (2λ2 − 3λ + 1) f (z) where β ≥ 0, − 1 ≤ α < 1, 0 ≤ λ ≤ 1 and z ∈ U. We also let T S (λ, α, β) = S (λ, α, β) ∩ T . We note that, by specializing the parameters λ, α and β , we obtain the following subclasses studied by various authors. (1) T S (0, α, 0) = T (α) and T S (1, α, 0) = K(α) (Silverman [19]); (2) T S (1/2, α, 0) = P (α) (Al-Amiri [1], Gupta and Jain [7] and Sarangi and Uralegaddi [14]); (3) T S (1/2, α, β) = T R(α, β) (Rosy [13] and Stephen and Subramanian [22]); (4) T S (0, α, β) = T S (α, β) and T S (1, α, β) = UCV (α, β)(Bharati et al. [2]); (5) T S (0, 0, β) = T S p (β) (Subramanian et al. [23]); (6) T S (1, 0, β) = UCV (β) (Subramanian et al. [24]). The Poisson distribution, derived in 1837 by a French mathematician Simé on Denis Poisson, is a discrete probability distribution that is used to express the probability of observing a number of events in a given interval of time or space if these events occur with a known average rate and independently of the time since the last event. A variable X is said to be Poisson distributed if it takes the values 0, 1, 2, 3, · · · with −m −m −m probabilities e−m , m e 1! , m 2 e 2! , m 3 e 3! , . . . respectively, where m is called the parameter. Thus m r e−m , r = 0
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