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Abstract
The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes SP_{p}(α,\b{eta}) and UCV_{p}(α,\b{eta}) of uniformly spirallike functions. Further, we consider properties of a special function related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.
Results & Lemmas (15)
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Lemma 1.2.
Lemma 1.2. [23] A function f of the form (2) is in T SPp(α, β) if and only if it satisfies ∞ X n=2 (2n −cos α −β) |an| ≤cos α −β (|α| < π/2;…
Lemma 1.2. [23] A function f of the form (2) is in T SPp(α, β) if and only if it satisfies ∞ X n=2 (2n −cos α −β) |an| ≤cos α −β (|α| < π/2 ; 0 ≤β < 1). (6) In particular, when β = 0, we obtain a necessary and sufficient condition for a function f of the form (2) to be in the class T SPp(α) is that ∞ X n=2 (2n −cos α) |an| ≤cos α (|α| < π/2). (7)
Lemma 1.3.
Lemma 1.3. [23] A function f of the form (2) is in UCT p(α, β) if and only if it satisfies ∞ X n=2 n(2n −cos α −β) |an| ≤cos α −β (|α| <…
Lemma 1.3. [23] A function f of the form (2) is in UCT p(α, β) if and only if it satisfies ∞ X n=2 n(2n −cos α −β) |an| ≤cos α −β (|α| < π/2 ; 0 ≤β < 1). (8) In particular, when β = 0, we obtain a necessary and sufficient condition for a function f of the form (2) to be in the class UCT p(α) is that ∞ X n=2 n(2n −cos α) |an| ≤cos α (|α| < π/2). (9)
Lemma 1.4.
Lemma 1.4. [3] If f ∈Rτ(A, B) is of the form (1), then |an| ≤(A −B)|τ| n, n ∈N 1. The result is sharp.
Lemma 1.4. [3] If f ∈Rτ(A, B) is of the form (1) , then |an| ≤(A −B)|τ| n , n ∈N\{1}. The result is sharp.
Theorem 2.1.
Theorem 2.1. We have Φm q ∈T SPp(α, β) if and only if 2q m 1 −q + (2 −cos α −β) [1 −(1 −q)m] ≤cos α −β. (14)
Theorem 2.1. We have Φm q ∈T SPp(α, β) if and only if 2q m 1 −q + (2 −cos α −β) [1 −(1 −q)m] ≤cos α −β. (14)
Theorem 2.2.
Theorem 2.2. We have Φm q ∈UCT p(α, β) if and only if 2q2 m(m + 1) (1 −q)2 + (6 −cos α −β) q m 1 −q + (2 −cos α −β) [1 −(1 −q)m] ≤cos α −β.…
Theorem 2.2. We have Φm q ∈UCT p(α, β) if and only if 2q2 m(m + 1) (1 −q)2 + (6 −cos α −β) q m 1 −q + (2 −cos α −β) [1 −(1 −q)m] ≤cos α −β. (17)
Theorem 3.1.
Theorem 3.1. Let m > 1. If f ∈Rτ(A, B), then Im q f(z) ∈T SPp(α, β) if (A−B)|τ| 2 h 1 −(1 −q)m i −cos α + β q(m −1) [(1 −q) −(1 −q)m −q(m…
Theorem 3.1. Let m > 1. If f ∈Rτ(A, B), then Im q f(z) ∈T SPp(α, β) if (A−B)|τ| 2 h 1 −(1 −q)m i −cos α + β q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1 −q)m] ≤cos α−β. (19)
Theorem 3.2.
Theorem 3.2. If f ∈Rτ(A, B), then Im q f is in UCT p(α, β) if (A −B)|τ| 2q m 1 −q + (2 −cos α −β) [1 −(1 −q)m] ≤cos α −β. (21) 4.…
Theorem 3.2. If f ∈Rτ(A, B), then Im q f is in UCT p(α, β) if (A −B)|τ| 2q m 1 −q + (2 −cos α −β) [1 −(1 −q)m] ≤cos α −β. (21) 4. Properties of a special function
Theorem 4.1.
Theorem 4.1. If the function Gm q is given by Gm q (z):= Z z 0 Φm q (t) t dt, z ∈U, (22) then Gm q ∈UCT p(α, β) if and only if the…
Theorem 4.1. If the function Gm q is given by Gm q (z) := Z z 0 Φm q (t) t dt, z ∈U, (22) then Gm q ∈UCT p(α, β) if and only if the inequality (14) holds.
Theorem 4.2.
Theorem 4.2. If m > 1, then the function Gm q ∈T SPp(α, β) if and only if 2 h 1 −(1 −q)m i −cos α + β q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1…
Theorem 4.2. If m > 1, then the function Gm q ∈T SPp(α, β) if and only if 2 h 1 −(1 −q)m i −cos α + β q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1 −q)m] ≤cos α −β. The proof of Theorem 4.2 is lines similar to the proof of Theorem 4.1, so we omitted the proof of Theorem 4.2. 5. Corollaries and consequences By specializing the parameter β = 0 in Theorems 2.1-4.2, we obtain the following corol- laries.
Corollary 5.1.
Corollary 5.1. We have Φm q ∈T SPp(α) if and only if 2q m 1 −q + (2 −cos α) [1 −(1 −q)m] ≤cos α. (24)
Corollary 5.1. We have Φm q ∈T SPp(α) if and only if 2q m 1 −q + (2 −cos α) [1 −(1 −q)m] ≤cos α. (24)
Corollary 5.2.
Corollary 5.2. We have Φm q ∈UCT p(α) if and only if 2q2 m(m + 1) (1 −q)2 + (6 −cos α) q m 1 −q + (2 −cos α) [1 −(1 −q)m] ≤cos α. (25)
Corollary 5.2. We have Φm q ∈UCT p(α) if and only if 2q2 m(m + 1) (1 −q)2 + (6 −cos α) q m 1 −q + (2 −cos α) [1 −(1 −q)m] ≤cos α. (25)
Corollary 5.3.
Corollary 5.3. Let m > 1. If f ∈Rτ(A, B), then Im q f ∈T SPp(α) if (A−B)|τ| 2 h 1 −(1 −q)m i − cos α q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1…
Corollary 5.3. Let m > 1 . If f ∈Rτ(A, B), then Im q f ∈T SPp(α) if (A−B)|τ| 2 h 1 −(1 −q)m i − cos α q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1 −q)m] ≤cos α. (26)
Corollary 5.4.
Corollary 5.4. If f ∈Rτ(A, B), then Im q f ∈UCT p(α) if (A −B)|τ| 2q m 1 −q + (2 −cos α) [1 −(1 −q)m] ≤cos α. (27)
Corollary 5.4. If f ∈Rτ(A, B), then Im q f ∈UCT p(α) if (A −B)|τ| 2q m 1 −q + (2 −cos α) [1 −(1 −q)m] ≤cos α. (27)
Corollary 5.5.
Corollary 5.5. The function Gm q ∈UCT p(α) if and only if the inequality (24) holds.
Corollary 5.5. The function Gm q ∈UCT p(α) if and only if the inequality (24) holds.
Corollary 5.6.
Corollary 5.6. If m > 1, then the function Gm q ∈T SPp(α) if and only if 2 h 1 −(1 −q)m i − cos α q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1 −q)m]…
Corollary 5.6. If m > 1, then the function Gm q ∈T SPp(α) if and only if 2 h 1 −(1 −q)m i − cos α q(m −1) [(1 −q) −(1 −q)m −q(m −1)(1 −q)m] ≤cos α. References [1] R. M. El-Ashwah, W. Y. Kota, Some condition on a Poisson distribution series to be in subclasses of univalent functions, Acta Universitatis Apulensis, No. 51/2017, pp. 89-103. [2] N. E. Cho, S. Y. Woo and S. Owa, Uniform convexity properties for hypergeometric functions, Fract. Cal. Appl. Anal., 5(3) (2002), 303–313. [3] K.K. Dixit and
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