Abstract
In this paper, we find the necessary and sufficient condi-
tions, inclusion relations for Poisson distribution series K(m, z) = z +
∞
P
n=2
mn−1
(n−1)!e−mzn to be in the subclasses S(k, λ) and C(k, λ) of analytic
functions with negative coefficients. Further, we obtain necessary and
sufficient conditions for the integral operator G(m, z) =
Rz
0
F(m,t)
t
dt to
be in the above classes.
1
Results & Lemmas (15)
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Lemma 1
Lemma 1 [3] A function f of the form (2) is in S(k, λ) if and only if it satisfies ∞ X n=2 [n((1 −λ) + k(1 + λ)) −(1 −λ)(1 −k)] |an| ≤2k (3)…
Lemma 1 [3] A function f of the form (2) is in S(k, λ) if and only if it satisfies ∞ X n=2 [n((1 −λ) + k(1 + λ)) −(1 −λ)(1 −k)] |an| ≤2k (3) where 0 < k ≤1 and 0 ≤λ < 1. The result is sharp.
Lemma 2
Lemma 2 [3] A function f of the form (2) is in C(k, λ) if and only if it satisfies ∞ X n=2 n[n((1 −λ) + k(1 + λ)) −(1 −λ)(1 −k)] |an| ≤2k…
Lemma 2 [3] A function f of the form (2) is in C(k, λ) if and only if it satisfies ∞ X n=2 n[n((1 −λ) + k(1 + λ)) −(1 −λ)(1 −k)] |an| ≤2k (4) where 0 < k ≤1 and 0 ≤λ < 1. The result is sharp.
Lemma 3
Lemma 3 [2] If f ∈Rτ(A, B) is of the form, then |an| ≤(A −B)|τ| n, n ∈N − 1. The result is sharp. 2 The necessary and sufficient conditions
Lemma 3 [2] If f ∈Rτ(A, B) is of the form, then |an| ≤(A −B)|τ| n , n ∈N −{1}. The result is sharp. 2 The necessary and sufficient conditions
Theorem 1
Theorem 1 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then F(m, z) is in S(k, λ) if and only if ((1 −λ) + k(1 + λ))mem ≤2k. (5)
Theorem 1 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then F(m, z) is in S(k, λ) if and only if ((1 −λ) + k(1 + λ))mem ≤2k. (5)
Theorem 2
Theorem 2 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then F(m, z) is in C(k, λ) if and only if ((1 −λ) + k(1 + λ))m2em + 2(1 + 2k + kλ −λ)mem ≤2k. (9)
Theorem 2 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then F(m, z) is in C(k, λ) if and only if ((1 −λ) + k(1 + λ))m2em + 2(1 + 2k + kλ −λ)mem ≤2k. (9)
Corollary 1
Corollary 1 If m > 0 and 0 < k ≤1, then F(m, z) is in S(k) if and only if (1 + k)mem ≤2k. (11)
Corollary 1 If m > 0 and 0 < k ≤1, then F(m, z) is in S(k) if and only if (1 + k)mem ≤2k. (11)
Corollary 2
Corollary 2 If m > 0 and 0 < k ≤1, then F(m, z) is in C(k) if and only if (1 + k)m2em + 2(1 + 2k)mem ≤2k. (12) 3 Inclusion properties
Corollary 2 If m > 0 and 0 < k ≤1, then F(m, z) is in C(k) if and only if (1 + k)m2em + 2(1 + 2k)mem ≤2k. (12) 3 Inclusion properties
Theorem 3
Theorem 3 Let m > 0, 0 < k ≤1 and 0 ≤λ < 1. If f ∈Rτ(A, B), then I(m, z)f is in S(k, λ) if and only if (A −B) |τ| ((1 −λ) + k(1 + λ))(1…
Theorem 3 Let m > 0, 0 < k ≤1 and 0 ≤λ < 1. If f ∈Rτ(A, B), then I(m, z)f is in S(k, λ) if and only if (A −B) |τ| ((1 −λ) + k(1 + λ))(1 −e−m) +(1 −λ)(k −1) m (1 −e−m(1 + m)) ≤2k. (13)
Theorem 4
Theorem 4 Let m > 0, 0 < k ≤1 and 0 ≤λ < 1. If f ∈Rτ(A, B), then F(m, z)f is in C(k, λ) if and only if (A −B) |τ| [((1 −λ) + k(1 + λ))m +…
Theorem 4 Let m > 0, 0 < k ≤1 and 0 ≤λ < 1. If f ∈Rτ(A, B), then F(m, z)f is in C(k, λ) if and only if (A −B) |τ| [((1 −λ) + k(1 + λ))m + 2k(1 −e−m)] ≤2k. (15)
Corollary 3
Corollary 3 Let m > 0 and 0 < k ≤1. If f ∈Rτ(A, B), then I(m, z)f is in S(k) if and only if (A −B) |τ| (1 + k)(1 −e−m) + (k −1) m (1…
Corollary 3 Let m > 0 and 0 < k ≤1. If f ∈Rτ(A, B), then I(m, z)f is in S(k) if and only if (A −B) |τ| (1 + k)(1 −e−m) + (k −1) m (1 −e−m(1 + m)) ≤2k. (16)
Corollary 4
Corollary 4 Let m > 0 and 0 < k ≤1. If f ∈Rτ(A, B), then I(m, z)f is in C(k) if and only if (A −B) |τ| [(1 + k)m + 2k(1 −e−m)] ≤2k. (17) 4…
Corollary 4 Let m > 0 and 0 < k ≤1. If f ∈Rτ(A, B), then I(m, z)f is in C(k) if and only if (A −B) |τ| [(1 + k)m + 2k(1 −e−m)] ≤2k. (17) 4 An integral operator In this section, we obtain the necessary and sufficient conditions for the integral operator G(m, z) defined by G(m, z) = Z z 0 F(m, t) t dt (18)
Theorem 5
Theorem 5 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then the integral operator G(m, z) defined by (18) is in C(k, λ) if and only if (5) is satisfied.
Theorem 5 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then the integral operator G(m, z) defined by (18) is in C(k, λ) if and only if (5) is satisfied.
Theorem 6
Theorem 6 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then the integral operator G(m, z) defined by (18) is in S(k, λ) if and only if ((1 −λ) + k(1 +…
Theorem 6 If m > 0, 0 < k ≤1 and 0 ≤λ < 1, then the integral operator G(m, z) defined by (18) is in S(k, λ) if and only if ((1 −λ) + k(1 + λ))(1 −e−m) + (1 −λ)(k −1) m (1 −e−m −me−m) ≤2k. By taking λ = 0 in Theorems 5 and 6, we obtain the following corollaries.
Corollary 5
Corollary 5 If m > 0 and 0 < k ≤1, then the integral operator defined by (18) is in C(k) if and only if (11) is satisfied.
Corollary 5 If m > 0 and 0 < k ≤1, then the integral operator defined by (18) is in C(k) if and only if (11) is satisfied.
Corollary 6
Corollary 6 If m > 0 and 0 < k ≤1,then the integral operator defined by (18) is in S(k) if and only if (1 + k)(1 −e−m) + (k −1) m (1 −e−m…
Corollary 6 If m > 0 and 0 < k ≤1,then the integral operator defined by (18) is in S(k) if and only if (1 + k)(1 −e−m) + (k −1) m (1 −e−m −me−m) ≤2k. Acknowledgements The author would like to thank the referee for his helpful comments and sug- gestions. References [1] N. E. Cho, S. Y. Woo, S. Owa, Uniform convexity properties for hyper- geometric functions, Fract. Cal. Appl. Anal., 5 (3) (2002), 303–313. [2] K. K. Dixit, S. K. Pal, On a class of univalent functions related to complex order, India
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