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Abstract
The aim of the present article is to introduce and study new
subclass of Janowski type functions defined using notions of Janowski
functions and (j, k)-symmetrical functions. Certain interesting coefficient
inequalities, sufficiency criteria, distortion theorem, neighborhood prop-
erty are investigated for this class.
1
Results & Lemmas (7)
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Theorem 1
Theorem 1 [10] For every mapping f: D 7→C, where D is a k-fold symmetric set, there exists exactly the sequence of (j, k)- symmetrical…
Theorem 1 [10] For every mapping f : D 7→C, where D is a k-fold symmetric set, there exists exactly the sequence of (j, k)- symmetrical functions fj,k, f(z) = k−1 X j=0 fj,k(z) where fj,k(z) = 1 k k−1 X v=0 ε−vjf(εvz), (4)
Lemma 1
Lemma 1 [3] Let p(z) = 1 + P∞ n=1 pnzn ∈P[A, B], then for n ≥1, |pn| ≤(A −B). 2 Main results
Lemma 1 [3] Let p(z) = 1 + P∞ n=1 pnzn ∈P[A, B], then for n ≥1, |pn| ≤(A −B). 2 Main results
Theorem 2
Theorem 2 If f ∈S(j,k)(A, B), then for n ≥2, −1 ≤B < A ≤1. |an| ≤ n−1 Y m=1 δm,j[(A −B) −1] + m m + 1 −δm+1,j, where δn,j by (5).
Theorem 2 If f ∈S(j,k)(A, B), then for n ≥2, −1 ≤B < A ≤1. |an| ≤ n−1 Y m=1 δm,j[(A −B) −1] + m m + 1 −δm+1,j , where δn,j by (5).
Theorem 3
Theorem 3 Let f(z) = z + P∞ n=2 anzn, be analytic in U,for (−1 ≤B < A ≤ 1), we have ∞ X n=2 (n −δn,j) + |Aδn,j −Bn| |an| ≤(A −B). Then,…
Theorem 3 Let f(z) = z + P∞ n=2 anzn, be analytic in U,for (−1 ≤B < A ≤ 1), we have ∞ X n=2 {(n −δn,j) + |Aδn,j −Bn|} |an| ≤(A −B). Then, f(z) ∈S(j,k)(A, B).
Theorem 4
Theorem 4 Let f(z) ∈S(j,k)(A, B), for (−1 < B < A ≤1), then |z| − i X n=2 |an||z|n −τi|z|i+1 ≤|f(z)| ≤|z| + i X n=2 |an||z|n + τi|z|i+1,…
Theorem 4 Let f(z) ∈S(j,k)(A, B), for (−1 < B < A ≤1), then |z| − i X n=2 |an||z|n −τi|z|i+1 ≤|f(z)| ≤|z| + i X n=2 |an||z|n + τi|z|i+1, where τi = (A −B) −Pi n=2 {(n −δn,j) + |Aδn,j −Bn|} |an| {(i + 1)(1 −|B|) −[1 −|A|]δi+1,j} .
Theorem 5
Theorem 5 For (−1 < B < A ≤1), S(j,k)(A, B) ⊆Nρ(e), where ρ = (A −B) 2(1 −|B|) −(1 −|A|)δ2,j + 1 2(1 −|B|) −(1 −|A|)δ2,j .
Theorem 5 For (−1 < B < A ≤1), S(j,k)(A, B) ⊆Nρ(e), where ρ = (A −B){2(1 −|B|) −(1 −|A|)δ2,j + 1} 2(1 −|B|) −(1 −|A|)δ2,j .
Theorem 6
Theorem 6 Let g ∈S(j,k)(A, B), and suppose that η = 1 − ρ 2(1 −|B|) −(1 −|A|)δ2,j 2 2(1 −|B|) −(1 −|A|)δ2,j −(A −B), (8) then Nρ(g)…
Theorem 6 Let g ∈S(j,k)(A, B), and suppose that η = 1 − ρ{2(1 −|B|) −(1 −|A|)δ2,j} 2{2(1 −|B|) −(1 −|A|)δ2,j −(A −B)}, (8) then Nρ(g) ⊆S(j,k)(A, B, η).
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