Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
Using the q-derivative operator in conjunction with the principle of
subordination between analytic functions, we introduce two subclasses of analytic
functions in the open unit disk U. We investigate convolution properties and
coefficient estimates for these subclasses.
Mathematics Subject Classification (2010): 30C45, 30C50.
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 2.1.
Theorem 2.1. If f(z) ∈A, then f(z) ∈Kq[b; A, B] if and only if 1 z f(z) ∗z + [1 −(1 + M (θ)) (q + 1)]qz2 (1 −z)(1 −qz)(1 −q2z) ̸= 0,…
Theorem 2.1. If f(z) ∈A, then f(z) ∈Kq[b; A, B] if and only if 1 z f(z) ∗z + [1 −(1 + M (θ)) (q + 1)]qz2 (1 −z)(1 −qz)(1 −q2z) ̸= 0, (2.1) where the symbol ∗stands for the convolution between two power series and M (θ) = M b;A,B(θ) = 1 b e−iθ + B A −B
Theorem 2.2.
Theorem 2.2. If f(z) ∈A, then f(z) ∈Sq[b; A, B] if and only if 1 z f(z) ∗z −(1 + M (θ)) qz2 (1 −z)(1 −qz) ̸= 0, (2.7) where M (θ) is…
Theorem 2.2. If f(z) ∈A, then f(z) ∈Sq[b; A, B] if and only if 1 z f(z) ∗z −(1 + M (θ)) qz2 (1 −z)(1 −qz) ̸= 0, (2.7) where M (θ) is given by (2.2).
Theorem 2.3.
Theorem 2.3. If f(z) ∈A, then f(z) ∈Kq[b; A, B] if and only if 1 − ∞ X k=2 [k]q ([k]q −1)(e−iθ + B) −(A −B)b (A −B)b akzk−1 ̸= 0 for all θ.…
Theorem 2.3. If f(z) ∈A, then f(z) ∈Kq[b; A, B] if and only if 1 − ∞ X k=2 [k]q ([k]q −1)(e−iθ + B) −(A −B)b (A −B)b akzk−1 ̸= 0 for all θ. (2.9)
Theorem 2.4.
Theorem 2.4. If f(z) ∈A, then f(z) ∈Sq[b; A, B] if and only if 1 − ∞ X k=2 ([k]q −1)(e−iθ + B) −(A −B)b (A −B)b akzk−1 ̸= 0 for all θ.…
Theorem 2.4. If f(z) ∈A, then f(z) ∈Sq[b; A, B] if and only if 1 − ∞ X k=2 ([k]q −1)(e−iθ + B) −(A −B)b (A −B)b akzk−1 ̸= 0 for all θ. (2.10)
Theorem 2.5.
Theorem 2.5. If f(z) ∈A satisfies the inequality ∞ X k=2 [k]q h ([k]q −1)(1 + |B|) + (A −B) |b| i |ak| ≤(A −B) |b|. (2.11) then f(z) ∈Kq[b;…
Theorem 2.5. If f(z) ∈A satisfies the inequality ∞ X k=2 [k]q h ([k]q −1)(1 + |B|) + (A −B) |b| i |ak| ≤(A −B) |b| . (2.11) then f(z) ∈Kq[b; A, B].
Theorem 2.6.
Theorem 2.6. If f(z) ∈A satisfies ∞ X k=2 h ([k]q −1)(1 + |B|) + (A −B) |b| i |ak| ≤(A −B) |b|. then f(z) ∈Sq[b; A, B].
Theorem 2.6. If f(z) ∈A satisfies ∞ X k=2 h ([k]q −1)(1 + |B|) + (A −B) |b| i |ak| ≤(A −B) |b| . then f(z) ∈Sq[b; A, B].
Function classes studied:
Related Papers