Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Ma-Minda φ-classes studied in this paper:
Abstract
The primary objective of this paper is to establish the sharp estimates
of the pre-Schwarzian norm for functions f in the class S∗(φ) and C(φ) when φ(z) =
1/(1 −z)s with 0 < s ≤1 and φ(z) = (1 + sz)2 with 0 < s ≤1/
√
2, where S∗(φ) and
C(φ) are the Ma-Minda type starlike and Ma-Minda type convex classes associated
with φ, respectively.
Results & Lemmas (4)
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 3.1.
Theorem 3.1. Let f ∈S∗ hyp. Then the pre-Schwarzian norm satisfies the following sharp inequality ∥Pf∥≤ sts(1 + ts) + (1 + ts)(1…
Theorem 3.1. Let f ∈S∗ hyp. Then the pre-Schwarzian norm satisfies the following sharp inequality ∥Pf∥≤ sts(1 + ts) + (1 + ts)(1 −ts)1−s −(1 −t2 s) ts for s ∈(0, 1) 4 for s = 1, where ts ∈(0, 1) is the unique root of the equation (1 −t)−s st2(1 −t)s + t2(1 −t)s + st2 + (1 −t)s + st −t2 −1
Theorem 3.2.
Theorem 3.2. Let f ∈S∗ L. Then the pre-Schwarzian norm satisfies the following inequality ∥Pf∥≤2s(1 −t2 s) 1 −sts + (1 −t2 s) (1 + sts)2…
Theorem 3.2. Let f ∈S∗ L. Then the pre-Schwarzian norm satisfies the following inequality ∥Pf∥≤2s(1 −t2 s) 1 −sts + (1 −t2 s) (1 + sts)2 −1 ts , where ts ∈(0, 1) is the unique positive root of the equation −3s4t4 + 2s3t3 + s4 + 7s2
Theorem 3.3.
Theorem 3.3. For any g ∈Chyp, the pre-Schwarzian norm satisfies the following sharp inequality ∥Pg∥≤ (1 + rs)(1 −rs)1−s −(1 −r2 s) rs…
Theorem 3.3. For any g ∈Chyp, the pre-Schwarzian norm satisfies the following sharp inequality ∥Pg∥≤ (1 + rs)(1 −rs)1−s −(1 −r2 s) rs for 0 < s < 1 2 for s = 1, where rs ∈(0, 1) is the unique root of the equation (1 −r)−s r2(1 −r)s + r2s −r2 + (1 −r)s + rs −1
Theorem 3.4.
Theorem 3.4. For any g ∈CL, the pre-Schwarzian norm satisfies the following sharp inequality ∥Pg∥≤ 2 √ 3s2 + 4 + 4 3s2 + 2 √ 3s2 + 4…
Theorem 3.4. For any g ∈CL, the pre-Schwarzian norm satisfies the following sharp inequality ∥Pg∥≤ 2 √ 3s2 + 4 + 4 3s2 + 2 √ 3s2 + 4 −4 27s .
Related Papers