Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ We examine the properties of the class $\mathcal{C}(\varphi)$ defined as $\mathcal{C}(\varphi) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=1+z+ m/n\, \, z^2, \text{ with } 2m \le n,\text{ for } m, n \in \mathbb{N} \right\},$ and compute the sharp second and third Hankel determinants for the functions in $\mathcal{C}(\varphi
Results & Lemmas (3)
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.
Lemma 3.5 · coeff
Lemma 3.5. [1] Let and define (i) If AC > 0, then (ii) If AC < 0, then where where We shall also use the following coefficient…
Lemma 3.5. [1] Let $A, B, C \in \mathbb{R}$ and define
$$Y(A, B, C) := \max_{z \in \overline{\mathbb{D}}} (|A + Bz + Cz^2| + 1 - |z|^2).$$
(i) If AC > 0, then
$$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & \text{if } |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & \text{if } |B| < 2(1 - |C|). \end{cases}$$
(ii) If AC < 0, then
$$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & \text{if } -4AC(C^2-1) \leq B^2 \text{ and } |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & \text{if } B^2 < \min\{4(1+|C|)^2, -4AC(C^2-1)\}, \\ R(A,B,C), & \text{otherwise,} \end{cases}$$
where
where
$$R(A, B, C) = \begin{cases} |A| + |B| + |C|, & \text{if } |C| (|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & \text{if } |AB| \le |C| (|B| - 4|A|), \\ (|A| + |C|) \sqrt{1 - \frac{B^2}{4AC}}, & \text{otherwise.} \end{cases}$$
We shall also use the following coefficient representation of Schwarz function. The next well-known lemma by Prokhorov and Szynal [7] will be crucial for our result.
Lemma 3.6
Lemma 3.6. Let be a Schwarz function, that is, w is analytic in, w(0) = 0, and |w(z)| < 1 for all. If, then there exist complex numbers…
Lemma 3.6. Let $w(z) = \sum_{n=1}^{\infty} c_n z^n$ be a Schwarz function, that is, w is analytic in $\mathbb{D}$ , w(0) = 0, and |w(z)| < 1 for all $z \in \mathbb{D}$ . If $c_1 \geq 0$ , then there exist complex numbers $\gamma, \eta, \rho$ with $|\gamma| \leq 1$ , $|\eta| \leq 1$ , $|\rho| \leq 1$ , such that
$$c_2 = (1 - c_1^2)\gamma,$$
$$c_3 = (1 - c_1^2) (\eta (1 - |\gamma|^2) - c_1 \gamma^2),$$
$$c_4 = (1 - c_1^2) (c_1^2 \gamma^3 - (1 - |\gamma|^2) (2c_1 \gamma \eta + \overline{\gamma} \eta^2) + (1 - |\gamma|^2) (1 - |\eta|^2)\rho).$$
Theorem 3.1 · coeff
Theorem 3.1. Let. Then the second Hankel determinant satisfies Moreover, the estimate is sharp.
Theorem 3.1. Let $f \in \mathcal{C}(\varphi)$ . Then the second Hankel determinant satisfies
$$|H_2(2)| = |a_2 a_4 - a_3^2| \le \begin{cases} \frac{1}{36}, & 0 \le t \le \frac{1}{4}, \\ \frac{1}{144} \left( 4 + \frac{(4t-1)^2}{8 + 20t - 16t^2} \right), & \frac{1}{4} \le t \le \frac{1}{2}. \end{cases}$$
Moreover, the estimate is sharp.
Coefficient bounds & claims (4)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H2(2) ≤ 1/36 for class C(phi) (sharp) [Theorem 3.1]
coefficient_bound
H2(2) ≤ (1/144)*(4 + (4*t-1)**2/(8 + 20*t - 16*t**2)) for class C(phi) (sharp) [Theorem 3.1]
coefficient_bound
H3(1) ≤ 1/144 for class C(phi) (sharp) [Theorem 3.2]
function_family
Class C(phi): f in A with 1 + z*f''(z)/f'(z) subordinate to phi(z) = 1+z+(m/n)*z^2, where 2m <= n, m,n in N
Related Papers