Abstract
Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ In the present paper, we consider $\mathcal{C}(\varphi) := \left\{ f \in \mathcal{A} : 1+zf''(z)/f'(z) \prec \varphi(z):=(1+z/2)^2 \right\}$, as subclass of convex functions and compute the sharp second and third Hankel determinants for functions in $\mathcal{C}(\varphi)$.
Results & Lemmas (2)
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 3.3
Lemma 3.3. Let be a Schwarz function, that is, is analytic in,, and for all. If, then there exist complex numbers with,,, such that
Lemma 3.3. Let $\omega(z) = \sum_{n=1}^{\infty} c_n z^n$ be a Schwarz function, that is, $\omega$ is analytic in $\mathbb{D}$ , $\omega(0) = 0$ , and $|\omega(z)| < 1$ 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
$$c_2 = (1 - c_1^2)\gamma,$$
$$c_3 = (1 - c_1^2) \left( \eta (1 - |\gamma|^2) - c_1 \gamma^2 \right),$$
$$c_4 = (1 - c_1^2) \left( c_1^2 \gamma^3 - (1 - |\gamma|^2) \left( 2c_1 \gamma \eta + \overline{\gamma} \eta^2 \right) + (1 - |\gamma|^2) (1 - |\eta|^2) \rho \right)$$
Theorem 3.2
Theorem 3.2. Let. Then, where denotes the third Hankel determinant. The result is sharp.
Theorem 3.2. Let $f \in C(\varphi)$ . Then $|H_3(1)| \le 1/144$ , where $H_3(1)$ denotes the third Hankel determinant. The result is sharp.
Coefficient bounds & claims (3)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_2(2) ≤ 1/36 for class C(phi) (sharp) [Theorem 3.1]
coefficient_bound
H_3(1) ≤ 1/144 for class C(phi) (sharp) [Theorem 3.2]
function_family
Class C(phi): f in A such that 1 + z f''(z)/f'(z) subordinate to phi(z) = (1 + z/2)^2
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