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Abstract
Let $f$ be analutic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bound of Hankel determinant of the second order for the class of analytic unctions satisfying \[ \left|\arg \left[\left(\frac{z}{f(z)}\right)^{1+α}f'(z) \right] \right|<γ\fracπ{2} \quad\quad (z\in\mathbb D),\] for $0<α<1$ and $0<γ\leq1$.
Results & Lemmas (1)
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 1 · coeff
Theorem 1. Let belongs to the class A and satisfy the condition (1.1). Then we have the next sharp estimation: where and.
Theorem 1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ belongs to the class A and satisfy the condition (1.1). Then we have the next sharp estimation:
$$|H_2(2)| = |a_2 a_4 - a_3^2| \le \left(\frac{2\gamma}{2-\alpha}\right)^2,$$
where $0 < \alpha < 2 - \sqrt{2}$ and $0 < \gamma \le \frac{1}{2}(\alpha^2 - 4\alpha + 2)$ .
Definitions (1)
Def 1
Definition 1. Let. Then the qth Hankel determinant of f is defined for, and by Thus, the second Hankel determinant is.
Definition 1. Let $f \in \mathcal{A}$ . Then the qth Hankel determinant of f is defined for $q \geq 1$ , and $n \geq 1$ by
$$H_q(n) = \begin{vmatrix} a_n & a_{n+1} & \dots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \dots & a_{n+q} \\ \vdots & \vdots & & \vdots \\ a_{n+q-1} & a_{n+q} & \dots & a_{n+2q-2} \end{vmatrix}.$$
Thus, the second Hankel determinant is $H_2(2) = a_2 a_4 - a_3^2$ .
Coefficient bounds & claims (3)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_2(2) = |a_2 a_4 - a_3^2| ≤ (2*gamma/(2-alpha))**2 for class class satisfying arg[(z/f(z))^{1+alpha} f'(z)] < gamma*pi/2 (sharp) [Theorem 1]
function_family
Class class satisfying arg[(z/f(z))^{1+alpha} f'(z)] < gamma*pi/2: f in A satisfies |arg[(z/f(z))^{1+alpha} f'(z)]| < gamma*pi/2, for 0 < alpha < 1, 0 < gamma <= 1
function_family
Class S*_beta (strongly starlike of order beta): |arg(zf'(z)/f(z))| < beta*pi/2
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