Abstract
Coefficient bounds and sharp bounds of second Hankel determinant of inverse logarithmic coefficients for normalized analytic univalent functions and harmonic mappings with specified conditions.
Results & Lemmas (8)
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 · coeff
Theorem 2.1. Let M > 0, and be of the form (1.1). For, we have and. The result is sharp for the functions and, where the functions are…
Theorem 2.1. Let M > 0, $\alpha \in (0,1]$ and $f = h + \overline{g} \in \mathcal{D}^0_{\mathcal{H}}(\alpha, M)$ be of the form (1.1). For $n \geq 2$ , we have $|a_n| \leq M/\left(n + (n^2 - 2n)\alpha\right)$ and $|b_n| \leq M/\left(n + (n^2 - 2n)\alpha\right)$ . The result is sharp for the functions $f_1$ and $f_2$ , where the functions are given by $f_1(z) = z + Mz^n/\left(n + (n^2 - 2n)\alpha\right)$ and $f_2(z) = z + M\overline{z^n}/\left(n + (n^2 - 2n)\alpha\right)$ for $n \geq 2$ .
Theorem 2.2
Theorem 2.2. The harmonic map belongs to if, and only if, the function belongs to for each with.
Theorem 2.2. The harmonic map $f = h + \overline{g}$ belongs to $\mathcal{D}^0_{\mathcal{H}}(\alpha, M)$ if, and only if, the function $F_{\varepsilon} = h + \varepsilon g$ belongs to $\mathcal{D}(\alpha, M)$ for each $\varepsilon$ with $|\varepsilon| = 1$ .
Theorem 2.3
Theorem 2.3. Let M > 0, and be of the form (1.1). Then, <span id="page-3-1"></span> (2.2) For each,, equality occurs for the function f…
Theorem 2.3. Let M > 0, $\alpha \in (0,1]$ and $f = h + \overline{g} \in \mathcal{D}^0_{\mathcal{H}}(\alpha, M)$ be of the form (1.1). Then,
<span id="page-3-1"></span>
$$|z| - \frac{M|z|^2}{2} \le |f(z)| \le |z| + \frac{M|z|^2}{2}.$$
(2.2)
For each $z \in \mathbb{D}$ , $z \neq 0$ , equality occurs for the function f given by $f(z) = z + Mz^2/2$ or its suitable rotations.
Theorem 2.4
Theorem 2.4. If for M > 0 and, then, with equality for the function.
Theorem 2.4. If $f \in \mathcal{D}^0_{\mathcal{H}}(\alpha, M)$ for M > 0 and $\alpha \in (0, 1]$ , then $J_f(z) \leq (1 + M|z|)^2$ , with equality for the function $f(z) = z + Mz^2/2$ .
Theorem 2.5 · coeff
Theorem 2.5. Let with g'(0) = 0 be given by (1.1). If <span id="page-6-0"></span> then.
Theorem 2.5. Let $f = h + \overline{g} \in \mathcal{H}$ with g'(0) = 0 be given by (1.1). If
<span id="page-6-0"></span>
$$\sum_{n=2}^{\infty} (n + (n^2 - 2n)\alpha) (|a_n| + |b_n|) \le M, \tag{2.10}$$
then $f \in \mathcal{D}^0_{\mathcal{H}}(\alpha, M)$ .
Lemma 2.1 · coeff
Lemma 2.1. [6] Let be given by (5.2). If, then f is starlike in.
Lemma 2.1. [6] Let $f = h + \overline{g}$ be given by (5.2). If $\sum_{n=2}^{\infty} n(|a_n| + |b_n|) \le 1$ , then f is starlike in $\mathbb{D}$ .
Theorem 2.6
Theorem 2.6. Let M > 0, and be given by (1.1). Then f is starlike in, where is the smallest root of the equation
Theorem 2.6. Let M > 0, $\alpha \in (0,1]$ and $f = h + \overline{g} \in \mathcal{D}^0_{\mathcal{H}}(\alpha, M)$ be given by (1.1). Then f is starlike in $|z| \leq r_1$ , where $r_1 \in (0,1)$ is the smallest root of the equation
$$2Mr_{2}F_{1}\left(1,\frac{1}{\alpha};1+\frac{1}{\alpha};r\right)-1=0.$$
Theorem 5.1
Theorem 5.1. Let with. Then <span id="page-10-1"></span> (5.1) The inequality (5.1) is sharp.
Theorem 5.1. Let $f \in \mathcal{P}(M)$ with $0 < M \le 1/\log 4 (\approx 0.7213475)$ . Then
<span id="page-10-1"></span>
$$\left| H_{2,1}\left(F_{f^{-1}}/2\right) \right| \le \begin{cases} M^2/36, & 0 < M \le \frac{1}{39}(6 + \sqrt{114}) \\ \frac{M^2}{144}\left(39M^2 - 12M + 2\right), & \frac{1}{39}(6 + \sqrt{114}) < M \le 1/\log 4. \end{cases}$$
(5.1)
The inequality (5.1) is sharp.
Function classes studied:
Coefficient bounds & claims (5)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
|a_n| ≤ M/(n + (n**2 - 2*n)*alpha) for class D^0_H(alpha, M) (sharp) [Theorem 2.1]
coefficient_bound
D^0_H(alpha, M): |z| - M|z|^2/2 <= |f(z)| <= |z| + M|z|^2/2. For each z in D, z != 0, equality occurs for the function f given by f(z) = z + Mz^2/2 or its suitable rotations. (sharp) [Theorem 2.3]
coefficient_bound
H_{2,1}(F_{f^{-1}}/2) ≤ M**2/36 for class P(M) (sharp) [Theorem 5.1]
function_family
Class D^0_H(alpha, M): Harmonic f=h+g in H with |(1-alpha)h'(z)+alpha*z*h''(z)-(1-alpha)| <= M - |(1-alpha)g'(z)+alpha*z*g''(z)| with g'(0)=0, M>0, alpha in (0,1]
function_family
Class P(M): Analytic h in A satisfying Re(zh''(z)) > -M for 0 < M <= 1/log4, z in D
Related Papers