Abstract
In this paper, sharp bounds are established for the second Hankel determinant of logarithmic coefficients for normalised analytic functions satisfying certain differential inequality.
Results & Lemmas (5)
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Theorem 2.1
Theorem 2.1. Let be given by (1.3). If, then sharp bound on Hankel determinant for is given by (2.1) Above inequality is sharp due to the…
Theorem 2.1. Let $F_f$ be given by (1.3). If $f \in \mathcal{S}_s^*$ , then sharp bound on Hankel determinant for $F_f$ is given by
$$|H_{2,1}(F_f)| \le \frac{1}{4}.$$
(2.1)
Above inequality is sharp due to the function
<span id="page-3-3"></span>
$$\tilde{f}(z) = \int_0^z (1+t^2)/(1-t^2)^2 dt. \tag{2.2}$$
Theorem 2.2
Theorem 2.2. Let be given by (1.3). If, then sharp bound on Hankel determinant for is given by <span id="page-5-4"></span> (2.7) Above…
Theorem 2.2. Let $F_f$ be given by (1.3). If $f \in \mathcal{F}_2$ , then sharp bound on Hankel determinant for $F_f$ is given by
<span id="page-5-4"></span>
$$|H_{2,1}(F_f)| \le \frac{1}{4}.$$
(2.7)
Above inequality is sharp due to the function given in Theorem 2.1.
Theorem 2.3
Theorem 2.3. Suppose and be given by (1.3). If, then sharp bound on Hankel determinant for is given by <span id="page-7-3"></span> (2.14)…
Theorem 2.3. Suppose $X(x) = -48x^4 - 96x^3 - 392x^2 + 24x + 357$ and $F_f$ be given by (1.3). If $f \in \mathcal{F}_1$ , then sharp bound on Hankel determinant for $F_f$ is given by
<span id="page-7-3"></span>
$$|H_{2.1}(F_f)| \le X(\eta)/2304,$$
(2.14)
where $\eta \approx 0.0302$ is the unique real root of the equation X'(x) = 0. This result is sharp due to the extremal function
<span id="page-7-4"></span>
$$\tilde{f}(z) = \int_0^z p_1(t)/(1-t)dt,$$
(2.15)
where $p_1(z)$ is defined in (2.13).
Theorem 2.4
Theorem 2.4. Suppose. Let be given by (1.3). If, then sharp bound on Hankel determinant for is given by <span id="page-9-0"></span> (2.18)…
Theorem 2.4. Suppose $X(x) = -176x^4 - 224x^3 - 264x^2 + 328x + 469$ . Let $F_f$ be given by (1.3). If $f \in \mathcal{F}_3$ , then sharp bound on Hankel determinant for $F_f$ is given by
<span id="page-9-0"></span>
$$|H_{2,1}(F_f)| \le X(\eta)/2304,$$
(2.18)
where $\eta \approx 0.3737$ is the unique real root of the equation X'(x) = 0. This result is sharp for the function
$$f(z) = \int_0^z p_2(t)/(1-t+t^2)dt,$$
where $p_2(z)$ is given by (2.13).
Theorem 2.5
Theorem 2.5. Suppose. Let be given by (1.3). If, then sharp bound on Hankel determinant for is given by <span id="page-10-4"></span> (2.24)…
Theorem 2.5. Suppose $48(17+x)X(x) = (1+x)(x^4+20x^3-114x^2+4x+125)$ . Let $F_f$ be given by (1.3). If $f \in \mathcal{F}_4$ , then sharp bound on Hankel determinant for $F_f$ is given by
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$$|H_{2,1}(F_f)| \le X(\eta),$$
(2.24)
where $\eta \approx 0.381$ is the unique real root of the equation X'(x) = 0. This result is sharp with extremal function
<span id="page-10-5"></span>
$$\tilde{f}(z) = \int_0^z p_3(t)/(1-t)^2 dt,$$
(2.25)
where $p_3(z)$ is defined in (2.13).
Function classes studied:
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