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Abstract
In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w) = \log\!\left(\frac{f^{-1}(w)}{w}\right) = 2\sum_{n=1}^{\infty}Γ_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficien
Results & Lemmas (5)
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Lemma 2.3
Lemma 2.3. [21] Let be given by 1.4. Then Moreover, for v < 0 or v > 1, equality holds if and only if or one of its rotations. For 0 < v <…
Lemma 2.3. [21] Let $p \in \mathcal{P}$ be given by 1.4. Then
$$|c_2 - vc_1^2| \le \begin{cases} -4v + 2, & v < 0, \\ 2, & 0 \le v \le 1, \\ 4v - 2, & v > 1. \end{cases}$$
Moreover, for v < 0 or v > 1, equality holds if and only if
$$h(z) = \frac{1+z}{1-z}$$
or one of its rotations.
For 0 < v < 1, equality holds if and only if
$$h(z) = \frac{1+z^2}{1-z^2}$$
or one of its rotations.
Lemma 2.4
Lemma 2.4. [27] Let J, K, and L be numbers such that,, and. Let be of the form (1.4) and define a function by Then and where M = |4K + 2L|.
Lemma 2.4. [27] Let J, K, and L be numbers such that $J \geq 0$ , $K \in \mathbb{C}$ , and $L \in \mathbb{R}$ . Let $p \in \mathcal{P}$ be of the form (1.4) and define a function by
$$\Phi(c_1, c_2) = |Kc_1^2 + Lc_2| - |Jc_1|.$$
Then
$$\Phi(c_1, c_2) \le \begin{cases} |4K + 2L| - 2J, & \text{if } |2K + L| \ge |L| + J, \\ 2|L|, & \text{otherwise.} \end{cases}$$
and
$$-\Phi(c_1, c_2) \le \begin{cases} 2J - M, & \text{when } J \ge M + 2|L|, \\ 2J\sqrt{\frac{2|L|}{M + 2|L|}}, & \text{when } J^2 \le 2|L|(M + 2|L|), \\ 2|L| + \frac{J^2}{M + 2|L|}, & \text{otherwise} \end{cases}$$
where M = |4K + 2L|.
Theorem 3.2
Theorem 3.2. Let, and let (n = 1, 2, 3) be defined by (1.3). Then Both inequalities are sharp.
Theorem 3.2. Let $f \in C_e$ , and let $\Gamma_n$ (n = 1, 2, 3) be defined by (1.3). Then
$$-\frac{1}{2\sqrt{7}} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{12}.$$
Both inequalities are sharp.
Theorem 3.3
Theorem 3.3. Let be of the form (1.1). Then The bound is sharp.
Theorem 3.3. Let $f \in C_e$ be of the form (1.1). Then
$$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{85}{12096} \approx 0.007027.$$
The bound is sharp.
Theorem 3.4 · coeff
Theorem 3.4. Let and. Suppose that is defined by (1.1). Then the generalized Fekete–Szegö functional satisfies the sharp double inequality…
Theorem 3.4. Let $\lambda \in \mathbb{C}$ and $\mu > 0$ . Suppose that $f(z) \in \mathcal{C}_e$ is defined by (1.1). Then the generalized Fekete–Szegö functional
$$F_{\lambda,\mu}(f) := |a_3 - \lambda a_2^2| - \mu |a_2|$$
satisfies the sharp double inequality
$$\mathcal{B}_L \le F_{\lambda,\mu}(f) \le \mathcal{B}_U,\tag{3.25}$$
where the sharp upper bound $\mathcal{B}_U$ is given by
$$\mathcal{B}_{U} = \begin{cases} \frac{1}{6}, & \text{if } |1 - \lambda| < \frac{2}{3} + 2\mu, \\ \frac{1}{4}|1 - \lambda| - \frac{1}{2}\mu, & \text{if } |1 - \lambda| \ge \frac{2}{3} + 2\mu, \end{cases}$$
and the sharp lower bound $\mathcal{B}_L$ is given by
$$\mathcal{B}_{L} = \begin{cases} \frac{1}{4}|1 - \lambda| - \frac{1}{2}\mu, & \text{if } |1 - \lambda| \leq \mu - \frac{2}{3}, \\ -\frac{1}{2}\mu\sqrt{\frac{2}{3|1 - \lambda| + 2}}, & \text{if } |1 - \lambda| \geq \frac{9\mu^{2} - 4}{6}, \\ -\frac{1}{6} - \frac{3\mu^{2}}{4(3|1 - \lambda| + 2)}, & \text{if } \mu - \frac{2}{3} < |1 - \lambda| < \frac{9\mu^{2} - 4}{6}. \end{cases}$$
Moreover, the intermediate parameter region
$$|\mu - \frac{2}{3} < |1 - \lambda| < \frac{9\mu^2 - 4}{6}$$
is nonempty if and only if $\mu > \frac{2}{3}$ . All the estimates are sharp.
Function classes studied:
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