Abstract
In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the strongly starlike and strongly convex functions of order alpha.
Results & Lemmas (5)
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Lemma 2.1
Lemma 2.1. [12] For a function of the form (2.1), the sharp inequality holds for each. Equality holds for the function p(z) = 1 + z/1 - z.
Lemma 2.1. [12] For a function $p \in \mathcal{P}$ of the form (2.1), the sharp inequality holds for each $n \geq 1$ . Equality holds for the function p(z) = 1 + z/1 - z.
Lemma 2.2
Lemma 2.2. [18, 19] If is of the form (2.1) with. Then there exits such that and Next we recall the following well-known result due to Choi…
Lemma 2.2. [18, 19] If $p \in \mathcal{P}$ is of the form (2.1) with $c_1 \geq 0$ . Then there exits $z, w \in \mathbb{D}$ such that
$$2c_2 = c_1^2 + (4 - c_1^2)z$$
and
$$4c_3 = c_1^3 + 2(4 - c_1^2)c_1z - c_1(4 - c_1^2)z + 2(4 - c_1^2)(1 - |x|^2)w.$$
Next we recall the following well-known result due to Choi et al. [11]. Lemma 2.3 plays an important role in the proof of our main results.
Lemma 2.3
Lemma 2.3. Let A, B, C be real numbers and (i) If, then (ii) If AC < 0, then where
Lemma 2.3. Let A, B, C be real numbers and
$$Y(A,B,C) := \max_{z \in \overline{\mathbb{D}}} \left( \left| A + Bz + Cz^2 \right| + 1 - |z|^2 \right).$$
(i) If $AC \geq 0$ , then
$$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$
(ii) If AC < 0, then
$$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & -4AC(C^{-2}-1) \le B^2 \land |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & B^2 < \min\{4(1+|C|)^2, -4AC(C^{-2}-1)\}, \\ R(A,B,C), & otherwise, \end{cases}$$
where
$$R(A,B,C) = \begin{cases} |A| + |B| + |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ (|A| + |C|)\sqrt{1 - \frac{B^2}{4AC}}, & otherwise. \end{cases}$$
Theorem 3.1
Theorem 3.1. Let given by (1.1) then <span id="page-5-3"></span>(3.1) The inequality is sharp.
Theorem 3.1. Let $f \in \mathcal{K}_{\alpha}$ given by (1.1) then
<span id="page-5-3"></span>(3.1)
$$|H_{2,1}(F_{f^{-1}}/2)| \le \begin{cases} \frac{\alpha^2}{36}, & 0 < \alpha \le 1/3, \\ \frac{\alpha^2(17 + 18\alpha + 13\alpha^2)}{144(4 + 6\alpha + \alpha^2)}, & 1/3 < \alpha \le 1. \end{cases}$$
The inequality is sharp.
Theorem 3.3
Theorem 3.3. Let given by (1.1) then <span id="page-10-1"></span>(3.15) where is the unique root in (0,1) of the equation. The inequality…
Theorem 3.3. Let $f \in \mathcal{S}_{\alpha}^*$ given by (1.1) then
<span id="page-10-1"></span>(3.15)
$$|H_{2,1}(F_{f^{-1}}/2)| \leq \begin{cases} \frac{\alpha^2}{4}, & 0 < \alpha < 1/5, \\ \frac{\alpha^2(2 + 5\alpha + 15\alpha^2)}{7 + 30\alpha + 35\alpha^2}, & 1/5 \leq \alpha \leq \alpha', \\ \frac{\alpha^2}{36}(4 + 35\alpha^2), & \alpha' < \alpha \leq 1. \end{cases}$$
where $\alpha' = 0.390595...$ is the unique root in (0,1) of the equation $44 + 60\alpha + 155\alpha^2 - 1050\alpha^3 - 1225\alpha^4 = 0$ . The inequality (3.15) is sharp.
Function classes studied:
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