Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
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 (7)
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.
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 (3.1) The inequality is sharp.
Theorem 3.1. Let $f \in \mathcal{K}_{\alpha}$ given by (1.1) then
(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.
Corollary 3.2 · coeff
Corollary 3.2. If, then We next find the sharp bound for the second Hankel determinant of inverse logarithmic coefficient of functions in.
Corollary 3.2. If $f \in \mathcal{K}$ , then
$$|H_{2,1}(F_f/2)| \le \frac{1}{33}$$
We next find the sharp bound for the second Hankel determinant of inverse logarithmic coefficient of functions in $\mathcal{S}_{\alpha}^*$ .
Theorem 3.3
Theorem 3.3. Let given by (1.1) then (3.15) where is the unique root in (0,1) of the equation. The inequality (3.15) is sharp.
Theorem 3.3. Let $f \in \mathcal{S}_{\alpha}^*$ given by (1.1) then
(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.
Corollary 3.4
Corollary 3.4. If, then Acknowledgment: The research of the first named author is supported by SERB-CRG, Govt. of India and the second…
Corollary 3.4. If $f \in \mathcal{S}^*$ , then
$$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{13}{12}.$$
Acknowledgment: The research of the first named author is supported by SERB-CRG, Govt. of India and the second named author's research work is supported by UGC-SRF.
Function classes studied:
Related Papers