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Abstract
In the present work, we propose to investigate the second Hankel determinant inequalities for certain class of analytic and bi-univalent functions. Some interesting applications of the results presented here are also discussed.
Results & Lemmas (3)
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Lemma 1.1
Lemma 1.1. [24] If the function is given by the series then the following sharp estimate holds:
Lemma 1.1. [24] If the function $p \in \mathcal{P}$ is given by the series
$$(1.3) p(z) = 1 + c_1 z + c_2 z^2 + c_3 z^3 + \cdots,$$
then the following sharp estimate holds:
$$|c_k| < 2, \qquad k = 1, 2, \cdots.$$
Lemma 1.2
Lemma 1.2. [11] If the function is given by the series (1.3), then for some x, z with and. Inspired by the works of [7, 28] we consider the…
Lemma 1.2. [11] If the function $p \in \mathcal{P}$ is given by the series (1.3), then
$$2c_2 = c_1^2 + x(4 - c_1^2)$$
$$4c_3 = c_1^3 + 2c_1(4 - c_1^2)x - c_1(4 - c_1^2)x^2 + 2(4 - c_1^2)(1 - |x|^2)z$$
for some x, z with $|x| \le 1$ and $|z| \le 1$ .
Inspired by the works of [7, 28] we consider the following subclass of the function class $\sigma$ .
For $0 \le \alpha \le 1$ and $0 \le \beta < 1$ , a function $f \in \sigma$ given by (1.1) is said to be in the class $\mathcal{M}^{\alpha}_{\sigma}(\beta)$ if the following conditions are satisfied:
$$\Re\left((1-\alpha)\frac{zf'(z)}{f(z)} + \alpha\left(1 + \frac{zf''(z)}{f'(z)}\right)\right) \ge \beta \qquad (z \in \mathbb{U})$$
and for $g = f^{-1}$
$$\Re\left((1-\alpha)\frac{wg'(w)}{g(w)} + \alpha\left(1 + \frac{wg''(w)}{g'(w)}\right)\right) \ge \beta \qquad (w \in \mathbb{U}).$$
The class was introduced and studied by Li and Wang [16], further the study was extended by Ali et al. [2]. In this paper we shall obtain the functional $H_2(2)$ for functions f belongs to the class $\mathcal{M}^{\alpha}_{\sigma}(\beta)$ and its special classes.
Theorem 2.1
Theorem 2.1. Let f of the form (1.1) be in. Then
Theorem 2.1. Let f of the form (1.1) be in $\mathcal{M}^{\alpha}_{\sigma}(\beta)$ . Then
$$|a_{2}a_{4}-a_{3}^{2}| \leq \begin{cases} \frac{4(1-\beta)^{2}}{3(1+\alpha)^{3}(1+3\alpha)} \left[4(1-\beta)^{2}+(1+\alpha)^{2}\right] ;\\ \beta \in \left[0,1-\frac{(1+\alpha)[3(1+3\alpha)+\sqrt{9(1+3\alpha)^{2}-48(1+\alpha)(1+3\alpha)+128(1+2\alpha)^{2}}]}{16(1+2\alpha)}\right]\\ \frac{(1-\beta)^{2}}{(1+\alpha)(1+3\alpha)} \frac{\left[(1-\beta)^{2}(1+3\alpha)(13+7\alpha)-12(1-\beta)(1+\alpha)(1+2\alpha)(1+3\alpha)-4(1+\alpha)^{2}(9\alpha^{2}+8\alpha+2)\right]}{[16(1-\beta)^{2}(1+2\alpha)-6(1-\beta)(1+\alpha)(1+3\alpha)](1+2\alpha)+(1+\alpha)^{2}[3(1+\alpha)(1+3\alpha)-8(1+2\alpha)^{2}]}\\ \beta \in \left(1-\frac{(1+\alpha)[3(1+3\alpha)+\sqrt{9(1+3\alpha)^{2}+128(1+2\alpha)^{2}}]}{32(1+2\alpha)},1\right). \end{cases}$$
Function classes studied:
Coefficient bounds & claims (2)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
M^alpha_sigma(beta): |a2a4 - a3^2| <= 4(1-beta)^2 / (3(1+alpha)^3(1+3alpha)) * [4(1-beta)^2 + (1+alpha)^2] for beta in [0, beta1); and a more complex expression for beta in [beta2, 1), where beta1, beta2 are thresholds involving alpha. [Theorem 2.1]
function_family
Class M^alpha_sigma(beta): Bi-Mocanu-convex functions of order beta: both f and g=f^{-1} satisfy Re((1-alpha)*z*f'(z)/f(z) + alpha*(1+z*f''(z)/f'(z))) >= beta, with 0 <= alpha <= 1, 0 <= beta < 1
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