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Abstract
In the present investigation the authors obtain upper bounds for the second Hankel determinant of the classes bi-starlike and bi-convex functions of order beta.
Results & Lemmas (6)
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 1.1
Lemma 1.1. [22] If the function is given by the series then the sharp estimate holds.
Lemma 1.1. [22] If the function $p \in \mathcal{P}$ is given by the series
$$(1.2) p(z) = 1 + c_1 z + c_2 z^2 + \dots$$
then the sharp estimate $|c_k| \leq 2 \ (k = 1, 2, ...)$ holds.
Lemma 1.2
Lemma 1.2. [10] If the function is given by the series (1.2), then for some x, z with and.
Lemma 1.2. [10] If the function $p \in \mathcal{P}$ is given by the series (1.2), then
$$(1.3) 2c_2 = c_1^2 + x(4 - c_1^2)$$
$$(1.4) 4c_3 = c_1^3 + 2(4 - c_1^2)c_1x - c_1(4 - c_1^2)x^2 + 2(4 - c_1^2)\left(1 - |x|^2\right)z,$$
for some x, z with $|x| \le 1$ and $|z| \le 1$ .
Theorem 2.1 · coeff
Theorem 2.1. Let f(z) given by (1.1) be in the class,. Then
Theorem 2.1. Let f(z) given by (1.1) be in the class $\mathcal{S}^*_{\sigma}(\beta)$ , $0 \leq \beta < 1$ . Then
$$|a_2 a_4 - a_3^2| \le \begin{cases} \frac{4}{3} (1 - \beta)^2 (4\beta^2 - 8\beta + 5), & \beta \in \left[0, \frac{29 - \sqrt{137}}{32}\right] \\ (1 - \beta)^2 \left(\frac{13\beta^2 - 14\beta - 7}{16\beta^2 - 26\beta + 5}\right), & \beta \in \left(\frac{29 - \sqrt{137}}{32}, 1\right). \end{cases}$$
Corollary 2.2 · coeff
Corollary 2.2. Let f(z) given by (1.1) be in the class. Then Our second main result for the class is following:
Corollary 2.2. Let f(z) given by (1.1) be in the class $\mathcal{S}_{\sigma}^*$ . Then
$$\left| a_2 a_4 - a_3^2 \right| \le \frac{20}{3}.$$
Our second main result for the class $\mathcal{K}_{\sigma}(\beta)$ is following:
Theorem 2.3 · coeff
Theorem 2.3. Let f(z) given by (1.1) be in the class,. Then
Theorem 2.3. Let f(z) given by (1.1) be in the class $\mathcal{K}_{\sigma}(\beta)$ , $0 \leq \beta < 1$ . Then
$$\left| a_2 a_4 - a_3^2 \right| \le \frac{(1-\beta)^2}{24} \left( \frac{5\beta^2 + 8\beta - 32}{3\beta^2 - 3\beta - 4} \right)$$
Corollary 2.4 · coeff
Corollary 2.4. Let f(z) given by (1.1) be in the class. Then
Corollary 2.4. Let f(z) given by (1.1) be in the class $K_{\sigma}$ . Then
$$\left| a_2 a_4 - a_3^2 \right| \le \frac{1}{3}.$$
Function classes studied:
Coefficient bounds & claims (8)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
H_2(2) = |a_2*a_4 - a_3^2| ≤ 4*(1-beta)**2*(4*beta**2 - 8*beta + 5)/3 for class S*_sigma(beta) [Theorem 2.1]
coefficient_bound
H_2(2) = |a_2*a_4 - a_3^2| for beta=0 ≤ 20/3 for class S*_sigma [Corollary 2.2]
coefficient_bound
H_2(2) = |a_2*a_4 - a_3^2| ≤ (1-beta)**2*(5*beta**2+8*beta-32)/(24*(3*beta**2-3*beta-4)) for class K_sigma(beta) [Theorem 2.3]
coefficient_bound
H_2(2) = |a_2*a_4 - a_3^2| for beta=0 ≤ 1/3 for class K_sigma [Corollary 2.4]
coefficient_bound
|a_3 - mu*a_2^2| (Fekete-Szego, from cited Zaprawa [28]) ≤ 1-beta for class S*_sigma(beta) [Introduction (citing Zaprawa 2014)]
coefficient_bound
|a_3 - mu*a_2^2| (Fekete-Szego, from cited Zaprawa [28]) ≤ (1-beta)/3 for class K_sigma(beta) [Introduction (citing Zaprawa 2014)]
function_family
Class S*_sigma(beta): bi-starlike functions of order beta: both f and f^{-1} satisfy Re(zf'/f) > beta, 0<=beta<1
function_family
Class K_sigma(beta): bi-convex functions of order beta: both f and f^{-1} satisfy Re(1+zf''/f') > beta, 0<=beta<1
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