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Abstract
Following the trend of coefficient bound problems in Geometric Function Theory, in the present paper, we obtain the sharp bound of $|H_3(1)|$ for the class $\mathcal{S}^*$, of starlike functions and $\mathcal{SL}_q^*$, of $q$- starlike functions related with lemniscate of Bernoulli. Bound on the initial class is also an improvement over the existing known bound and the bound on the latter class generalizes the prior known outcome. Further, we determine the extremal functions to prove the sharpne
Results & Lemmas (4)
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Lemma 1.1
Lemma 1.1. Let and of the form. Then and for some, and such that, and.
Lemma 1.1. Let $p \in \mathcal{P}$ and of the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then
$$2p_2 = p_1^2 + \lambda(4 - p_1^2),$$
$$4p_3 = p_1^3 + 2p_1(4 - p_1^2)\lambda - p_1(4 - p_1^2)\lambda^2 + 2(4 - p_1^2)(1 - |\lambda|^2)\mu$$
and
$$8p_4 = p_1^4 + (4 - p_1^2)\lambda(p_1^2(\lambda^2 - 3\lambda + 3) + 4\lambda) - 4(4 - p_1^2)(1 - |\lambda|^2)(p_1(\lambda - 1)\mu + \overline{\lambda}\mu^2 - (1 - |\mu|^2)\delta),$$
for some $\delta$ , $\lambda$ and $\mu$ such that $|\delta| \leq 1$ , $|\lambda| \leq 1$ and $|\mu| \leq 1$ .
Theorem 2.1 · coeff
Theorem 2.1. Let and of the form. Then we have
Theorem 2.1. Let $q \in (0,1)$ and $f \in \mathcal{SL}_q^*$ of the form $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ . Then we have
$$|H_3(1)| \le \frac{(1+q)^2}{16q^2(1+q+q^2)^2}.$$
Corollary 2.2
Corollary 2.2. Let. Then. Moreover, extremal functions also coincide in the case of.
Corollary 2.2. Let $f \in \mathcal{SL}^*$ . Then $|H_3(1)| \leq 1/36$ .
Moreover, extremal functions also coincide in the case of $q \to 1^-$ .
Theorem 2.3 · coeff
Theorem 2.3. Let of the form. Then the sharp bound for third order Hankel determinant for such functions is given by
Theorem 2.3. Let $f \in \mathcal{S}^*$ of the form $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ . Then the sharp bound for third order Hankel determinant for such functions is given by
$$|H_3(1)| \le 4/9. \tag{2.7}$$
Function classes studied:
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