An improvement of the estimates of the modulus of the Hankel determinants of second and third order for the class $\mathcal{S}$ of univalent functions

Theorem 1. For the class S we have $$|H_2(2)| \le A$$ , where $1 \le A \le \frac{11}{3} = 3,666...$ and $$|H_3(1)| \le B$$ , where $\frac{4}{9} \le B \le \frac{32 + \sqrt{285}}{15} = 3.258796 \cdots$ In this paper we improve these results by proving:

Theorem 2. For the class S we have the next estimations: - (i) $|H_2(2)| \le 1.3614...$ ; - (i) $|H_3(1)| \le 1.6787...$ The proof of this theorem will make use mainly the notations and results given in the book of N.A. Lebedev ([3]). For an univalent function f from S we have $$\log \frac{f(t) - f(z)}{t - z} = \sum_{p,q=0}^{\infty} \omega_{p,q} t^p z^q,$$ where $\omega_{p,q}$ are the so-called Grunsky's coefficients such that $\omega_{p,q} = \omega_{q,p}$ . This coefficients satisfy the Grunsky's inequality ([1, 3]): (3) $$\sum_{q=1}^{\infty} q \left| \sum_{p=1}^{\infty} \omega_{p,q} x_p \right|^2 \le \sum_{p=1}^{\infty} \frac{|x_p|^2}{p},$$ where $x_p$ are arbitrary complex numbers such that last series converges. Next, it is well-known that if (4) $$f(z) = z + a_2 z^2 + a_3 z^3 + \dots$$ belongs to $\mathcal{S}$ , then also does (5) $$f_2(z) = \sqrt{f(z^2)} = z + c_3 + c_5 z^5 + \dots$$ Then, for the function $f_2$ the appropriate Grunsky's coefficients are of the form $\omega_{2p-1,2q-1}^{(2)}$ and the inequality (3) appears to be (6) $$\sum_{q=1}^{\infty} (2q-1) \left| \sum_{p=1}^{\infty} \omega_{2p-1,2q-1}^{(2)} x_{2p-1} \right|^2 \le \sum_{p=1}^{\infty} \frac{|x_{2p-1}|^2}{2p-1}.$$ Finally, from [3, p.57] we have that the coefficients $a_2, a_3, a_4$ of f can be expressed by Grunsky's coefficients $\omega_{2p-1,2q-1}^{(2)}$ of $f_2$ given by (5) as: $$a_{2} = 2\omega_{11},$$ $$a_{3} = 2\omega_{13} + 3\omega_{11}^{2},$$ $$a_{4} = 2\omega_{33} + 8\omega_{11}\omega_{13} + \frac{10}{3}\omega_{11}^{3}$$ $$a_{5} = 2\omega_{35} + 8\omega_{11}\omega_{33} + 5\omega_{13}^{2} + 18\omega_{11}^{2}\omega_{13} + \frac{7}{3}\omega_{11}^{4}$$ $$0 = 3\omega_{15} - 3\omega_{11}\omega_{13} + \omega_{11}^{3} - 3\omega_{33}$$ $$0 = \omega_{17} - \omega_{35} - \omega_{11}\omega_{33} - \omega_{13}^{2} + \frac{1}{3}\omega_{11}^{4}.$$ Here and in the rest of the paper, for simplicity of the expressions, we omit upper index "(2)" in $\omega_{2p-1,2q-1}^{(2)}$ . We note that in the book [3] there exists a typing mistake for the coefficient $a_5$ . Namely, instead of the therm $5\omega_{13}^2$ , there is $5\omega_{15}^2$ . Also, from (6) for $x_{2p-1} = 0$ , p = 3, 4, ... we have (8) $$|\omega_{11}x_1 + \omega_{31}x_3|^2 + 3|\omega_{13}x_1 + \omega_{33}x_3|^2$$ $$+5|\omega_{15}x_1 + \omega_{35}x_3|^2 + 7|\omega_{17}x_1 + \omega_{37}x_3|^2 \le |x_1|^2 + \frac{|x_3|^2}{3}.$$ From (8), for $x_1 = 1$ and $x_3 = 0$ , since $\omega_{31} = \omega_{13}$ , we have the next inequalities $$|\omega_{11}|^2 + 3|\omega_{13}|^2 + 5|\omega_{15}|^2 + 7|\omega_{15}|^2 \le 1$$ , and further $$\begin{aligned} |\omega_{11}|^2 &\leq 1, \\ |\omega_{11}|^2 + 3|\omega_{13}|^2 &\leq 1, \\ |\omega_{11}|^2 + 3|\omega_{13}|^2 + 5|\omega_{15}|^2 &\leq 1. \end{aligned}$$ This leads to: $$|\omega_{11}| \leq 1,$$ $$|\omega_{13}| \leq \frac{1}{\sqrt{3}} \sqrt{1 - |\omega_{11}|^2},$$ $$|\omega_{15}| \leq \frac{1}{\sqrt{5}} \sqrt{1 - |\omega_{11}|^2 - 3|\omega_{13}|^2},$$ $$|\omega_{17}| \leq \frac{1}{\sqrt{7}} \sqrt{1 - |\omega_{11}|^2 - 3|\omega_{13}|^2 - 5|\omega_{15}|^2}.$$ We note that we can get the first inequality from (9) using the fact $$|a_2| = |2\omega_{11}| \le 2 \quad \Rightarrow \quad |\omega_{11}| \le 1$$ (see (7)).