On certain applications of Grunsky coefficients in the theory of univalent functions

Theorem 1. Let $f \in \mathcal{S}$ and be given by (1). Then - (i) $|\gamma_3| \le 0.5566178...$ ([20]); (ii) $|\gamma_4| \le 0.51059...$ ([16]).

Theorem 2. [19] Let $\gamma_2$ , $\gamma_3$ and $\gamma_4$ be the logarithmic coefficients of function $f \in \mathcal{S}$ . Then $$|\gamma_3| - |\gamma_2| \le \frac{1}{\sqrt{5}}$$ and $|\gamma_4| - |\gamma_3| \le \frac{1}{\sqrt{7}}$

Theorem 3. [15] For the class S we have $$|H_{2,2}(f)| \le A$$ , where $1 \le A \le \frac{11}{3} = 3,66...$ and $$|H_{3,1}(f)| \le B$$ , where $\frac{4}{9} \le B \le \frac{32 + \sqrt{285}}{15} = 3.258796 \cdots$ In [21] the authors improved these results by proving:

Theorem 4. [21] For the class S we have the next estimations: - (i) $|H_{2,2}(f)| \leq 1.3614...$ ; - $(ii) |H_{3,1}(f)| \leq 1.6787...$ The modulus of $H_{2,3}(f)$ was estimated In [17] for the case of vanishing and non-vanishing second coefficient:

Theorem 5. [17] Let $f \in \mathcal{S}$ is given by (1). Then - (i) $|H_{2,3}(f)| \le 2.02757...$ if $a_2 = 0$ ; (ii) $|H_{2,3}(f)| \le 4.8986977...$ for every $f \in \mathcal{S}$ . In the next theorem we give the upper bound of the modulus of the second and the third Hankel determinant for the inverse functions of the functions from the class $\mathcal{S}$ (sharp for the second order determinant). The proof of the later is included since it differs from the previous ones.

Theorem 6. ([23]). Let $f \in \mathcal{S}$ and $f^{-1}$ be its inverse function. Then - (i) $|H_{2,2}(f^{-1})| \leq 3$ and the result is the best possible; - $|H_{3,1}(f^{-1})| \leq 2.36639...$

Theorem 7. Let $f \in S$ . Then the following estimates are sharp: - (i) $|H_{2,2}(f^{-1}) H_{2,2}(f)| \le 4$ ; - (ii) $|H_{3,1}(f^{-1}) H_{3,1}(f)| \le 1.$ Proof. (i) From (25) we receive that for any function $f \in \mathcal{S}$ , $$|H_{2,2}(f^{-1}) - H_{2,2}(f)| = |a_2|^2 |a_3 - a_2^2| \le 4,$$ since $|a_2| \le 2$ and $|a_3 - a_2^2| \le 1$ (see [2]). The above estimate is sharp since for the Koebe function we have $H_{2,2}(k) = 2 \cdot 4 - 3^2 = -1$ and $H_{2,2}(k^{-1}) = (-2) \cdot (-14) - 5^2 = 3$ , so that $H_{2,2}(k^{-1}) - H_{2,2}(k) = 4$ . (ii) From the definition of the third Hankel determinant we have $$H_{3,1}(f^{-1}) = A_3(A_2A_4 - A_3^2) - A_4(A_4 - A_2A_3) + A_5(A_3 - A_2^2).$$ Using the relations (22), after some calculations, we receive $$H_{3,1}(f^{-1})$$ = $a_3(a_2a_4 - a_3^2) - a_4(a_4 - a_2a_3) + a_5(a_3 - a_2^2) - (a_3 - a_2^2)^3$ = $H_{3,1}(f) - (a_3 - a_2^2)^3$ , and further, for $f \in \mathcal{S}$ , $$|H_{3,1}(f^{-1}) - H_{3,1}(f)| = |a_3 - a_2^2|^3 \le 1,$$ for every $f \in \mathcal{S}$ . The above inequality is also sharp as the function $f_1(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \cdots$ with $a_2 = a_4 = 0$ and $a_3 = a_5 = 1$ shows. Indeed, from (22) we receive $A_2 = A_4 = 0$ , $A_3 = -1$ , $A_5 = 2$ , i.e., $H_{3,1}(f_1) = 0$ , $H_{3,1}(f_1^{-1}) = -1$ and $|H_{3,1}(f_1^{-1}) - H_{3,1}(f_1)| = 1$ . The following estimate (non-sharp) of the modulus of $H_{3,2}(f^{-1})$ was given in [19]:

Theorem 8. [19] Let $f \in \mathcal{S}$ is given by (1) and let $a_2 = 0$ . Then $$|H_{3,2}(f^{-1})| \le \frac{\sqrt{3}}{6\sqrt{7}} + 2\sqrt{3} = 3.57321\dots$$ Finally, in [20] an estimate of the second Hankel dterminant for the logarithmic coefficients wa given:

Theorem 9. ([20]). Let $f \in \mathcal{S}$ is given by (1). Then $$|H_{2,1}(F_f/2)| = |\gamma_1 \gamma_3 - \gamma_2^2| \le \frac{1}{3}.$$

Theorem 10. Let $f \in \mathcal{S}$ and be given by (1) with $a_2 = 0$ . Then - (i) $|a_3| \le 1$ ([14]), - (ii) $|a_4| \le \frac{2}{2} = 0.666 \dots ([14]),$ - (iii) $|a_5| \leq \frac{3}{4} + \frac{1}{\sqrt{7}} = 1.12796...$ ([22]), - $(iv) |H_{2,2}(f)| \le 1 ([14]),$ - (v) $|H_{3,1}(f)| \le 1.026 \dots (22)$ .

Theorem 11. ([22]). Let $f \in \mathcal{S}$ and be given by (1), with $a_3 = 0$ . Then - $(i) |a_2| \leq 1,$ - (ii) $|a_4| \le \frac{1}{4}\sqrt{\frac{21}{5}} + \frac{5}{8} = 1.1373...,$ - $(iii) |a_5| \le 1.674896577...,$ - $(iv) |H_{2,2}(f)| \leq 1.1373...,$ A long standing problem in the theory of univalent functions is to find sharp upper and lower bounds for $|a_{n+1}| - |a_n|$ , when $f \in \mathcal{S}$ . Since the Keobe function has coefficients $a_n = n$ , it is natural to conjecture that $||a_{n+1}| - |a_n|| \le 1$ . As early as 1933, this was shown to be false even when n=2, when Fekete and Szegö [3] obtained the sharp bounds $$-1 \le |a_3| - |a_2| \le \frac{3}{4} + e^{-\lambda_0} (2e^{-\lambda_0} - 1) = 1.029...,$$ where $\lambda_0$ is the unique value of $\lambda$ in $0 < \lambda < 1$ , satisfying the equation $4\lambda = e^{\lambda}$ . Hayman [6] showed that if $f \in \mathcal{S}$ , then $||a_{n+1}| - |a_n|| \leq C$ , where C is an absolute constant. The exact value of C is unknown, the best estimate to date being C = 3.61... [4], which because of the sharp estimate above when n = 2, cannot be reduced to 1.

Theorem 12. Let $f \in \mathcal{S}$ and be given by (1). Then - (i) $|a_4| |a_3| \le 1.75185...$ ([22]), (ii) $|a_5| |a_3| \le \frac{2}{\sqrt{7}} = 0.7559...$ if f is an odd function ([22]), - $(iii) |a_2a_3 a_4| \le 2.10064... ([20]),$ - $(iv) |a_5| |a_4| \le 2.3297 \dots ([16]).$ Remark 2. We believe that $|a_2a_3 - a_4| \leq 2$ is true for the class S.