The Krzyz conjecture and convex univalent functions

We obtained the sharp estimates of the moduli of the initial Taylor coefficients for functions $f$ of the class $B$ of bounded nonvanishing functions in the unit circle. Two types of estimates are obtained: one for ``large'' values of $|f(0)|$ and another one for ``small'' values of $|f(0)|$. The first type of estimates is asymptotic in the sense that it applies to an increasing number of initial coefficients as $|f(0)|$ increases. Similarly, the second type of estimates is asymptotic in the sense that it applies to an increasing number of initial coefficients as $|f(0)|$ decreases. Both types of estimates are deduced using methods of subordinate function theory and the Caratheodory-Toeplitz theorem for the Caratheodory class. This became possible due to the relation we found between the coefficients of convex univalent functions (class $S^0$) and the coefficients of the majorizing functions in the studied subclasses of the class $B$. The bounds for the applicability of the method are provided depending on $|f(0)|$ and on the coefficient number. The obtained results are applied to the theory of Laguerre polynomials. These results are compared with the previously known ones. The methods outlined here can be applied to arbitrary classes of subordinate functions.