Sharp estimate of the third coefficient for bounded non-vanishing holomorphic functions with real coefficients

Let $\Omega_0^r$ be a class of holomorphic functions $\omega$ in the unit disk $\Delta,$ with real coefficients, and such that $|\omega(z)|<1,$ $\omega(0)=0,$ $z\in\Delta.$ The coefficients problem in the class $\Omega_0^r$ is formulated as follows: find the necessary and sufficient conditions to be imposed on the real numbers $\{\omega\}_1, \{\omega\}_2,\ldots$ in order for the series $\{\omega\}_1 z+\{\omega\}_2 z^2+\ldots$ to be the Taylor series of a function in the class $\Omega_0^r.$

   The class $B^r$ consists of holomorphic functions $f$ in $\Delta$ with real coefficients and such that    $0<|f(z)|\leq 1,$ $z\in\Delta.$ The classes $B_t^r,$ $t\geq 0,$ are defined as the sets of functions $f\in B^r$ such that $f(0)=e^{-t}.$ The problem of obtaining a sharp estimation of $|\{f\}_n|,$ $n\in\mathbb N,$ on the class $B^r$ or $B_t^r$ is commonly referred to as the Krzyz problem (for the class $B^r$ or $B_t^r$). It~is clear that the union of all classes $B_t^r$ exhausts the class $B^r$ up to rotations in the plane of variable $w$ ($w=f(z)$).

 Based on the solution of the coefficients problem for the class $\Omega_0^r,$ the problem of obtaining a sharp estimation of the functional $|\{f\}_3|$ on the classes $B_t^r$ for every $t\geq 0$ is solved by transitioning to the functional over the class $\Omega_0^r,$ after which the problem is reduced to finding the global constrained extremum of a function of two real variables with inequality-type constraints.

The extreme functions are found in two forms: as a convex combination of Schwartz kernels related to the Caratheodory class, and as Blaschke products related to the class
$\Omega_0^r.$