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

Let Ω_0^r be a class of holomorphic functions ω in the unit disk ∆, with real coefficients, and such that |ω(z)|<1, ω(0)=0, z∈∆. The coefficients problem in the class Ω_0^r is formulated as follows: find the necessary and sufficient conditions to be imposed on the real numbers {ω}_1,{ω}_2,… in order for the series {ω}_1 z+{ω}_2 z^2+⋯ to be the Taylor series of a function in the class Ω_0^r. The class B^r consists of holomorphic functions f in ∆ with real coefficients and such that 0<|f(z)|≤1, z∈∆. The classes B_t^r, t≥0, are defined as the sets of functions f∈B^r such that f(0)=e^(-t). The problem of obtaining a sharp estimation of |{f}_n |, n∈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 Ω_0^r the problem of obtaining a sharp estimation of the functional |{f}_3 | on the classes B_t^r for every t≥0 is solved by transitioning to the functional over the class Ω_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 Ω_0^r.