On the coefficient bodies of bounded real non-vanishing univalent functions (Q2769218)

Let \(S(b)=\{f|f(z)=b (z+a_2z^2+ \dots,|z|< 1,|f(z)|<1,0 <b<1\}\), \(S_R(b) \subset S(b)\) denote the real subclass with all the \(a_v\)-coefficients real, \(S_R(B)= \{F\mid F(z)=B+ A_1+\cdots, |z|<1\), \(F(z)\neq 0\), \(0<B<1\), \(A_1>0\}\), \(S_R'(b)\subset S(b)\) is a subclass with all the \(A\)-coefficients real. The problem of finding the coefficient bodies within the class \(S_R(b)\) was investigated by the author and other mathematicians. In this paper the author presents analogous results concerning the class \(S_R'(B)\). He established a connection between the coefficients of functions of both classes which enabled him to find relations between coefficient bodies in \(S_R(b)\) and \(S_R'(B)\).