On the monotonicity of some functionals in the family of univalent functions

LetS denote the class of regular and univalent functions in |z|<1 with the normalizationf(0)=0,f′(0)=1. Denoted _f=inf _f∈s {|α||f(z)≠ α, |z|<1} and letS(d)={f¦f∈S,d _f=d, 1/4≦d≦1}. The analytic functionf(z) is univalent in |z|<1 if and only if $$log\frac{{f(z) - f(\zeta )}}{{z - \zeta }} = \sum\limits_{m,n = 0}^\infty {d_{mn} z^m \zeta ^n } $$ converges in the bicylinder |z|<1, |ξ|<1. LetC _mn =√mnd _mn andC _nn (d)= Max _fεS(d){Re(C _nn )}. The paper deals with the monotonicity ofc _nn(d) and related functionals.