Let M denote the class of functions \(w=f(z)\) meromorphic and univalent in the disc \(U=\{z:| z| <1\}\), whereas S - the class of regular functions f(z)\(\in M\) such that \(f(0)=0\), \(f'(0)=1.\)
The author determines a sharp estimate from below of the geometric mean value of \[ | f'(z_ k)/(f(z_ k)-w_ 0)|,\quad k=1,...,n\quad (n\geq 2),\quad f(z)\in M, \] where \(w_ 0\) and \(z_ k\) are any points satisfying the conditions \(| z_ k| =r\), \(0<r<1\), \[ \arg [(f(z_ k)-w_ 0)/(f(z_ 1)-w_ 0]=2\pi (k-1)/n \] (k\(=1,...n)\). He also gives a new piece of information in the form of two theorems on the behaviour of level lines under a conformal mapping.