Some classes of analytic functions, connected with the Schwarzian derivative (Q1078358)

Let \(\{\) f,z\(\}\) denote the Schwarzian derivative of f, that is \[ \{f,z)\}=f\prime''(z)/f'(z)-3/2(f''(z)/f'(z))^ 2, \] and let F(z) be a given function holomorphic in the unit disc \(E=\{z:\) \(| z| <1\}\) and satisfying \(| F(z)| \leq F(| z|)\) for all z in E. The authors introduce the class Q(F) of functions f(z) (analytic or meromorphic in E) which satisfy \(| \{f,z\}| \leq F(r)\quad (| z| =r).\) If f(z)\(\in Q(F)\) then f(z) is either holomorphic in E; or meromorphic in E, in which case it has only poles of order one there. In addition, \(Q_ 0(F)\) is the subclass of Q(F) consisting of functions f(z) which satisfy the conditions (a) \(f(0)=0\), \(f'(0)=1\); (b) on every circle \(| z| =r\) in the disc \(| z| <r_ 0\leq 1\) in which f(z) is holomorphic, the absolute extremal points of the same kind for the functions \(| f'(z)|\) and \(W=Re[1+zf''(z)/f'(z)]\) coincide. For functions of \(Q_ 0(F)\) having a common disc of holomorphy \(| z| <r_ 0\leq 1\) best possible upper and lower bounds for \(| f(z)|\), \(| f'(z)|\) and W are obtained, together with an equation which determines the radius of the largest disc centre 0 in which these functions are univalent and convex.