On weighted compositions preserving the Carath\'eodory class
  We characterize in various ways the weighted composition transformations which preserve the class ${\mathcal P}$ of normalized analytic functions in the disk with positive real part. We analyze the meaning of the criteria obtained for various special cases of symbols and identify the fixed points of such transformations. 
Irina Ar\'evalo, Rodrigo Hern\'andez, Mar\'ia J. Mart\'in, Dragan   Vukoti\'c
math.CV