We present self-similar semi-analytical solutions obtained in the framework of general relativistic magnetohydrodynamics (GRMHD) which describe steady and axisymmetric outflows from the system of a hot coronal magnetosphere of a Schwarzschild black hole and its surrounding accretion disk. The model allows to extend previous non relativistic MHD studies for coronal winds from young stars to spine jets from Active Galactic Nuclei surrounded by disk-driven outflows, The collimation depends critically on an energetic integral measuring the efficiency of the magnetic rotator, similarly to the non relativistic case. The outflows are thermally driven and magnetically (thermally) collimated if the magnetic rotator is efficient (inefficient). It is also shown that relativistic effects affect quantitatively the depth of the gravitational well and the coronal temperature distribution in the launching region of the outflow. Similarly to previous analytical and numerical studies, relativistic effects tend to increase the efficiency of the thermal driving but reduce the effect of magnetic self-collimation.