Growth Estimates for a Class of Subharmonic Functions in a Half Plane
  A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates $u(z)= o(y^{1-\alpha}|z|^{m+\alpha})$ at infinity in the upper half plane ${\bf C}_{+}$, which generalizes the growth properties of analytic functions and harmonic functions. 
Pan Guoshuang and Deng Guantie