Relative order of entire and meromorphic functions (Q2762358)

Let \(f\) be a meromorphic function in the plane and \(g\) be an entire function. The relative order of \(f\) with respect to \(g\) is \(\rho_g(f)= \inf\{\mu> 0|T(r,f)< T(r^\mu,g)\) for all large \(r\}\) where \(T\) is the Nevanlinna characteristic function. The authors provide an integral representation for \(\rho_g(f)\) and show \(\rho_g(f)=\rho_g(f')\) for \(f\) a transcendental meromorphic function. Also they discuss \(\rho_g(f_1+f_2)\) and \(\rho_g (f_1\cdot f_2)\) for \(f_1\) and \(f_2\) meromorphic and some restriction on \(g\).