  We investigate the ultraviolet and infrared fixed point structure of
gauge-Yukawa theories featuring a single gauge coupling, Yukawa coupling and
scalar self coupling. Our investigations are performed using the two loop gauge
beta function, one loop Yukawa beta function and one loop scalar beta function.
We provide the general conditions that the beta function coefficients must
abide for the theory to be completely asymptotically free while simultaneously
possessing an infrared stable fixed point. We also uncover special trajectories
in coupling space along which some couplings are both asymptotically safe and
infrared conformal.
