We present a detailed study of a three-level quantum heat engine operating at maximum efficient power function, a trade-off objective function defined by the product of the efficiency and power output of the engine. First, for near equilibrium conditions, we find general expression for the efficiency and establish universal nature of efficiency at maximum power and maximum efficient power... Then in the high temperature limit, optimizing with respect to one parameter while constraining the other one, we obtain the lower and upper bounds on the efficiency for both strong as well as weak matter-field coupling conditions. Except for the weak matter-field coupling condition, the obtained bounds on the efficiency exactly match with the bounds already known for some models of classical heat engines. Further for weak matter-field coupling, we derive some new bounds on the the efficiency of the the engine which lie beyond the range covered by bounds obtained for strong matter-field coupling. We conclude by comparing the performance of our three-level quantum heat engine in maximum power and maximum efficient power regimes and show that the engine operating at maximum efficient power produces at least $88.89\%$ of the maximum power output while considerably reducing the power loss due to entropy production. (read more)