A study is carried out of the external quantum efficiency (EQE) of organic light-emitting diodes (OLEDs) based on blue-emitting fluorene polymers containing hole-transporting units. For that purpose, 10 polymers containing a systematically varied amount of two different (benzidine-based and phenoxazine-based) aromatic amine comonomers are synthesized. For all copolymers, measurements are carried out of the electronic level structure and photoluminescence, the current density in hole-only and electron-only sandwich-type devices, and the EQE of OLEDs. The EQE is found to display a distinct maximum at a composition-dependent peak voltage. With decreasing hole mobility, which can be achieved in independent ways by either varying the total amine concentration or by varying the ratio of the two amine concentrations, the maximum EQE increases and the peak voltage decreases. The maximum EQE is well predicted from the hole mobility, provided that a correction is made for the composition-dependent radiative decay efficiencies. These results are explained from a numerical drift-diffusion device model, which shows that the efficiency peak is predominantly due to a voltage dependence of the light-outcoupling efficiency. It is shown that tuning the voltage dependence of the efficiency by a variation of the hole mobility makes it possible to optimize in OLED device applications the power efficiency obtained at the required brightness level.