Heterogeneous processors (e.g., ARM?s big.LITTLE) improve performance in power-constrained environments by executing applications on the ?little? low-power core and move them to the ?big? high-performance core when there is available power budget. The total time spent on the big core depends on the rate at which the application dissipates the available power budget. When applications with different big-core power consumption characteristics concurrently execute on a heterogeneous processor, it is best to give a larger share of the power budget to applications that can run longer on the big core, and a smaller share to applications that run for a very short duration on the big core. This article investigates mechanisms to manage the available power budget on power-constrained heterogeneous processors. We show that existing proposals that schedule applications onto a big core based on various performance metrics are not high performing, as these strategies do not optimize over an entire power period and are unaware of the applications? power/performance characteristics. We use linear programming to design the DPDP power management technique, which guarantees optimal performance on heterogeneous processors. We mathematically derive a metric (Delta Performance by Delta Power) that takes into account the power/performance characteristics of each running application and allows our power-management technique to decide how best to distribute the available power budget among the co-running applications at minimal overhead. Our evaluations with a 4-core heterogeneous processor consisting of big.LITTLE pairs show that DPDP improves performance by 16% on average and up to 40% compared to a strategy that globally and greedily optimizes the power budget. We also show that DPDP outperforms existing heterogeneous scheduling policies that use performance metrics to decide how best to schedule applications on the big core.