The radio resource allocation in cognitive radio networks is an essential problem where the secondary users (SUs) use the spectrum allocated to primary users (PUs) without causing much interference to PUs. This paper studies optimal power and bandwidth allocation in a cognitive radio network under Rician and Nakagami-m fading channels. The performance metric for the network used is the ergodic-capacity of all the SUs. Further, this paper investigates the optimal power allocation schemes to achieve the primary capacity bounds of a secondary network with Rician and Nakagami-m fading channels. Specifically, the ergodic-capacity is considered. The methodology used involves deriving closed-form results for both Rician and Nakagami-m scenarios. Besides, the peak/average transmit-power constraints at the SUs and the peak/average interference power constraint imposed by the PU. The equations of optimal power allocations are also formulated under peak-power and peak-interference constraints. Further, the analysis is done for a network of SUs. Simulation results depicted with figures and tables show that the optimal power and bandwidth allocation for Rician and Nakagami-m fading channels. The investigation and analysis on optimal power and bandwidth allocation can be used for future reference of resource allocation in the cognitive radio networks over Rician and Nakagami-m fading channels.