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The Bode-Fano integral can be used as an objective tool for assessing the bandwidth of antennas, and especially schemes for bandwidth improvement. Results for U-slot and E-slot dual resonant patch antennas suggest that the Fano integral is invariantly related to the overall volume. The Bode-Fano and Youla theories of broadband matching have been applied to the narrowband and wideband lumped equivalent circuit of microstrip antennas to calculate the maximum achievable return loss-bandwidth product of linearly polarized microstrip antennas. Curves are presented showing the relation between the antenna bandwidth, maximum achievable return loss, and parameters of the equivalent circuit. It has been shown that creating parallel slots on the patch despite all potential advantages, may reduce the potential bandwidth of patch antennas.