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This paper applies the theory of Youla and Carlin et al. to a simplified circuit model of a dual-band PIFA (Planar Inverted-F Antenna) and gives the minimum simultaneously achievable amplitude of reflection coefficient Su for two given non-contiguous bands. A good agreement to the obtained limits is shown in simulations with high-order matching circuits. The limits are then applied to a dual-band mobile phone numerical model. The corresponding circuit model is found by a least-square based optimization program. The accuracy of the obtained matching limits depends on how closely the impedance of the antenna-chassis combination is modeled by the impedance of the circuit model, or more exactly, by the resemblance of the quality factors of the two systems. It is also noteworthy that the Su for two given bands are "exchangeable quantities" , trying to reduce the Su for one band leads to an increase of Su in the other band.