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We present some interesting antenna designs that exploit the anomalous resonance features of a subwavelength elliptical patch antenna filled with a double-positive dielectric shell and a μ-negative metamaterial core, geometry that we have introduced in a recent paper. With appropriate filling ratio and magnetic material properties, the elliptical patch may be squeezed to an arbitrary subwavelength size, still keeping its overall radiation properties and dual-mode orthogonal operation. We show here how, by varying the patch eccentricity, it is possible to combine the two resonances and achieve significant bandwidth enhancement, in principle overcoming the Chu limit on bandwidth for single-mode electrically small antennas. The antenna may also be tailored to operate as a dual-band electrically small antenna, and intriguing polarization properties may be envisioned by coupling the two orthogonal modes.