Antennas are provided for electronic devices such as portable computers. An electronic device may have a housing in which an antenna is mounted. The housing may be formed of conductive materials. A dielectric antenna window may be mounted in the housing to allow radio-frequency signals to be transmitted from the antenna and to allow the antenna to receive radio-frequency signals. Near-field radiation limits may be satisfied by reducing transmit power when an external object is detected in the vicinity of the dielectric antenna window and the antenna. A proximity sensor may be used in detecting external objects. A parasitic antenna resonating element may be interposed between the antenna resonating element and the dielectric antenna window to minimize near-field radiation hotspots. The parasitic antenna resonating element may be formed using a capacitor electrode for the proximity sensor. A ferrite layer may be interposed between the parasitic element and the antenna window.