We report quasi-simultaneous GMRT observations of seven extragalactic radio sources at 150, 325, 610 and 1400 MHz, in an attempt to accurately define their radio continuum spectra, particularly at frequencies below the observed spectral turnover. We had previously identified these sources as candidates for a sharply inverted integrated radio spectrum whose slope is close to, or even exceeds $alpha_c$ = +2.5, the theoretical limit due to synchrotron self-absorption (SSA) in a source of incoherent synchrotron radiation arising from relativistic particles with the canonical (i.e., power-law) energy distribution. We find that four out of the seven candidates have an inverted radio spectrum with a slope close to or exceeding +2.0, while the critical spectral slope $alpha_c$ is exceeded in at least one case. These sources, together with another one or two reported in very recent literature, may well be the archetypes of an extremely rare class, from the standpoint of violation of the SSA limit in compact extragalactic radio sources. However, the alternative possibility that free-free absorption is responsible for their ultra-sharp spectral turnover cannot yet be discounted.
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