We present the first results of a systematic search for the rare extragalactic radio sources showing an inverted (integrated) spectrum, with spectral index $alpha ge +2.0$, a previously unexplored spectral domain. The search is expected to yield strong candidates for $alpha ge +2.5$, for which the standard synchrotron self-absorption (characterized by a single power-law energy distribution of relativistic electron population) would not be a plausible explanation, even in an ideal case of a perfectly homogeneous source of incoherent synchrotron radiation. Such sharply inverted spectra, if found, would require alternative explanations, e.g., free-free absorption, or non-standard energy distribution of relativistic electrons which differs from a power-law (e.g., Maxwellian). The search was carried out by comparing two sensitive low-frequency radio surveys made with sub-arcminute resolution, namely, the WISH survey at 352 MHz and TGSS/DR5 at 150 MHz. The overlap region between these two surveys contains 7056 WISH sources classified as `single and brighter than 100 mJy at 352 MHz. We focus here on the seven of these sources for which we find $alpha > +2.0$. Two of these are undetected at 150 MHz and are particularly good candidates for $alpha > +2.5$. Five of the seven sources exhibit a `Gigahertz-Peaked-Spectrum (GPS).
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