  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).
