  We calculate the flux of $\gamma$-rays emitted from primordial black holes
(PBHs) which are formed by a ``blue'' power-law spectrum of density
fluctuations in the early universe. Gamma-ray emission from such PBHs may
contribute significantly to the observed extragalactic diffuse $\gamma$-ray
background (DGB). Using the observed DGB flux from the imaging Compton
Telescope (COMPTEL) and the Energetic Gamma Ray Experiment Telescope (EGRET) as
the upper limit of $\gamma$-ray flux from PBHs, we derive the upper limit on
the spectral index $n$ of the density fluctuations. The range of initial PBH
masses which can contribute to the DGB is $2\times 10^{13}\g - 5\times
10^{14}\g$, corresponding to a cosmic reheating temperature of $7\times
10^{7}\gev - 4\times 10^{8} \gev$. In this range, we find the upper limit to be
$n \lsim 1.23-1.25$. This limit is stronger than those derived from the energy
density in PBHs or PBH relics and matches the value of $n$ required to explain
the cosmic microwave background anisotropy.
