Denote by
$\mathbb{P}$
the set of all prime numbers and by
$P(n)$
the largest prime factor of positive integer
$n\geq 1$
with the convention
$P(1)=1$
. In this paper, we prove that, for each
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$
, there is a constant
$c(\unicode[STIX]{x1D702})>1$
such that, for every fixed nonzero integer
$a\in \mathbb{Z}^{\ast }$
, the set
$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$
has relative asymptotic density one in
$\mathbb{P}$
. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’,
J. Aust. Math. Soc.82 (2015), 133–147], Theorem 1.1, which requires
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$
in place of
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$
.