The lower bound of shifted primes with large prime factors

In this paper, we consider the asymptotic density of $T_c(x):=\mathrm{\#}\{p\le x:P^{+}(p-1)\ge p^c\}$, where $P^{+}(n)$ denote the largest prime factor of $n$. We show that for $x\to \mathrm{\infty }$, for any $0<c\le 1\mathrm{/}2$, one has \begin{align*} \mathop{\lim\inf}{x\rightarrow\infty}\frac{T_c(x)}{π(x)}&\geq\max\left(1-\frac{16}{5}ρ\left(\frac{1}{c}\right),1-c+g©\right), \end{align*} where $g(c)>0$ for $11\mathrm{/}32<c\le 1\mathrm{/}2$, and $g(c)=0$ for $0<c\le 11\mathrm{/}32$. In particular, we have $g(1\mathrm{/}2)>0.0436$. This improves a previous result by Liu-Wu-Xi (2020), who showed \begin{align*} \mathop{\lim\inf}{x\rightarrow\infty}\frac{T_c(x)}{π(x)}&\geq\max\left(1-4ρ\left(\frac{1}{c}\right),1-c\right), \end{align*} for $0<c\le 1\mathrm{/}2$.