Note on shifted primes with large prime factors

We denote by $P^{+}(n)$ the largest prime factor of the integer $n$. In 1935, Erd\H os studied the quantity $T_c(x)$ defined by $$T_c(x)=\mid \{p\le x:P^{+}(p-1)\ge p^c\}\mid ,$$ and he proved $$\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{T_c(x)}{\pi (x)}\to 0,\text{as }c\to 1.$$ Recently, Ding gave a quantitative form of Erd\H os’ result, showing that $$\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{T_c(x)}{\pi (x)}\le 8(c^{-1}-1)\mathrm{.}$$ holds for $8\mathrm{/}9<c<1$. In this paper, we improve Ding’s upper bound to $$\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{T_c(x)}{\pi (x)}\le -\frac{7}{2}\log c$$ for $e^{-\frac{2}{7}}<c<1$.