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Let 
$\mathcal {P}$ be the set of primes and 
$\pi (x)$ the number of primes not exceeding x. Let 
$P^+(n)$ be the largest prime factor of n, with the convention 
$P^+(1)=1$, and 
$ T_c(x)=\#\{p\le x:p\in \mathcal {P},P^+(p-1)\ge p^c\}. $ Motivated by a conjecture of Chen and Chen [‘On the largest prime factor of shifted primes’, Acta Math. Sin. (Engl. Ser.) 33 (2017), 377–382], we show that for any c with 
$8/9\le c<1$, 
$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\le 8(1/c-1), \end{align*} $$
which clearly means that 
$$ \begin{align*} \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0 \quad \text{as } c\rightarrow 1. \end{align*} $$
