Note on shifted primes with large prime factors

For any $0<c<1$ let Tc(x)=|≤ x: p∈ P, P⁺(p-1)≥ pᶜ\|, where P is the set of primes and P⁺(n) denotes the largest prime factor of n. Erd os proved in 1935 that x→ ∞Tc(x)/π(x)→ 0, as~c→ 1, where $π(x)$ denotes the number of primes not exceeding x. Recently, Ding gave a quantitative form of Erd os' result and showed that for $8/9< c<1$ we have x→ ∞Tc(x)/π(x)≤ 8(c⁻¹-1). In this article, Ding's bound is improved to x→ ∞Tc(x)/π(x)≤ -7/2log c for e-2/7< c<1.