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Let p_n be the nth prime number. In 2000, Papaitopol proved that the inequality p_1\cdots p_n>p_{n+1}^{n-\pi (n)} holds, for all n\geq 2, where \pi (x) is the prime counting function. In 2021, Yang and Liao tried to sharpen this inequality by replacing n-\pi (n) by n-\pi (n)+\pi (n)/\pi (\log n)-2\pi (\pi (n)), however there is a small mistake in their argument. In this paper, we exploit properties of the logarithm error term in inequalities of the type p_1\cdots p_n>p_{n+1}^{k(n,x)}, where k(n,x)=n-\pi (n)+\pi (n)/\pi (\log n)-x\pi (\pi (n)). In particular, we improve Yang and Liao estimate, by showing that the previous inequality at x=1.4 holds for all n\geq 21. [ABSTRACT FROM AUTHOR]