Mathematical analysis establishes new bounds for the Wallis ratio, indicating sharper constants and inequalities.
In this paper, we establish several new bounds for the Wallis ratio W_n = (2n-1)!!/(2n)!! in the form of {1}{√π G_n(a,b)} , where G_n(a, b) = n + 1/4 + b/n+a with a > -1 and b > -5(a+1)/4 . In particular, we find that {1}{√π G_n(1/4, 1/32)} < W_n < {1}{√π G_n(a_0, 1/32)} for n ∈ N with the best constants 1/4 and a_0 = 5(16 - 5π)/16π , where the lower and upper bounds are the sharpest.
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Yang et al. (2025) studied this question.
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