In this paper, we show that if λ 1 , λ 2 , λ 3 , λ 4 , λ 5 are nonzero real numbers not all of the same sign, η is real, 0 < σ < 1 720 , and at least one of the ratios λ i /λ j (1 ≤ i < j ≤ 5) is irrational, then the inequality |λ
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