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By using new inequalities involving powers of rational functions, we give another proof of an important Spira's relation for the Riemann zeta-function |ζ(1 −s)| |ζ(s)| in the strip 0 < ℜs < 1/2, |ℑs| 12. Moreover, we establish a sufficient condition of the validity of the Riemann hypothesis in terms of the derivative of |ζ(s)| 2 with respect to ℜs and conjecture its necessity.

Another proof of Spira's inequality and its application to the Riemann hypothesis