On an inequality for the Riemann zeta-function in the critical strip

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |\zeta(1-s)| <= |\zeta(s)| in the strip 0< Re s<1/2,\ |\Im s| >= 12. Moreover, we establish a sufficient condition of the validity of the Riemann hypothesis in terms of the derivative with respect to Re s of |\zeta(s)|^2 and conjecture its necessity.