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In the range $1<x<3/2$, the known bounds $$-\frac{\zeta'(x)}{\zeta(x)}<\frac{1}{x-1}<\zeta(x)$$ imply that $$-\frac{\zeta'(x)}{\zeta^2(x)}<1<\frac{1}{x-1/2}.$$ In the range $3/2\leq x<2$, we have that $$-\frac{\zeta'(x)}{\zeta^2(x)}\leq-\frac{\zeta'(3/2)}{\zeta^2(3/2)}<\frac{2}{3}<\frac{1}{x-1/2}.$$ In the range $2\leq x<3$, we have that $$-\frac{\zeta'(x)}{\zeta^2(x)}\leq-\frac{\zeta'(2)}{\zeta^2(2)}<\frac{2}{5}<\frac{1}{x-1/2}.$$ In the range $3\leq x$, the known bound $\psi(r)<1.04 r$ for the second Chebyshev function implies that $$-\frac{\zeta'(x)}{\zeta^2(x)}<-\frac{\zeta'(x)}{\zeta(x)}=x\int_2^\infty\frac{\psi(r)}{r^{x+1}}\,dr<\frac{2.08x}{x-1}2^{-x}<\frac{1}{x-1/2}.$$ The quoted bounds can be found in Delange (1987), Montgomery-Vaughan: Multiplicative number theory I, and Rosser-Schoenfeld (1962).