$x,y,z$ are in G.P
$\large\frac{y}{x}=\frac{z}{y}=$$\log _e\big(\large\frac{y}{x}\big)=$$\log_e\big(\large\frac{z}{y}\big)$
$ln \;y-ln \;x=ln\; z-ln\; y$
$ln\;x,ln\; y,ln \;z$ are in A.P
$\large\frac{1}{1+ln\;x}.\frac{1}{1+ln\;y}.\frac{1}{1+ln\;z}$ are in H.P
Hence (c) is the correct answer.