It is known that Teichmüller distance ($d\_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d\_{WP}$) is not complete. 


See for example the article 


Wolpert, Scott. Noncompleteness of the Weil-Petersson metric for Teichmüller space. Pacific J. Math. 61 (1975), no. 2, 573--577
,


in which Wolpert constructs an the explicit path on the Teichmüller space, which diverges in $d\_{Teich}$, but converges in $d\_{WP}$.


My question is : is it true in general that there is a positive constant $C$ such that $d\_{WP} < C d\_{Teich}$?



Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?
Yes, this is a [result of Michele Linch, 1974.](http://www.math.uchicago.edu/~mduchin/viewing/thesis/linch-twometrics.pdf)


