  Some differential implications of classical Marx-Strohh\"acker theorem are
extended for multivalent functions. These results are also generalized for
functions with fixed second coefficient by using the theory of first order
differential subordination which in turn, corrects the results of Selvaraj and
Stelin [On multivalent functions associated with fixed second coefficient and
the principle of subordination, Int. J. Math. Anal. {\bf 9} (2015), no.~18,
883--895].
