Symbolic minimum value

I have the following polynomial expression v1[z]:

-((5 L^3 q z)/(12 J Y)) + (L^2 q z^2)/(4 J Y) - (L q z^3)/(12 J Y) 

and I would like to symbolically evaluate its minimum in 0<z<L, with q>0, L>0, J>0 and Y>0. How can I do this?

I have tried the following numerical way:

minv1 = Minimize[{v1[z] /. q -> 1 /. L -> 1 /. J -> 1 /. Y -> 1,z > 0, z < 1}, z ]