Is there a way to do this with Mathematica shorthand notation?

You could use the operator form of Map instead:

SecondFn @ Map[FirstFn] @ {2, 3, 5}

SecondFn[{FirstFn[2], FirstFn[3], FirstFn[5]}]

comp = SecondFn @* Map[FirstFn];
comp @ {2, 3, 5}

SecondFn[{FirstFn[2], FirstFn[3], FirstFn[5]}]

FirstFn /@ a // SecondFn

SecondFn[{FirstFn[2], FirstFn[8], FirstFn[5]}]

A solution using only prefix notation is

List /* f @@ g /@ {x,y,z}
(* Out: f[{g[x],g[y],g[z]}] *)

Edit. A slightly nicer version (cf the order of the functions) uses Composition (@*) instead of the above RightComposition (/*):

f @* List @@ g /@ {x,y,z}
(* Out: f[{g[x],g[y],g[z]}] *)

This is still less elegant than the (rightfully) accepted answer, but it has the benefit of allowing one to use any function rather than just List.

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