The function that you have **is convex** for unitarily invariant norms, but for the (basis dependent) elementwise absolute value, it can clearly break as a trivial counterexample below shows.

\begin{equation*}
 X = \begin{pmatrix}
 2 & 0 & 0\\\\
 0 & 1 & 0\\\\
 0 & 0 & 1
\end{pmatrix},\qquad Y = \begin{pmatrix}
 10  &    9 &    5\\\\
     9 &   10 &    5\\\\
     5 &    5 &    4
\end{pmatrix}
\end{equation*}

Now, simply define $Z = 0.5(X+Y)$, and consider $g(H)=\|H^{-1}\|_1$ as your function. 
Then, we see that $g(Z) = 3.1692$, while $0.5(g(X)+g(Y)) = 3$, clearly violating convexity.