If the arcs bend the other way, then "yes";
otherwise there is an obvious counterexample. 

Note that the statement holds if $c\in [a,b]$.
Moreover, if the arcs $ac$ and $bc$ are tangent to the arc $ab$ at $a$ and $b$ respectively, then equality holds.
Furthermore, note that we can assume that $ac$ and $bc$ are tangent to $ab$.

On the following diagram, if arcs look tangent, then they are.

[![enter image description here][1]][1]

Observe that 
$|ab|=|ac|+|cx|+|xb|\geqslant |ac|+|cb|$, where $|{*}{*}|$ denotes the length of the arc.


  [1]: https://i.sstatic.net/J9n02.png