On the number of distinct values of a class of functions with finite   domain
  By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. The upper bound is also connected to triangular numbers, and a slight improvement to this bound could be obtained by resolving a problem on them. In the final section, we consider implications of our bounds in various settings, including finite fields, coding theory and additive combinatorics. In particular, we obtain the first non-trivial upper bound for the image set of a planar function over a finite field; this bound is better than the bound implied by the Dembowski-Ostrom conjecture. 
Robert Coulter and Steven Senger
math.CO math.GR math.NT