  Thue equations and their relative and inhomogeneous extensions are well known
in the literature. There exist methods, usually tedious methods, for the
complete resolution of these equations. On the other hand our experiences show
that such equations usually do not have extremely large solutions. Therefore in
several applications it is useful to have a fast algorithm to calculate the
"small" solutions of these equations. Under "small" solutions we mean the
solutions, say, with absolute values or sizes $\leq 10^{100}$. Such algorithms
were formerly constructed for Thue equations, relative Thue equations. The
relative and inhomogeneous Thue equations have applications in solving index
form equations and certain resultant form equations. It is also known that
certain "totally real" relative Thue equations can be reduced to absolute Thue
equations (equations over $\Bbb Z$).
  As a common generalization of the above results, in our paper we develop a
fast algorithm for calculating "small" solutions (say with sizes $\leq
10^{100}$) of inhomogeneous relative Thue equations, more exactly of certain
inequalities that generalize those equations. We shall show that in the
"totally real" case these can similarly be reduced to absolute inhomogeneous
Thue inequalities. We also give an application to solving certain resultant
equations in the relative case.
