On the Converse Stoltz Theorem
We consider the problem when the converse generalized Stoltz theorem is true. We obtain some conditions on the comparison sequence under which the complete or weak version of such a result is true on the family of all positive sequences. The same problem is solved for the important subclass of convex sequences which used in the theories of ultradifferential and entire functions. It is proved that our results cannot be improved. We construct some examples showing that the estimates found out for the class of convex sequences are better than the similar estimates for the class of all positive sequences.