Irreducible constituents of products of characters

In this article, we give some results about the relationship between the structure of a finite solvable group G and the kernels of irreducible constituents of the character where chi is an irreducible character of G. For example, we obtain some results for the inequality which is known as the Taketa inequality, where dl(G) is the derived length of G and is the number of distinct irreducible complex character degrees of G