On the Coincidence Theorem

We are proving Coincidence theorem due to Walsh for the case when the total degree of a polynomial is less than the number of arguments. Also, the following result has been proven: if p (z) is a complex polynomial of degree n, then closed disk D that contains at least n-1 of its zeros (counting multiplicity) contains at least -2k+12 zeros of its k-th derivative, provided that the arithmetical mean of these zeros is also centre of D. We also prove a variation of the classical composition theorem due to Szegö.