This thesis deals with analytic functions as well as multivalent functions de-
�ned on the unit disk U. In most cases, these functions are assumed to be normalized,
either of the form
f(z) = z +
1X
k=2
akzk;
or
f(z) = zp +
1X
k=1
ak+pzk+p;
p a �xed positive integer. Let A be the class of functions f with the �rst normalization,
while Ap consists of functions f with the latter normalization. Five
research problems are discussed in this work.
First, let f(q) denote the q-th derivative of a function f 2 Ap. Using the theory
of di�erential subordination, su�cient conditions are obtained for the following
di�erential chain to hold:
f(q)(z)
�(p; q)z
