A certain q-Ruscheweyh type derivative operator and its applications involving multivalent functions

In the present paper, by using the concept of convolution and q -calculus, we define a certain q -derivative (or q -difference) operator for analytic and multivalent (or p -valent) functions. This presumably new q -derivative operator is an extension of the known q -analogue of the Ruscheweyh derivative operator. We also give some interesting applications of this q -derivative operator for multivalent functions by using the method of differential subordination. Relevant connections with a number of earlier works on this subject are also pointed out.