Region of variability for certain classes of univalent functions satisfying differential inequalities

For complex numbers alpha, beta and M is an element of R with 0 < M <= vertical bar alpha vertical bar and vertical bar beta vertical bar <= 1, let B (alpha, beta, M) be the class of analytic and univalent functions f in the unit disk D with f(0) = 0, f '(0) = alpha and f ''(0) M beta satisfying vertical bar zf ''(z) vertical bar <= M, z is an element of D. Let P(alpha, M) be the another class of analytic and univalent functions in D with f(0)=0, f '(0)=alpha satisfying Re(zf ''(z)) > -M, z is an element of D, where alpha is an element of C\{0}, 0 < M <= 1/log 4. For any fixed z(0) is an element of D, and lambda is an element of D we shall determine the region of variability V(j) (j=1, 2) for f '(z(0)) when f ranges over the class S(j) (j=1, 2), where S(1) = {f is an element of B(alpha, beta, M) : f'''(0) = M(1 - vertical bar beta vertical bar(2))lambda} and S(2) = {f is an element of P(alpha, M) : f ''(0) = 2M lambda}. In the final section we graphically illustrate the region of variability for several sets of parameters.