Some applications of differential subordination on analytic functions

Let R-lambda(n)(s; h) denote the class of analytic functions f = {f(1), f(2) ,..., fs} on the unit disk U satisfying z[D(lambda)(n)fi(z)]'/1/s Sigma(s)(j=1) D-lambda(n) fj(z) < h(z) ((f)i is an element of A; i = 1, 2, ..., s; z is an element of U), where z(-1) Sigma(s)(j=1) D-lambda(n) f(j)(z) not equal 0 and D-lambda(n) (n is an element of N-0; lambda >= 0), is the extended Ruscheweyh derivative defined by D-lambda(n) f(z) = z + Sigma(infinity)(k=2)[1 + lambda (k - 1)]C(n,k)a(k)z(k), and h is convex univalent in U with h(0) = 1. Also let F = {F-1, F-2 ,..., F-s}, where F-i(z) = gamma+1/z(gamma) integral(z)(0) t(gamma-1) f(i)(t)dt (gamma is an element of C; Re(gamma) > 0; i=1,2, ..., s). It is proved that F is an element of R-lambda(n)(s;h) whenever f is an element of R-lambda(n)(s;h) and also that R-lambda(n+1)(s;h) subset of R-lambda(n)(s; h) Three more such classes denoted by Q(lambda)(n)(s; h), H-lambda(n)(s; a; h) and R-lambda(n)(s; alpha, h) are introduced and studied here by subordination methods and convolutions.