Some results on generalized Cesàro stable of Janowski function

Let g(1) and g(2) be any two analytic functions defined in the unit disc which are normalized by the condition g(1)(0) = 1 = g(2)(0) and sigma(b-1,c)(n)(z) be the n th Ces & agrave;ro mean of type (b - 1,c) for 1 + b > c > 0. Then g(1) is generalized Ces & agrave;ro stable with respect to g(2), whenever sigma(b-1,c)(n)(g(1), z) / g(1)(z) < 1/g(2)(z) (z is an element of Delta, n is an element of N-0), where sigma(b-1,c)(n)(g(1), z) = 1/B-n & sum;(n )(j=0)B(n-j)b(j)z(j )= sigma(b-1,c)(n)(z)& lowast;g(z). The main aim of this article is to prove that ((Az + 1)/(Bz + 1))(eta) is generalized Ces & agrave;ro stable with respect to (1/(Bz + 1))(eta)) but not with respect to itself for -1 <= B < A <= 0 and 0 < eta <= 1. As an application, we obtain new and existing results on Ces & agrave;ro stability and stability.