Abstract
Let A(n) denote the class of analytic functions f in the open unit disk U = {z : vertical bar z vertical bar < 1} normalized by f (0) = f' (0) - 1 = 0. In this paper, we introduce and study the classes S-n,S- mu(gamma, alpha, beta, lambda, (sic)) and R-n,R- mu(gamma, alpha, beta, lambda, (sic)) of functions f is an element of A(n) with (mu)z(D-lambda((sic)+2) (alpha, omega) f (z))' + (1 - mu)z(D-lambda((sic)+1) (alpha, omega) f (z))' not equal 0 and satisfy some conditions available in literature, where f is an element of A(n), alpha, omega, lambda, mu >= 0, (sic) is an element of N boolean OR{0}, z is an element of U, and D-lambda(m) (alpha, omega) f (z) : A -> A, is the linear fractional differential operator, newly defined as follows
D-lambda(m) (alpha, omega) f (z) = z + Sigma(infinity)(k = 2) a(k) (1 + (k - 1)lambda omega(alpha))(m) z(k).
Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integralmeans inequalities and inclusion for the functions included in the classes S-n,S- mu(gamma, alpha, beta, lambda, (sic), omega) and R-n,R- mu(gamma, alpha, beta, lambda, (sic), omega) are given.