New extensions for classes of analytic functions associated with close-to-convex and starlike of order α

Let A be the class of functions f(z) with f(0) = 0 and f'(0) = 1 in the open unit disk U. For f(z) is an element of A, f(z) is said to be in the class A(beta(1), beta(2), beta(3); lambda) if it satisfies vertical bar beta(1)z(f(z)/z)' + beta(2)z(2)(f(z)/z)'' + beta(3)z(3)(f(z)/z)'''vertical bar <=lambda (z is an element of U) for some complex numbers beta(1), beta(2) and beta(3), and for some real lambda > 0. Some interesting properties for f(z) is an element of A(beta(1), beta(2), beta(3); lambda) associated with close-to-convex functions and starlike functions of order alpha are discussed. (C) 2011 Elsevier Ltd. All rights reserved.