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

Let AA be the class of functions f(z)f(z) with f(0)=0f(0)=0 and f′(0)=1f′(0)=1 in the open unit disk UU. For f(z)∈Af(z)∈A, f(z)f(z) is said to be in the class A(β1,β2,β3;λ)A(β1,β2,β3;λ) if it satisfies |β1z(f(z)z)′+β2z2(f(z)z)″+β3z3(f(z)z)‴|≦λ(z∈U) for some complex numbers β1β1, β2β2 and β3β3, and for some real λ>0λ>0. Some interesting properties for f(z)∈A(β1,β2,β3;λ)f(z)∈A(β1,β2,β3;λ) associated with close-to-convex functions and starlike functions of order αα are discussed.