On classes of analytic functions related to conic domains

We introduce classes of analytic functions related to conic domains, using a new linear multiplier fractional differential operator Dλn,α (n∈N0={0,1,…}n∈N0={0,1,…}, 0⩽α<10⩽α<1, λ⩾0λ⩾0), which is defined asD0f(z)=f(z),D0f(z)=f(z),Dλ1,αf(z)=(1−λ)Ωαf(z)+λz(Ωαf(z))′=Dλα(f(z)),Dλ2,αf(z)=Dλα(Dλ1,αf(z)),⋮Dλn,αf(z)=Dλα(Dλn−1,αf(z)), whereΩαf(z)=Γ(2−α)zαDzαf(z), and Dzα is the known fractional derivative. Basic properties of these classes are studied, such as inclusion relations and coefficient bounds. Various known or new special cases of our results are also pointed out.