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For an arbitrary operator T on Hilbert space, we study the maps ?Φ: f(T) → f(?T) and?Φ: f(T) → f(?T), where?T and?T are the Aluthge and Duggal transforms of T, respectively, and f belongs to the algebra Hol(σ(T)). We show that both maps are (contractive and) completely contractive algebra homomorphisms. As applications we obtain that every spectral set for T is also a spectral set for T and T, and also the inclusion W(f(T?)) - U W(f(T?)) - C W(f(T)) - relating the numerical ranges of f(T), f(T?), and f(T?).

Complete contractivity of maps associated with the Aluthge and Duggal transforms