In this paper, we develop the injective and interpolative procedures to generate holomorphic Lipschitz ideals \(\mathcal {A}^{\mathcal {H}L_0}\) . Based on the injective procedure of Pietsch for operator ideals, the concept of injective hull of \(\mathcal {A}^{\mathcal {H}L_0}\) , denoted by \((\mathcal {A}^{\mathcal {H}L_0})^{\textrm{inj}}\) , is introduced and characterized in terms of a domination property. A description of the closed injective hull of \(\mathcal {A}^{\mathcal {H}L_0}\) is established in terms of an Ehrling-type inequality. Building upon the interpolative procedure of Matter for operator ideals, we also present the concept of interpolative hull of \(\mathcal {A}^{\mathcal {H}L_0}\) , denoted by \((\mathcal {A}^{\mathcal {H}L_0})_{\sigma }\) for \(\sigma \in [0,1)\) . We prove that \((\mathcal {A}^{\mathcal {H}L_0})_{\sigma }\) is an injective holomorphic Lipschitz ideal which is located between the injective hull and the closed injective hull of \(\mathcal {A}^{\mathcal {H}L_0}\) . We describe the (closed) injective hull of holomorphic Lipschitz ideals generated by composition and duality with Banach operator ideals \(\mathcal {A}\) , and these descriptions are applied to concrete examples of holomorphic Lipschitz ideals.
