Apparatus is provided that includes a valve prosthesis for attachment to a native valve complex of a subject. The prosthesis is configured to assume a compressed delivery state and an uncompressed implantation state. The prosthesis includes a support frame, which is shaped so as to define an upstream inlet having upstream-most portions that are tapered in an upstream direction toward a central longitudinal axis of the prosthesis when the prosthesis assumes the compressed delivery state. The prosthesis further includes a flexible prosthetic heart valve component, coupled to the support frame. Other embodiments are also described.