Regularity bounds for curves by minimal generators and Hilbert function

Let $rho_C$ be the regularity of the Hilbert function of a projective curve $C$ in $mathbb P^n_K$ over an algebraically closed field $K$ and $alpha_1,...,alpha_{n-1}$ be minimal degrees for which there exists a complete intersection of type $(alpha_1,...,alpha_{n-1})$ containing the curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $max{rho_C+1,alpha_1+...+alpha_{n-1}-(n-2)}$. We study and, for space curves, refine the above bound providing several examples.