A full-twist inequality for the $\nu^+$-invariant

Hom and Wu introduced a knot concordance invariant called $\nu^+$, which dominates many concordance invariants derived from Heegaard Floer homology. In this paper, we give a full-twist inequality for $\nu^+$. By using the inequality, we extend Wu's cabling formula for $\nu^+$ (which is proved only for particular positive cables) to all cables in the form of an inequality. In addition, we also discuss $\nu^+$-equivalence, which is an equivalence relation on the knot concordance group. We introduce a partial order on $\nu^+$-equivalence classes, and study its relationship to full-twists.