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Inequality on $t_ν(K)$ defined by Livingston and Naik and its applications

Let $D_+(K,t)$ denote the positive $t$-twisted double of $K$. For a fixed integer-valued additive concordance invariant $ν$ that bounds the smooth four genus of a knot and determines the smooth four genus of positive torus knots, Livingston and Naik defined $t_ν(K)$ to be the greatest integer $t$ such that $ν(D_+(K,t)) = 1$. Let $K_1$ and $K_2$ be any knots then we prove the following inequality : $t_ν(K_1) + t_ν(K_2) \leq t_ν(K_1 \# K_2) \leq min(t_ν(K_1) - t_ν(-K_2), t_ν(K_2) - t_ν(-K_1)).$ As an application we show that $t_τ(K) \neq t_s(K)$ for infinitely many knots and that their difference can be arbitrarily large, where $t_τ(K)$ (respectively $t_s(K)$) is $t_ν(K)$ when $ν$ is Ozváth-Szabó invariant $τ$ (respectively when $ν$ is normalized Rasmussen $s$ invariant).