Inertia of some conditionally negative definite matrices

Let \(f:[0,\infty )\rightarrow [0,\infty )\) be a non-linear operator monotone function and \(g:\mathbb {R}\rightarrow [0,\infty )\) be a cnd function such that \(g(x)=0\) only at \(x=0.\) Let \(p_1,p_2,\ldots ,p_n\) be any distinct real numbers. Then the matrix \((f(g(p_i-p_j)))\) is cnd, see (Kapil et al. in Mediterr J Math 15:199, 2018). We aim to obtain the inertia of such matrices and several other cnd matrices which are arising from the functions having the Weierstrass factorization. These generalize and subsume several existing results.