Conditionally Negative Definite Functions

Abstract

Let $$f:[0,\infty )\rightarrow [0,\infty )$$ be an operator monotone function and $$g: \mathbb {R}\rightarrow [0,\infty )$$ be a conditionally negative definite(in short cnd) function. We obtain that $$f\circ g:\mathbb {R}\rightarrow [0,\infty )$$ is also conditionally negative definite. This generalizes and subsumes several existing results. A versatile direct connection between cnd functions and functions having Weierstrass factorization is established and consequently a reasonable account for cnd functions is presented.