Conditionally Negative Definite Functions

  • Yogesh KapilEmail author
  • Rajinder Pal
  • Anchal Aggarwal
  • Mandeep Singh


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.


Conditionally negative definite Positive definite Infinitely divisible Operator monotone 

Mathematics Subject Classification

Primary 42A82 42B99 



The authors are grateful to a referee for the valuable comments and suggestions in preparing the final version.


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Authors and Affiliations

  1. 1.Department of MathematicsSant Longowal Institute of Engineering and TechnologyLongowalIndia

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