A note on Turán type and mean inequalities for the Kummer function
The main result is the following: Suppose \(a,b> 0\). Then for any \(n\in\mathbb{N}\) with \(a,b\geq n-1\) \[ \phi(a,a+ b,x^2)> \phi(a+n, a+ b,x)\phi(a- n,a+ n,x) \] for all nonzero \(x\in\mathbb{R}\). Moreover, these expressions coincide with value 1 when \(x= 0\) and asymptotically for any \(x\) when \(b\to\infty\) (\(\phi\) is the Kummer confluent hypergeometric function). Few more results are obtained by utilizing the above-mentioned result. A possible application to modeling credit risk is indicated.
Kummer confluent hypergeometric function
scientific article; zbMATH DE number 5362150
scientific article