Inequalities for Laguerre functions
If \(\Phi\) is the confluent hypergeometric function, then the Laguerre functions are defined by \[ L_{\nu}^{\mu}(z)= {\mu +\nu \choose \nu} \Phi (-\nu, 1+\mu;z). \] When \(\nu=n\), a nonnegative integer, \(L_{\nu}^{\mu}\) becomes a polynomial in \(z\) and it has the following nice inequality for \(z=x>0\): \(| L_{n}^{\mu}(x)| \leq {\mu +n \choose \mu} e^{x/2}.\) The author derives several new inequalities both for Laguerre functions \(L_{\nu}^{\mu}(x)\) and for Laguerre polynomials \(L_{n}^{\mu}(x)\). These results do not seem to be available in the literature. Some special inequalities are also observed for confluent hypergeometric functions. In fact, the author obtains an upper bound for the function \(e^{-x}| L_{\nu}^{\mu}(x)| \) whenever \(x>0\), \(\text{Re }\mu >-1/2\) and \(\text{Re }(\mu+\nu +1)>0\). Similarly, a lower bound for the function \(e^{\lambda x}| L_{\nu}^{\mu}(x)| \) is obtained if \(\text{Re }\mu >-1\), \(\text{Re }\nu >0\), \(\mu+\nu +1>0\), \(\lambda \in (0,1)\) and if \(x>0\) is sufficiently large.
confluent hypergeometric functions and fractional integration
scientific article; zbMATH DE number 1139491