Inequalities for confluent hypergeometric functions of two and three variables (Q1209783)

Two-sided inequalities for Appell's hypergeometric functions \(F_ 1\), \(F_ 2\), \(F_ 3\) were obtained by \textit{Y. L. Luke} [J. Approximation 11, 73-84 (1970; Zbl 0276.33009)]. The first author and \textit{J. P. Arya} obtained inequalities for confluent hypergeometric functions of two variables \(\phi_ 1\), \(\phi_ 2\), \(\phi_ 3\), \(\Xi_ 1\), \(\Xi_ 2\), \(\psi_ 1\), \(\psi_ 2\) [Indian J. Pure Appl. Math. 13, 491-500 (1982; Zbl 0493.33005)]. In this paper, inequalities of the type \[ e^{-\theta y}(1+\theta x)^{-b}<\phi_ 1(a,b;c;-x,-y)<1-\theta +\theta e^{- y}(1+x)^{-b}, \] where \(\theta =a/c\), \(c>a>0\), \(b>0\), \(x>0\), \(y>0\), are obtained for confluent hypergeometric functions of two and three variables through Euler and Laplace type integral representations. The bounds obtained in this paper hold in wider domains and also for both positive and negative real arguments.