Inequalities for the chi square distribution function
For \(x,a>0\), let \(\Gamma(x)= \int^\infty_0 e^{-t} t^{x-1} dt\) and \(\gamma(x,a)= \int^a_0 e^{-t} t^{x-1} dt\) denote the gamma function and the incomplete gamma function, respectively, and set \(P_a(x)= \gamma(x,a)/ \Gamma (x)\). \(P_a(x)\) has a close connection with probability theory, namely if \(\xi_1, \dots, \xi_n\) are i.i.d. random variables, each following a standard normal distribution, then the sum \(\chi_n^2 =\xi^2_1 +\cdots +\xi^2_n\) follows the chi-square distribution, and the probability that \(\chi^2_n\leq a\) is equal to \(P_{a/2} (n/2)\). Further, for \(u,v>0\) and \(\alpha \in(0,1)\), define \(M_t (u,v; \alpha)= (\alpha u^t+(1-\alpha) v^t)\) if \(t\neq 0\), \(=u^\alpha v^{1-\alpha}\) if \(t=0\), \(=u\wedge v\) if \(=-\infty\), \(=u\vee v\) if \(t=\infty\). The main result is that, for \(x,y>0\) \((x\neq y)\) and \(\alpha \in (0,1)\), the inequalities \[ M_r\bigl( P_a(x), P_a(y); \alpha\bigr)< P_a\bigl(\alpha x+(1-\alpha) y\bigr) <M_s \bigl( P_a(x), P_a(y); \alpha\bigr) \] are valid for all \(a > 0\) if and only if \(r\leq 0\) and \(s=\infty\). A by-product is that for \(a > 0\), \[ {n+1\over n+2}<{I_{n-1}(a) I_{n+1} (a)\over I_n(a)^2} <1,\quad n\geq 1, \] where \(I_n(a)=e^a-\sum^n_{k=0} a^k/k!\).
scientific article; zbMATH DE number 1215668