Some inequalities for the Chi square distribution function and the exponential functions
For \(u,x>0\) let \(\gamma(u)\) be the gamma function and let \(\gamma(u,x)=\int_ 0^ x e^{-t} t^{u-1} dt\) (incomplete Gamma function). Let \(P(u,x)=\gamma(u,x)/\Gamma(u)\). We prove several inequalities related to the function \(P\). Main ones are the following: \[ \begin{aligned} {{P(u+\beta,x)} \over {P^{1-\beta}(u,x)P^ \beta(u+1,x)}} &< (u+1)^ \beta {{\Gamma(u+1)} \over {\Gamma(u+\beta+1)}}, \qquad (\beta\in(0,1)),\\ {{P^{\beta-1}(u,x)P(u+\beta,x)} \over {P^ \beta(u+1,x)}} &> (u+1)^ \beta {{\Gamma(u+1)} \over {\Gamma(u+\beta+1)}}, \qquad (\beta\in(-u,0)\cup (1,+\infty)),\\ {{P(u,x)\Gamma(u+2,x)} \over {P^ 2(u+1,x)}} &< 1.\end{aligned} \] The first inequality is an improvement of an author's inequality [Publ. Elektrothehn. Fak., Univ. Beogr. Ser. Mat. 2, 89-94 (1991; Zbl 0748.60021)]. The second inequality for \(\beta=2\) and \(u\in \mathbb{N}\) reduces to an inequality for residual after the \(n\)th term of the Maclaurin expansion for \(e^ x\) [\textit{H. Alzer}, Arch. Math. 55, 462-464 (1990; Zbl 0723.26007)]. The third inequality gives rise to a conjecture that the function \(u\mapsto P(u,x)\) is log-concave.
scientific article; zbMATH DE number 446374