Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions
Let \(\Gamma(a)\) and \(\Gamma(a;z)\) denote the gamma and the incomplete gamma functions, respectively. Some monotonicity results for the gamma function are proved and a lower bound for the function \[ \int_0^xe^{-t^p}dt=[\Gamma(1/p)-\Gamma(1/p;x^p)]/p \] with \(p>1\) for \(0<x<[9(3p+1)/4(2p+1)]^{1/p}\) is extended to \(x>0\).
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