Strict power concavity of a convolution

We give a sufficient condition for the strict parabolic power concavity of the convolution in space variable of a function defined on $$\mathbb{R}$^n \times (0,+\infty)$andafunctiondefinedon$$\mathbb{R}$^n$. Since the strict parabolic power concavity of a function defined on $$\mathbb{R}$^n \times (0,+\infty)$ naturally implies the strict power concavity of a function defined on ${\mathbb{R}}^n$, our sufficient condition implies the strict power concavity of the convolution of two functions defined on ${\mathbb{R}}^n$. As applications, we show the strict parabolic power concavity and strict power concavity in space variable of the Gauss–Weierstass integral and the Poisson integral for the upper half-space.