The following result can be found in [this article](https://www.researchgate.net/publication/50887744_Blow-up_and_Large_Time_Behavior_of_Solutions_of_a_Weakly_Coupled_System_of_Reaction-Diffusion_Equations)


*(Jensen’s inequality)* Let $v = v(x, t)$ be any nonnegative function.
Then it holds that, for all $t > 0$,
$$[S(t)v(s)]^q \leq S(t)v^q(s)$$
if $q \geq 1,$ and
$$[S(t)v(s)]^q \geq S(t)v^q(s)$$
if $q \leq 1.$


The demonstration is omitted and I do not know how these inequalities are obtained. I was wondering if they remain true if $ S (t) $ is Schrödinger semigroup? Because the author uses them assuming that $S(t)$ is the heat semigroup.
If true, can you give me a reference for consultation?



Jensen’s inequality for Heat semigroup is valid for Schrödinger semigroup?
$$S(t)v(x)=\int\_E k(t,x,y)v(y) dy=\int\_E k(t,x,y)^{\frac1q+\frac{1}{q'}}v(y) dy \le\left (\int\_E k(t,x,y)v(y)^q dy\right )^{\frac1q} \left (\int\_E k(t,x,y) dy \right )^{\frac{1}{q'}}.$$
What you need is $\int\_E k(t,x,y) dy \le 1$.


