Let $\phi, a \in C^{\infty}([0,1])$ and assume $a(0)=1$. Suppose that
$$
 \int\_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt =0 \qquad \text{for all $\tau \in \mathbb R$}.
$$
Does it follow that $\phi$ is a constant in a neighborhood of $t=0$?


If the answer to the above question is affirmative I have a follow up question as follows: Suppose additionally that $b \in C^{\infty}([0,1]$ with $b(0)=1$ and instead of the latter equation there holds
$$\Bigg |\int\_0^1 e^{\tau \,\phi(t)}\,a(t)\,dt\Bigg| \leq \frac{1}{|\tau|} \Bigg|\int\_0^1 e^{\tau \,\phi(t)}\,b(t)\,dt\Bigg|$$
Does it again follow that $\phi(t)$ is constant in a neighborhood of $t=0$?



On an asymptotic integral
No. E.g., let $\phi(t)=(t-1/2)^2$ (so that $\phi$ is even about $1/2$) and $a(t)=2(1/2-t)$ (so that $a$ is odd about $1/2$).


