Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ where $M$ is a positive constant number. Does this imply that $f$ is identically zero?

The motivations comes from section 3 page 42 of the following paper, "Discussions and further Reseaches":

http://mcs.qut.ac.ir/article_243944.html

In a similar way we generalize the question as follows:

Let $\Delta$ be the Laplace operator associated to a Riemannian manifold. Assume that $f:M\to \mathbb{R}$ is a smooth function which satisfies $|\Delta(f)(x)|\leq M |f(x)|,\quad \forall x\in M$ where $M$ is positive constant. Does this imply that $f$ is identically zero(at least locally around the point $p$)?

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ where $M$ is a positive constant number. Does this imply that $f$ is identically zero?

The motivations comes from section 3 page 42 of the following paper, "A Dynamical Approach to Quasi Analytic type Problems

In a similar way we generalize the question as follows:

Let $\Delta$ be the Laplace operator associated to a Riemannian manifold. Assume that $f:M\to \mathbb{R}$ is a smooth function which satisfies $|\Delta(f)(x)|\leq M |f(x)|,\quad \forall x\in M$ where $M$ is positive constant. Does this imply that $f$ is identically zero(at least locally around the point $p$)?

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ where $M$ is a constant. Does this imply that $f$ is identically zero?

The motivations comes from section 3 page 42 of the following paper, "Discussions and further Reseaches":

http://mcs.qut.ac.ir/article_243944.html

In a similar way we generalize the question as follows:

Let $\Delta$ be the Laplace operator associated to a Riemannian manifold. Assume that $f:M\to \mathbb{R}$ is a smooth function which satisfies $|\Delta(f)(x)|\leq M |f(x)|,\quad \forall x\in M$ where $M$ is positive constant. Does this imply that $f$ is identically zero(at least locally around the point $p$)?

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ where $M$ is a positive constant number. Does this imply that $f$ is identically zero?

The motivations comes from section 3 page 42 of the following paper, "Discussions and further Reseaches":

http://mcs.qut.ac.ir/article_243944.html

In a similar way we generalize the question as follows:

Let $\Delta$ be the Laplace operator associated to a Riemannian manifold. Assume that $f:M\to \mathbb{R}$ is a smooth function which satisfies $|\Delta(f)(x)|\leq M |f(x)|,\quad \forall x\in M$ where $M$ is positive constant. Does this imply that $f$ is identically zero(at least locally around the point $p$)?