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Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials in which $f(r) = \sum_{j=0}^m c_j r^{2j}$ ?

PS: I asked this question on Math.SE but no one answered so it is posted here. This is a quite simple/straightforwad question (that a Freshman in math can fully understand) but it is surprising that till now, no one(me included) could answer it or at least give some idea.

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials(or an uniformly and absolutely converging power series) with only even powers, namely, $$f(r) = \sum_{j=0}^m c_j r^{2j} \quad(m\text{ can be}+\infty)\quad?$$

PS: I asked this question on Math.SE but no one answered so it is posted here. This is a quite simple/straightforwad question (that a Freshman in math can fully understand) but it is surprising that till now, no one(me included) could answer it or at least give some idea.

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials in which $f(r) = \sum_{j=0}^m c_j r^{2j}$ ?

PS: I asked this question on Math.SE but no one answered so it is posted here.

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials in which $f(r) = \sum_{j=0}^m c_j r^{2j}$ ?

PS: I asked this question on Math.SE but no one answered so it is posted here. This is a quite simple/straightforwad question (that a Freshman in math can fully understand) but it is surprising that till now, no one(me included) could answer it or at least give some idea.

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(B)$, where $B$ is the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials in which $f(r) = \sum_{j=0}^m c_j r^{2j}$ ?

PS: I asked this question on Math.SE but no one answered so it is posted here.

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$. Prove or disprove the following.

Given any positive integer $k$, $$\sup_{|\alpha|=k,x\in B} |D^\alpha f(x)| \leq \sup_{r < 1} \lvert f^{(k)}(r) \rvert,$$ where $\alpha$ is a multi-index and $D^\alpha f$ is the corresponding derivative of $f$. By $f^{(k)}(r)$, we mean the $k^{th}$ derivative of $f$ as a function of $r=|x|.$

I try some functions, taking second order derivatives, and the inequality holds for all of them. The case where $k=1$ is easy to prove but I can't prove for a general $k$.

Instead of a general smooth $f$, can we prove the assertion for polynomials in which $f(r) = \sum_{j=0}^m c_j r^{2j}$ ?

PS: I asked this question on Math.SE but no one answered so it is posted here.