Return to Question

Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold. How to find out whether the following true?

$\exists C,\epsilon>0$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-1-\epsilon}$

What I was thinking to try is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, see if it is finite. Well, it is (zero for $1+\epsilon \le a/2$). But that doesn't give a constant $C$ independent of $d$. Ideas would be appreciated.

EDIT: Replaced the condition as suggested by Yaakov Baruch's comment.

Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold (which can depend on $d$ in some mild way, say logarithmically). How to find out whether the following true?

$\exists C,\epsilon>0$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-1-\epsilon}$

What I was thinking to try is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, see if it is finite. Well, it is (zero for $1+\epsilon \le a/2$). But that doesn't give a constant $C$ independent of $d$. Ideas would be appreciated.

EDIT: Replaced the condition as suggested by Yaakov Baruch's comment.

Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold. How to find out whether the following true?

$\exists C,\epsilon>0$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-1-\epsilon}$

What I was thinking to try is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, see if it is finite. Well, it is (zero). But that doesn't give a constant $C$ independent of $d$. Ideas would be appreciated.

EDIT: Replaced the condition as suggested by Yaakov Baruch's comment.

Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold. How to find out whether the following true?

$\exists C,\epsilon>0$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-1-\epsilon}$

What I was thinking to try is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, see if it is finite. Well, it is (zero for $1+\epsilon \le a/2$). But that doesn't give a constant $C$ independent of $d$. Ideas would be appreciated.

EDIT: Replaced the condition as suggested by Yaakov Baruch's comment.

Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold. How to find out whether the following true?

$\exists C,\epsilon>0,\exists b \ge 1+\epsilon$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-b}$

What I was thinking to try is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, see if it is finite. Well, it is (zero). But that doesn't give a constant $C$ independent of $d$. Ideas would be appreciated.

EDIT: Replaced the $b\ge 2$ condition by $b\ge 1+\epsilon, \epsilon > 0$. (It is a condition needed in the original problem which this one would help solving.)

Let $t,d,a \ge 1$, of which $d$ can be unbounded, $a$ can be constrained to be larger than some threshold. How to find out whether the following true?

$\exists C,\epsilon>0$ constants independent of $d$ s.t. $\forall t>d\cdot C$, it holds that:

$\left(\frac{t}{d}\right)^{d/2}\left(\frac{d+a}{t+a}\right)^{(d+a)/2} \le C\cdot t^{-1-\epsilon}$

What I was thinking to try is to look at the limit of the fraction of the LHS/RHS when $t\rightarrow \infty$, see if it is finite. Well, it is (zero). But that doesn't give a constant $C$ independent of $d$. Ideas would be appreciated.

EDIT: Replaced the condition as suggested by Yaakov Baruch's comment.