Three equivalent inequalities with $ \rho :\ \mathbb{R}\to [0,+\infty ),\ \int_{-\infty }^{+\infty } \rho(t) \mathrm{d}t=1$

Question

Let $ \rho :\ \mathbb{R}\to [0,+\infty ),\ \int_{-\infty }^{+\infty } \rho(t) \mathrm{d}t=1,$ Prove that the following are equivalent: \begin{align*} 1) \ & \exists M,c>0\text{ s.t. }\int_{|t|>x} \rho (t) \mathrm{d}t \leq Me^{-cx},\ \forall x\geq0;\\ 2) \ & \exists a>0\text{ s.t.}\ \int_{-\infty }^{+\infty } e^{a|t|}\rho (t) \mathrm{d}t <\infty; \\ 3) \ & \exists K>0 \text{ s.t. }\left[\int_{-\infty }^{+\infty } |t|^p\rho (t) \mathrm{d}t\right]^{\frac{1}{p} }<Kp,\ \forall p\in \mathbb{N}.\\ \end{align*} I have achieved proving $1)\Leftrightarrow 2)$ and $1) \Rightarrow 3)$ with mainly Integration by parts, but I have no idea how to prove $3) \Rightarrow 1)$ or $3) \Rightarrow 2)$, does anyone know how?

Answer 1

From the power series of exp (and monotone convergence theorem), (3) implies $\int \exp(|t|/2Ke) \rho(t) dt \le \sum_{p=1}^\infty \frac{p^p}{p!(2e)^{p}} $ By Stirling $p!\sim Cp^{p+1/2} e^{-p}$ the sum is convergent. This is (2) with $\alpha=1/2Ke$.