Suppose $(a\_n)\_{n=1}^\infty$ and $(b\_n)\_{n=1}^\infty$ are two decreasing sequences of positive numbers such that 
$a\_1<b\_1$ and 
$$\sum\_{k=1}^\infty a^p\_k\leq \sum\_{k=1}^\infty b^p\_k<\infty$$ for $p=2,3,\cdots.$


Does it follow that 
$$\sum\_{k=1}^\infty a^p\_k\leq \sum\_{k=1}^\infty b^p\_k$$
for all real $p\geq 2$?


$\textbf{Remark 1:}$ Its transparent that the latter inequality cannot be violated by a non-integral sequence $(r\_n)$ that runs to infinity. Otherwise by continuity argument $a\_1=\max a\_n=\displaystyle\lim\_{n\to\infty}\|a\_n\|\_{r\_n}\geq\lim\_{n\to\infty}\|b\_n\|\_{r\_n} =\max b\_n=b\_1$!!



Extending inequality for $\ell^p$ from integer $p$ to real $p$
All we have to do is solve some equations. If I did it right, then: 


Let $q \approx 2.45$ satisfy $204q^4-580q^3+867q^2-4028=0$. Let
$$
r = \frac{578}{4323}q^3+\frac{2465}{12969}q^2-\frac{8056}{12969}q+\frac{41905}{2593}\approx 3.20 .
$$
Let
$$
a\_1 :=3 ,\qquad a\_2 := q,\qquad a\_3 := \frac{1}{2},
$$


$$
b\_1 := r,\qquad b\_2 := 2,\qquad b\_3 := 1.
$$


Then $f(p) := (b\_1^p+b\_2^p+b\_3^p) - (a\_1^p+a\_2^p+a\_3^p)$ is negative for $2\lt p\lt 3$, zero for $p=2,3$ and positive for $p\gt 3$ and $0 < p < 2$. In particular, $f(p) \ge 0$ for all natural numbers $p$. 


I did this by solving for $q,r$ in the two equations $f(2)=0, f(3)=0
$.


