Is this inequality on sums of powers of two sequences correct?

As a counter-example with $n=3$

$$a_1=6, a_2=42,a_3=52$$

$$b_1=12, b_2=22, b_3=66$$

Then

  • $a_1+a_2+a_3 = 100 = b_1+b_2+b_3$
  • $a_1^p+a_2^p+a_3^p \lt b_1^p+b_2^p+b_3^p$ for $p \gt 1$
  • $\sqrt{a_1}+\sqrt{a_2}+\sqrt{a_3} \lt 16.2 \lt \sqrt{b_1}+\sqrt{b_2}+\sqrt{b_3}$

though notice that $a_1^p+a_2^p+a_3^p \gt b_1^p+b_2^p+b_3^p$ for $0.851 \le p \lt 1$, and I suspect all such counterexamples with $n=3$ reverse the inequality in a small interval below $1$


Added

Perhaps a more interesting counter-example is

$$a_1=1, a_2=4,a_3\approx 5.3931524748543$$

$$b_1=2, b_2=2, b_3\approx 6.3931524748543$$

where $ 6.3931524748543$ is an approximation to the solution of $x^x=16 (x-1)^{x-1}$, so $\sum a_i = \sum b_i$ and $\prod a_i^{a_i} = \prod {b_i}^{b_i}$

This has $$a_1^p+a_2^p+a_3^p \le b_1^p+b_2^p+b_3^p$$ for all non-negative real $p$ (integer or not), and equality only when $p=0$ or $1$


I think in general, the claim may not hold without additional conditions. In particular, the following theorem may help obtain a counterexample (see: "An Inequality from Moment Theory" by G. Bennett; Positivity, 11 (2007), pp 231-238):

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Alternatively, for $n=2$ we have:

$$\begin{align} 0=(a_1+a_2)^2-(b_1+b_2)^2&=(a_1^2+a_2^2)-(b_1^2+b_2^2)+2a_1a_2-2b_1b_2 \\ &\leq 2a_1a_2-2b_1b_2 \end{align}.$$ So, $a_1a_2\geq b_1b_2$ and $2\sqrt{a_1a_2}\geq 2\sqrt{b_1b_2}$. Again, since $a_1+a_2=b_1+b_2$ we gather that $a_1+a_2+2\sqrt{a_1a_2}\geq b_1+b_2+2\sqrt{b_1b_2}$. Therefore, $(\sqrt{a_1}+\sqrt{a_2})^2\geq(\sqrt{b_1}+\sqrt{b_2})^2$ and thus $$\sqrt{a_1}+\sqrt{a_2}\geq \sqrt{b_1}+\sqrt{b_2}.$$

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Inequalities