A problem of Ghorbani (Q2276536)

Suppose that \(a_{1},\ldots,a_{N}\) and \(b_{1},\ldots,b_{N}\) are non-negative numbers and that \((1)\;\;\sum_{n=1}^{N}a_{n}^{p}\leq \sum_{n=1}^{N}b_{n}^{p} \) whenever \(p\) is a positive integer. A problem of Ghorbani asks whether we can deduce that \(\sum_{n=1}^{N}\sqrt{a_{n}}\leq \sum_{n=1}^{N}\sqrt{b_{n}} \) holds also for \(p=1/2\) if there is equality in (1) when \(p=1\) and \(p=2.\) The author proves that the answer is positive when \(N\leq 3.\) Moreover, for \(N=3\), if (1) is valid for \(p=\infty\) and it is also valid, with equality, when \(p=1\) and \(p=2,\) then it is valid for \(p=1/2\). Of course, for \(p=\infty\), the sum in (1) is replaced by \(\max \{a_{1},\ldots,a_{N}\}\). For \(N=4\), a counterexample is provided by the 4-tuples \(\left( 2,2,7,7\right)\) and \(\left( 1,4,5,8\right)\). To produce counterexamples in higher dimensions we have to use the transformation \(\left( a,b,c,d\right) \rightarrow \left(a,b,c,d,1,\ldots,1\right)\). The author proves also that there exist positive numbers \(a,b,c,d,w,x,y,z\) satisfying \(a^{2}+b^{2}+c^{2}+d^{2}=w^{2}+x^{2}+y^{2}+z^{2}\) such that the inequality \(a^{p}+b^{p}+c^{p}+d^{p}<w^{p}+x^{p}+y^{p}+z^{p}\) is valid whenever \(p<0,1<p<2\) or \(p>2,\) and reverses direction whenever \(0<p<1\).