A note on generalizations of Hölder inequalities via convex and concave functions (Q2639977)

Let us denote by \(X_{\uparrow}=(x_{(1)},x_{(2)},...,x_{(n)})\) the increasing rearrangement of the vector \(X=(x_ 1,x_ 2,...,x_ n)\). The main result of this note is given by: Theorem 1. Let \(f(x)\in C'\) be a concave function in \(0<a\leq x\). If the positive vectors X, D and Y satisfy: \(\sum^{n}_{i=1}x_ i=\sum^{n}_{i=1}d_ i=1\) and \(x_ i/y_ i\geq d_ i/y_ i\geq d_ j/y_ j\geq x_ j/y_ j\), \(i=1,...,m\); \(j=m+1,...,n\), then: \[ \sum^{n}_{i=1}y_ if(x_ i/y_ i)\leq \sum^{n}_{i=1}y_ if(d_ i/y_ i)\leq \sum^{n}_{i=1}y_{(i)}f(d_{(i)}/y_{(i)}). \] Taking \(f(x)=x^{1/p}\), with \(p>1\) and \(1/p+1/q=1\) it follows that: \[ \sum^{n}_{i=1}x_ i^{1/p}y_ i^{1/q}\leq \sum^{n}_{i=1}d^{1/p}_{(i)}y^{1/q}_{(i)} \] which allows to establish an upper bound in Hölder's inequality (less than one under additional constraints on the vectors X and Y).