On inverses of the Hölder inequality (Q1189692)

The author derives an inequality complementary to \[ x^{1/p} y^{1/q}\leq x/p+y/q. \] Using this inequality, then converse Hölder- type inequalities of the form \[ a\| f\|_ 1+b\| g\|_ 1\leq c\| f^{1/p} g^{1/q}\|_ 1\qquad\text{and}\qquad \| f\|_ p\| g\|_ q\leq d\| fg\|_ 1 \] are obtained, where the ranges of \(f\) and \(g\) are compact subintervals of \((0,\infty)\). The constants \(c\) and \(d\) then depend on \(a\), \(b\), \(p\), \(q\) and the endpoints of the intervals containing the ranges of \(f\) and \(g\). The results include a lot of earlier inequalities complementary to Hölder's one.