Sharp bounds for integral means (Q1856934)

A sample of the results obtained here is the following theorem: Let \((X,{\mathcal M},m)\) be a measure space, and let \(E\in{\mathcal M}\). A number \(M\) is an \((a,A)\)-bound for \(f: E\to\mathbb{R}\) if there exist functions \(f_1,f_2: E\to\mathbb{R}\) and numbers \(a\leq A\) such that \[ M= A-a,\;f(x)= f_1(x)- f_2(x);\tag{1} \] \[ a\leq f_1(x), f_2(x)\leq A,\tag{2} \] \[ m(E) \int_e f_1f_2 dm\geq \int_E f_1dm \int_E f_2 dm.\tag{3} \] If \(M\) is an \((a,A)\)-bound for \(f\), and \(\int_E f dm= 0\), then \(\int_E|f|dm\leq {M\over 2} m(E)\), where \(m(E)\) denotes the measure of \(E\). Then sharp bounds are deduced for the average of \(|f|^p\) over \(E\). A new proof for the famous Grüss inequality is deduced. A new estimate for the error term in the trapezoidal rule is also given.