Chebyshev inequality in function spaces (Q1189085)

This paper gives new variants, generalizations and abstractions of the well-known Chebyshev inequality for monotonic functions. For example, the following result was proved by reviewer's method: Let \(K\) be a positive continuous function on \(I^ 2\;(I=[0,a],a>0)\) and suppose \(f:I^ 2\to[0,\infty)\) is a continuous positive set function. a) If for all \(t\in I\) \(\int^ a_ t \int^ t_ 0 K(u,v)dv du = \int^ t_ 0 \int^ a_ tK(u,v)dv du\) then \[ \int^ a_ 0 \int^ a_ 0K(s,t)f(s,t)ds dt\leq\int^ a_ 0 \left(\int^ a_ 0K(s,t)ds\right)f(t,t)dt. \] b) If for all \(s,t\in I\) \(\int^ a_ 0K(u,t)du=Bw(t)\) and \(\int^ a_ 0K(s,v)dv=Bw(s)\), then \[ \int^ a_ 0 \int^ a_ 0K(s,t)f(s;t)ds dt\leq B\int^ a_ 0w(s)f(s,s)ds. \]