INTEGRAL INEQUALITIES FOR TWO FUNCTIONS.
Abstract
In the interval of 0 < or = x < or = 1, let f(x) be an arbitrary continuous, piecewise smooth function with f(0) = 0, and let h(x) be a positive concave function with piecewise smooth derivative h' (i.e., h'' < or = 0 where it exists) satisfying h'(0) < or = 0. Then the inequality L(h,f) =((the integral h(x) f' f' dx) / (the integral h(x) dx) (the integral f f dx)) > or = (pi)(pi)/4 holds, as the integral foes from 0 to 1. Furthermore, equality holds for h(x) equivalent to 1, and f(x) = sin (pi x/2). When h(x) is equivalent to 1, this inequality reduces to the Wirtinger inequality. The proof proceeds in ratchet fashion on the string of inequalities L(h,f) > or = L(h,f(h)) > or = L(h,f(h)) > or L(h,f(h)) > or = L(h = 1, sine (pi x/2)) = (pi)(pi)/4, where f(h) denotes the fundamental eigenfunction f belonging to the variational problem delta L - 0 for given h, and h denotes a concave function consisting of one or two straight line segments. Related inequalities are also proved.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1966
- Accession Number
- AD0801621
Entities
People
- B. A. Troesch
Organizations
- The Aerospace Corporation