INEQUALITY'S BLOG

April 2, 2011

Van Khea’s inequality and application

Filed under: Van Khea 01 — KKKVVV @ 9:37 pm

Theorem:

1) For a real convex function f(x), numbers a, b, c in its domain such that a\leq b\leq c and let m, n, p are positive real numbers such that m\geq n ; p\geq n then we have

m(c-b)f(a)-n(c-a)f(b)+p(b-a)f(c)\geq 0

Special case if m=n=p\neq 0 then we have (c-b)f(a)-(c-a)f(b)+(b-a)f(c)\geq 0

2) For a real concave function f(x), numbers a, b, c in its domain such that a\leq b\leq c and let m, n, p are positive real numbers such that m\leq n ; p\leq n then we have

m(c-b)f(a)-n(c-a)f(b)+p(b-a)f(c)\leq 0

Special case if m=n=p\neq 0 then we have (c-b)f(a)-(c-a)f(b)+(b-a)f(c)\leq 0

Proof

Suppose that c_1\in (a, b) ; c_2\in (b, c) then from Lagrange’s theorem we get:

\displaystyle f'(c_1)=\frac{f(b)-f(a)}{b-a} ; f'(c_2)=\frac{f(c)-f(b)}{c-b}

click here to download