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:

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