INEQUALITY'S BLOG

April 4, 2010

The new inequality by van khea

Filed under: Uncategorized — KKKVVV @ 7:52 pm

For a real convex fruction f:\mathbb{R}\longrightarrow \mathbb{R}_0^{+} numbers a, b, c in its domain a\leq b\leq c , and positive m\geq n, p\geq n so we can write that:

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

For a real concave  fruction f:\mathbb{R}\longrightarrow \mathbb{R}_0^{+} numbers a, b, c in its domain a\leq b\leq c, and positive m\leq n, p\leq n so we can write that:

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

Example 1: Let a\geq b\geq c>0 and k\geq 1. Prove that:

\displaystyle \frac{a^{k}b}{c}+\frac{b^{k}c}{a}+\frac{c^{k}a}{b}\geq a^k+b^k+c^k

Example 2: Let a\geq b\geq c>0 and k>0. Prove that:

\displaystyle \frac{a}{b^{k}}+\frac{b}{c^{k}}+\frac{c}{a^{k}}\leq \frac{b}{a^{k}}+\frac{b}{c^{k}}+\frac{a}{c^{k}}

Example 3: Let a\geq b\geq c>0 and k\in Z^{+}. Prove that:

If function f:R\longrightarrow R_{0}^{+} such that f''>0 then we have:

(a-b)^{k}(a-c)^{k}f(x)+(b-a)^{k}(b-c)^{k}f(y)+(c-a)^{k}(c-b)^{k}f(z)\geq 0; \forall{x,y,z\in I}

Example 4: Let a function f:R\longrightarrow R_{0}^{+} such that f''>0. Prove that for \alpha, \beta >0; \alpha+\beta=1 we have:

\alpha f(x)+\beta f(y)\geq f(\alpha x+\beta y)

Example 5: 

Let a function f:R\longrightarrow R_{0}^{+} such that f''>0. Prove that for \alpha, \beta >0; \alpha-\beta=1 we have:

\alpha f(x)-\beta f(y)\leq f(\alpha x-\beta y)

Example 6: Let f:R\longrightarrow R_{0}^{+} such that f''>0. Prove that for t\in (0, 1) then we have:

tf(x)+(1-t)f(y)\geq f(tx+(1-t)y)

Example 7: Let x, y, a, b>0; a+b=1 Prove that:

ax+by\geq x^ay^b

Example 8: Let \displaystyle x, y>0 ; p, q>1 ; \frac{1}{p}+\frac{1}{q}=1. Prove that:

\displaystyle \frac{x^{p}}{p}+\frac{y^{q}}{q}\geq xy

Example 9: Let f:R\longrightarrow R_{0}^{+} such that f''>0 and x\leq y\leq z or x\geq y\geq z. Prove that for p\geq n\geq 0 ; m\geq n\geq 0 ; m-n+p=1 then we have:

mf(x)-nf(y)+pf(z)\geq f(mx-ny+pz)

Example 10: Let a\geq b\geq c>0. Prove that:

b^{a-c}\geq a^{b-c}c^{a-b}

 

 

 

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