# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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### Prove that $(p+q)^m \leq p^m+q^m$

If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$ Trial: For $m=0$, $(p+q)^0=1 < 2= p^0+q^0$ and for $m=1$, $(p+q)^1=p+q =p^1+q^1$. So, For $m=0,1$ ...
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### Triangle inequality for subtraction?

Is the following inequality(that looks like the triangle inequality) valid: $|a - b| \leq |a| - |b|$ Why?
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### Inequality: $(x + y + z)^3 \geq 27 xyz$

Edit: $a,b,c$ and $x,y,z$ are positive, real numbers. Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$. ...
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### The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given a triangle $ABC$, and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$. When does equality hold?
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### How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

let $a,b,c>0$,and such $a+b+c=3$, show that $$\dfrac{2}{(a+b)(4-ab)}+\dfrac{2}{(b+c)(4-bc)}+\dfrac{2}{(a+c)(4-ac)}\ge 1$$ I think this inequality use this $$ab\le\dfrac{(a+b)^2}{4}$$
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### How might one prove the following inequality?

Let $r$ be a natural number. I wish to prove that $x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \cdots + \dfrac{x^r}{r} \leq x^{r+1} + \log(r+1)$ for all $x>0$. Some friends and I have tried using ...
### How prove this inequality $2a^ab^bc^cd^d\ge ac+bd$
Let $a,b,c,d$ be positive numbers such that $a+b+c+d=2$. Show that $$2a^ab^bc^cd^d\ge ac+bd$$ My try: I think maybe I can use this inequality $$(1+x)^n\ge 1+nx \hspace{12pt} (n>1)$$ then I can't ...
### Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
Let $f: [0,\infty) \to [0,\infty)$ be a continuous function such that $f(f(x)) = x^2, \forall x \in [0,\infty)$. Prove that $\displaystyle{\int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}}$. All I know ...