Articles of inequality

$f\left(\sum X_i\right) \leq \sum f(X_i)$, where $X_i\gt 0$; for what functions is this true?

In a previous post, the following inequality has been proven $${\left( {\sum\limits_{i = 1}^n {{W_i}} } \right)^a} \le \sum\limits_{i = 1}^n {{W_i}^a}$$ where $W_i\gt 0$, $0\lt a\lt 1$. I guess it is more correct to say that this is always greater, and it is valid for $a\gt 0$ not just $0\lt a \lt 1$. I […]

Ramsey lower bounds

I’m doing some self study on Ramsey graph theory. One of the first theorems concerning lower bounds shows that if $${n \choose k} \cdot \frac {1}{2^{( \frac {k}{2}-1)}} <1$$ then $N(k,k)> n$ In the derivation of a later result, the claim is made that due to stirlings formula, $${n \choose k} \cdot \frac {1}{2^{( \frac […]

Incorrect General Statement for Modulus Inequalities

$$|x + 1| = x$$ This, quite evidently, has no solution. Through solving many inequalities, I came to the conclusion that, If, $$|f(x)| = g(x)$$ Then, $$f(x) = ±g(x)$$ And this was quite successful in solving many inequalities. However, applying the above to this particular inequality: $$|x + 1| = x$$ $$x + 1 = […]

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ does not depend on $r>0$.

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can’t find a way to manipulate this expression and find that the new series is $< 2 \sqrt{x+1}$. Can someone show me the correct process? I failed […]

Proof of a trigonometric inequality with a parameter

Let $$ f_p(x)=\left(-p^3-3 p^2-3 p-1\right) \sin \left(x-\frac{3 x}{p}\right)+\left(3 p^3+3 p^2-15 p-23\right) \sin \left(x-\frac{x}{p}\right)+\left(-3 p^3+3 p^2+15 p-23\right) \sin \left(\frac{x}{p}+x\right)+\left(p^3-3 p^2+3 p-1\right) \sin \left(\frac{3 x}{p}+x\right). $$ I want to show $$ f_p(x)>0,\; x\in(0,\pi/2) $$ for any integer $p\ge2$. This trigonometric inequality has been verified by Mathematica using the Plot commend. However, I cannot give a rigorous proof […]

Prove that $1^2 + 2^2 + … + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + … + n^2$

I’m having trouble on starting this induction problem. The question simply reads : prove the following using induction: $$1^{2} + 2^{2} + …… + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + …… + n^{2}$$

How prove this inequality $\left+1>\frac{n^2}{\sqrt{3n^2-5}}$

let $k$ is postive integer,and for any postive integer $n\ge 2$, show that: $$\left[\dfrac{n}{\sqrt{3}}\right]+1>\dfrac{n^2}{\sqrt{3n^2-5}}>\dfrac{n}{\sqrt{3}}$$ where $[x]$ is the largest integer not greater than $x$

Show two functions are uniformly comparable on the real line

I am trying to find inequalities between this two functions $ f(k)=| (\lambda^2+k^2)^{\alpha/2}\cos ( \alpha \arctan \frac{|k|}{\lambda} )-\lambda^{\alpha} |$ and $ g(k)=|k|^{\alpha} $ such as, $$ c_1 g(k)\le f(k)\le c_2 g(k),$$ where $ 0<c_1<c_2 $, $ k \in \mathbb{R} $, and the parameters $\alpha$ and $\lambda $ fixed and $ 1<\alpha<2 $, $ \lambda>0 $. […]

Is $\lvert a-b\rvert\le\lvert a\rvert+\lvert b\rvert$ always true?

This question already has an answer here: Proof of triangle inequality 6 answers