# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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### Inequalities and averages

Dikshant writes down $2 k+1$ positive integers in a list where $k$ is a positive integer. The integers are not necessarily all distinct, but there are at least three distinct integers in the list. The ...
• 11
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### Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$

The problem Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$ My idea $(\sqrt{a}+\sqrt{b})^2=a+b+ 2\cdot \sqrt{ab}= 101 + 2\cdot \sqrt{ab}$ ...
• 833
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### How to prove this simple inequality based on convexity of $e^{x}$?

Suppose $\theta > 0$ and $x>0$. I would like to show that $$e^{\theta(x+1)} - e^{\theta x} - \frac{ e^{\theta x}-1}{x} \geq \frac{e^{\theta x}-1}{x} - (1-e^{-\theta})$$ Another way to put it: ...
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### fractional power function inequality

I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result. Let ...
### Differential inequation $f'(x) > f(x)/x$
I am interested in the following differential inequation, for $f : \mathbb{R} \to \mathbb{R}$: $$f'(x) > \dfrac{f(x)}{x}$$ I only care about this being satisfied at infinity, ie for $x \gg 1$. ...