# Tagged Questions

For questions about proving and manipulating functional inequalities.

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### Cauchy functional inequality

Given a function on a closed interval $f\colon I\subset \mathbb{R}\to \mathbb{R}$ with $$f(x+y) \leq f(x) + f(y).$$ Moreover, I know that $f$ is monotonic increasing continuous on all points except ...
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### Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
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### Extending continuous linear functional of the derivative to continuous linear functional of the function

Suppose given $f,g\in L^2(\mathbb{R})$ and $f', g' \in L^2(\mathbb{R})$, the linear functional defined by $$F(g):= \int_{\mathbb{R}} f'g' dx$$ is continuous with respect to the derivative, that is ...
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### Inequality related to Beta functions

Consider two monotonically decreasing functions $f_0(x)$ and $f_1(x)$ for which the following holds: $$1-t\leq f_0(t)\leq f_1(t)\leq1.$$ Let $n,m\in\mathbb{N}$ and $m\leq n$...
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### Solving a series of inequalities

If $x_1=1$ and $x_1,x_2,\ldots,x_{100}$ satisfy the following inequalities: $$(x_1 - 4x_2 + 3x_3 )\geqslant0\\ (x_2 - 4x_3 + 3x_4 )\geqslant0\\ \vdots\\ (x_{100} - 4x_1 + 3x_2 )\geqslant0$$ ...
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### Finding a lower bound of a function which is an inequality [closed]

A function $F(n)$ satisfies the recurrence $F(n) \le 7F(3n/2) + 3n$ for all $n \in \mathbb{N}$. Give a lower bound for $F(n)$.
### An inequality involved $L_p$ functions [duplicate]
If $p\ge 2$ & $f,g$ are $L_p$ functions, prove that : $||\frac {f+g}{2}||_p^p +||\frac {f-g}{2}||_p^p \le \frac {1}{2}[||f||^p_p +||g||_p^p ]$ At first glance , I thought it is easy noticing ...