# Tagged Questions

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### Simple inequality help

I need a function $f(x)$ that satisfies the properties bellow for all integers $k$ $$\frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \$$ $$\lim_{k \rightarrow \infty} f(k)=0$$ ...
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### Ceiling function inequality

In class, we used the fact that $\lceil{a + b \rceil} \geq \lceil{a}\rceil + \lfloor{b}\rfloor$. However, we weren't given a proof of this statement. I am interested to see how this works. Can anyone ...
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### Smooth function on $\mathbb R$ whose small increments are not controlled by the first derivative at infinity

I need some help in finding a (as simple as possible) smooth function $f:\mathbb R \rightarrow \mathbb R$ which does NOT satisfy the following: There exist a constant $C>0$, a compact ...
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### how to find arrange the following functions in increasing or decreasing order?

I have the following three functions $f_1(x) = \frac{1}{4} (8-3x + \sqrt{(x-2) (5x-14)}) (1-x)$ $f_2(x) = \frac{1}{8} (12-4x + \sqrt{2} \sqrt{(5x-14)(x-3)} + \sqrt{2} \sqrt{(x-2)(x-3)} )(1-x)$ ...
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### Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.
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### Minimizing a function over two variables

Given two natural numbers $i$ and $p$ such that $0 < i \leqslant 2^p$, let  \psi(p,i) := p - \alpha + 1 - \frac{1}{2^p}\left((2^p+i)\lg(2^p+i) - i\lg i - i + \alpha - \frac{2^p}{i+1} - ...
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### Mathematical function for four corelated attributes

I have $4$ attributes $A,B,C,D$ each of them takes value between $[0,1]$ The more $A$ and $B$, the more the function value is. The more $C$ and $D$, the less the function value is. if C or D equals ...
### A function that maps inequalities to $-1$, $0$, or $1$.
The Python computer language has a built-in operation cmp(a,b) that returns $-1$, $0$ or $1$, if $a<b$, $a=b$ or $a>b$, respectively. I'd like to know if ...
### Geometric intuition for the inequality $(f(y) - c) ( y - d ) \geq (f(d) - c) ( f^{-1}(c) - d )$
Good day to everyone. I am interested in the geometric intuition for the following statement: Let $f:\mathbb{R} \mapsto \mathbb{R}$ be a monotonically increasing, invertible function and \$c,d \in ...