# Tagged Questions

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### reference needed for Gamma function

Please help me to find a reference (book) for the following upper bound of Gamma function For $x \geq 1$ $$\Gamma(x)\leq x^{x-1}.$$ Thank you.
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### Upper bound for a gamma function

Let $n \in N$. How to find a non-asymptotic upper bound for $\Gamma(n)$ and $\Gamma(\frac n2+1)$? Thank you
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### inequality with gamma function

Help me please to prove the following inequality For $x,y>1, x \neq y$. $$\frac{1}{\Gamma(x)\Gamma(y)}\leq 2\sqrt{2\pi}\frac{\sqrt{x+y}}{\Gamma(x+y)}.$$ Thank you.
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### Expressing solution to an inequality with Lambert W function

I'm new to Lambert functions, any ideas on how to solve this are welcome: $$\theta \rho^{\theta}+r \theta>v$$ where $\theta \in \mathbb{R}^{+}, -1<r,v<1, \ 0<\rho<1$. I've tried ...
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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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### Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
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### Concave functions on discrete domain

We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property: $f(x+y) \leq f(x) + f(y)$ for all natural numbers $x$ and ...
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### finding bound for the integral

I am trying to get bound for the following integral $$\int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty$$ In particular, the bound of the form $\frac{constant}{r}$. Sorry, we can ...