# Tagged Questions

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### Gamma Function inequality .

Why does the following inequality hold :- $$\sqrt 2 \frac {\Gamma ((n+1)/2)}{\Gamma (n/2)} -\sqrt 2 \frac {\Gamma ((m+1)/2)}{\Gamma (m/2)} \ge \sqrt n - \sqrt m$$ provided $n \ge m \ge 1$ .
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### An inequality involving the Gamma function

Given $x\in\mathbb{R}$ ,$\forall x\ge1$ seems to hold the following inequality: $$\Gamma(x)+\Gamma\left(\frac{1}{x}\right)\le\Gamma\left(1+x+\frac{1}{x}\right)$$ where the sign of equality holds only ...
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### Show that $\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 … (2n-1)}{2\cdot 4\cdot 6 … (2n)} \le \frac1{\sqrt{n \pi}}$

Show that, if $n$ is a positive integer, $$\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 ... (2n-1)}{2\cdot 4\cdot 6 ... (2n)} \le \frac1{\sqrt{n \pi}} .$$ This result is in a current ...
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### Inequality $\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y-1)$

I need to prove $$\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y),$$ for $x>\alpha$ and $y>\beta$ with $0<\alpha,\beta \leq \frac{1}{2}$ constants, and $C(\alpha, \beta)$ is a ...
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### Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: $$I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt,$$ ...
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### Upper bound for the quotient of gamma functions?

I am looking for an upper bound for $$\frac{\Gamma(x+\beta)}{\Gamma(x)},\,\,\,\beta>0.$$ In this question it was shown that $$\frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta.$$ Then, I ...
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### Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
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### Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
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Does anyone know of any tight upper/lower bound for incomplete Gamma functions? i.e either of the following functions: $$\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t$$ or $$\gamma(s,x) ... 0answers 79 views ### Inequalities involving regularized incomplete Gamma functions I am new to the world of the Gamma functions and am wondering if there exist positive functions f_1(x)>0 and g_1(x)>0, and non-negative functions f_2(x)\geq0 and g_2(x)\geq0 such that ... 2answers 116 views ### About the use of Stirling approximation How to prove this inequality:$$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$Sry I forgot to mention that x>300 1answer 150 views ### Inequality for Gamma functions Let k, n ,m \in N and such that 0\leq k \leq n \leq m. When the following ineuality is true?$$ ...
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.