# 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 $x>300$

-

2
Try plugging in $x=1$ to see what happens. – Byron Schmuland Jan 28 at 13:41

1
@gauss115 - please revise the question, as Byron pointed out the statement as it stands is false. – nbubis Jan 28 at 13:54
@nbubis sry! The condition is $x>300$ – gauss115 Jan 28 at 14:00
How do you know this is true? What did you try to show this is true? – Did Jan 28 at 14:06

## 2 Answers

Another approach would be to use approximations. There is a quickly convergent version of Stirling's formula which goes like this:$$\ln{\Gamma(x)}=\left(x-\frac{1}{2}\right)\ln{x}-x+\frac{\ln{2\pi}}{2}+\frac{1}{12(x+1)}+O(x^{-2})$$ (see http://goo.gl/9hsnO). Derive upper and lower bound from this and plug back into your inequality. You'll end up with a relatively straightforward logarithmic inequality.

-

Simply take the derivative of your function: $$f'(x) = \psi(x)-\psi\left(\frac{x+1}{2}\right)$$ And then show that $f''(x) >0$ and that $f'(x)>2/3$ for say $x=20$. This shows that the derivative is larger for all $x>20$.

Now calculate $f(300) > 200$ to prove that $f(x) > 2x/3$ for all $x>300$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 10, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9170681238174438, "perplexity_flag": "head"}