A simple inequality, does it always hold?

Question

I am looking at the following inequality:

$$\Gamma(-\frac{x+1}{x})\lt x , \forall x\gt \frac{7}{2}$$

It seems that the LHS and RHS eventually diverge for large enough $x$, but I have failed in a proof of that inequality. Any help would be greatly appreciated. Thanks.

Answer 1

We shall show this using the helpful observation made by Claude Leibovici.

First, we shall show that $0 < z < 2/7$, $$ f(z) \equiv \log\Gamma(1 - z) - z + \frac{ z^2 } { 2} < 0. \qquad (1) $$ From the Bohr-Mollerup theorem, $\log\Gamma(1-z)$ is convex, so is $-z + z^2/2$. So $f(z)$ is convex, and for $z < a = 2/7$, we have $$ f(z) < \frac{z}{a} f(a) + \left(1 - \frac{z}{a}\right) f(0) = \frac{z}{a} f(a) < 0, $$ where we have used the fact $f(a) \approx -0.0012 < 0$.

Second, for $0 < z < 1$ $$ \begin{aligned} \log(1 + z) &= z - \frac{z^2}2 + \frac{z^3}{3} - \frac{z^4}{4} + \dots \\ &> z - \frac{z^2}{2} + \frac{z^3(1-z)}{4} + \frac{z^5(1-z)}{6} \dots \\ &> z - \frac{z^2}{2}. \qquad (2) \end{aligned} $$ Thus, from (1) and (2) we have for $0 < z < 2/7$, $$ \log \frac{\Gamma(1-z)}{1+z} < f(z) < 0, $$ or $$ \frac{\Gamma(1-z)}{1+z} < 1. $$

Finally, for $x > 7/2$, $$ \begin{aligned} \Gamma\left(-\frac{x+1}{x}\right) &= \frac{ \Gamma\left( -\frac 1 x \right) } {\left(-1 - \frac 1 x \right)} = \frac{ \Gamma\left( 1 -\frac 1 x \right) } {\left(-1 - \frac 1 x \right)\left(-\frac 1 x\right) } = x \, \frac{ \Gamma\left( 1 -\frac 1 x \right) } {1 + \frac{1}{x} } < x. \end{aligned} $$

Answer 2

I am not sure that this could be an answer but it is too long for a comment.

Considering for large values of $x$ $$\Gamma(-\frac{x+1}{x})=\Gamma(-1-y)$$ and developing as a Taylor series at $y=0$, we have $$\Gamma(-1-y)=\frac{1}{y}+(\gamma -1)+\frac{1}{12} \left(12-12 \gamma +6 \gamma ^2+\pi ^2\right) y+O\left(y^2\right)$$ which make $$\frac 1y-\Gamma(-1-y)=(1-\gamma )+\frac{1}{12} \left(-12+12 \gamma -6 \gamma ^2-\pi ^2\right) y+O\left(y^2\right)$$ where $\gamma $ is Euler-Mascheroni constant ($\approx 0.577216$). The next coefficient is $\approx -1.41184$.

Concerning the solution of the equation $$\Gamma(-\frac{x+1}{x})- x=0$$ I did not find any analytical solution but numerical methods give $$x\approx 3.29728$$