# On a growth condition satisfied by given functional :

Consider the following function :

$$f(x) = \sin^2(\frac{π\Gamma(x)}{2x})$$

Is following growth condition true ?:

$$\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy=o\left(\int_2^x f(t) dt\right)$$ ?

If yes , how to achieve it?

One can also ask for the following :

$$\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy=o(g_1(x))=O(g_2(x))$$

As $$x \to \infty$$.

what are some possible candidates for $$g_1(x)$$ and $$g_2(x)$$ ?Also can we find $$g_1(x)$$ and $$g_2(x)$$ for which bounds are sharp?

See this MSE post for more details

Accurate (upto 4 decimals) values of functional upto $$x=5$$:

Important: Any computational analysis expert can help me to compute them for larger $$x$$ upto 100 or 1000 or provide me with graph for large $$x$$ Answers and comments from them are welcome

• What do you get from Stirling (the sum over poles of $\Gamma'/\Gamma$ is up to a $\log(z)$ term a power series in $z^{-1}$ which gives very good approximations of $\sin(\Gamma)$) – reuns Jun 20 at 22:05
• Please avoid making several edits. – Aloizio Macedo Jun 29 at 10:12