Monotonic nature of the Gamma function

This study begins by examining the monotonic behavior of the function $\ln \Gamma (x+1)/\left (\ln \left (x^{2}+6\right )-\ln (x+6)\right )$ on the interval $x\in (-1,1)$, supported by well-established functional inequalities where the main idea of the proofs comes from the use of L’Hôpital-style rule for the monotonicity of a ratio of two functions. Our approach can be viewed as a slight modification of the technique presented in Zhao et al. (Publ. Math. (Debr.) 80(3–4):333–342, 2012). We further demonstrate the monotonicity of the function $[1,+\infty )\ni s\mapsto u(s)/\Gamma (s+\rho )\Gamma (s)$ where u is a positive-valued differentiable function on $[1,+\infty )$ and $\rho >-1$, punctuated by a selection of several special cases that highlight its unique characteristics.