The author proves some interesting inequalities involving the logarithmic derivative of the gamma function \(\psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}\), and its derivatives \(\psi_{n}(x)=\psi^{(n)}(x),\;n=1,2,3,\ldots\), known as polygamma functions. In particular, the author proves the following inequalities: For all positive integers \(n\) and positive real numbers \(x>0\), we have \[ (n-1)!\exp\big(-n\psi(x+1/2)\big)(-1)^{n-1}n\psi_{n}(x+1)\Bigg(1-\Big(\frac{\psi_{n}(x+1)}{\psi_{n}(x)}\Big)^{1/n}\Bigg). \] \[ (n!)^{\frac{1}{n+1}}[x-(x^{-1/n}+\alpha)^{-n}]^{-\frac{1}{n+1}}<\big((-1)^{n-1}\psi_{n}\big)^{-1}(x) \] \[ < (n!)^{\frac{1}{n+1}}[x-(x^{-1/n}+\beta)^{-n}]^{-\frac{1}{n+1}} \] where the constants \(\alpha=((n-1)!)^{-1/n}\) and \(\beta=(n!\zeta(n+1))^{-1/n}\) are best possible. \[ -\gamma+x\psi^{\prime}(x/2)<\psi(x+1)<-\gamma+x\psi^{\prime}(\sqrt{x+1}-1), \] where \(\gamma\) is Euler's constant. Finally, the author proves that for all positive integers \(n\) and positive real numbers \(x,y\), we have \[ (-1)^{n}\psi^{(n+1)}\Big(\frac{x+y}{2}\Big)<\frac{(-1)^{n}(\psi^{(n)}(x)-\psi^{(n)}(y))}{x-y}<(-1)^{n}\psi^{(n+1)}\big(S_{-(n+1)}(x,y)\big), \] where for \(p\in \mathbb{R}\) \[ S_{p}(a,b)=\Big(\frac{a^{p}-b^{p}}{p(a-b)}\Big)^{1/(p-1)} \] is Stolarsky's mean of \(a,b\). \[ \alpha-\log(e^{1/x}-1)<\psi(x)<\beta-\log(e^{1/x}-1), \] where the constants \(\alpha=-\gamma\) and \(\beta=0\) are best possible.