The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its logarithmic derivative $\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}$ is called the digamma function. In sequence, the functions $\psi'(z)$, $\psi''(z)$, $\psi'''(z)$, and $\psi^{(4)}(z)$ are called the trigamma, tetragamma, pentagamma, and hexagamma functions respectively. In a word, the derivatives $\psi^{(n)}(z)$ for $n\ge1$ are called polygamma functions.

How to prove that the determinant \begin{gather*} \begin{vmatrix} \psi'(x) & \psi'\bigl(x+\frac{1}{2}\bigr) & 0\\ \psi''(x) & \psi''\bigl(x+\frac{1}{2}\bigr) & \psi'(x)-\psi'\bigl(x+\frac{1}{2}\bigr)\\ \psi'''(x) & \psi'''\bigl(x+\frac{1}{2}\bigr) & 2\bigl[\psi''(x)-\psi''\bigl(x+\frac{1}{2}\bigr)\bigr] \end{vmatrix}\\ =\begin{vmatrix} \int_{0}^{\infty}p(t,x)\operatorname{d}t & \int_{0}^{\infty} p(t,x)\operatorname{e}^{-t/2}\operatorname{d}t & 0\\ \int_{0}^{\infty}p(t,x) t\operatorname{d}t & \int_{0}^{\infty} p(t,x)\operatorname{e}^{-t/2}t\operatorname{d}t & \int_{0}^{\infty} p(t,x)(1-\operatorname{e}^{-t/2})\operatorname{d}t\\ \int_{0}^{\infty} p(t,x) t^2\operatorname{d}t & \int_{0}^{\infty} p(t,x)\operatorname{e}^{-t/2}t^2\operatorname{d}t & 2\int_{0}^{\infty} p(t,x)(1-\operatorname{e}^{-t/2})t\operatorname{d}t \end{vmatrix} \end{gather*} is negative for $x>0$? where \begin{equation*} p(t,x)=\frac{t\operatorname{e}^{-xt}}{1-\operatorname{e}^{-t}}>0, \quad (t,x)\in(0,\infty)\times(0,\infty) \end{equation*} and \begin{equation} \psi^{(n)}(z)=(-1)^{n+1}\int_{0}^{\infty}\frac{t^{n}}{1-\operatorname{e}^{-t}}\operatorname{e}^{-zt}\operatorname{d}t, \quad \Re(z)>0, \quad n\in\mathbb{N}. \end{equation} This problem is related to the problems at the sites How to prove the convexity of a simple function involving a ratio of two polygamma functions? and How to prove convexity of a simple function involving a ratio of two polygamma functions?.

In order to answer this question, it is sufficient to show \begin{equation*} \begin{vmatrix} \frac{\psi'(x)}{\psi''(x)} & \frac{\psi'(x+\frac{1}{2})}{\psi''(x+\frac{1}{2})}\\ 1-\frac{1}{2}\frac{\psi'''(x)}{\psi''(x)}\frac{\psi'(x)-\psi'(x+\frac{1}{2})}{\psi''(x)-\psi''(x+\frac{1}{2})} & 1-\frac{1}{2} \frac{\psi'''(x+\frac{1}{2})}{\psi''(x+\frac{1}{2})}\frac{\psi'(x)-\psi'(x+\frac{1}{2})}{\psi''(x)-\psi''(x+\frac{1}{2})} \end{vmatrix} >0, \quad x>0. \end{equation*}

`\operatorname{d}t`

probably doesn't space as most people would want. See Should I \mathrm the d in my integrals? and questions linked there for best practices in TeXing integrals. (One common approach is to use $\mathrm dt$`\mathrm dt`

instead. Personally, I agree that the $\operatorname d$ is an operator, and so am stuck using the unsatisfactory ${\operatorname d}t$`{\operatorname d}t`

.) I didn't edit, in case this is really what you wanted. $\endgroup$`\textup{d}t`

, which doesn't include the extra spacing. $\endgroup$2more comments