Convexity of a scaled multivariate digamma function

Question

The problem...

Let $\psi_p(a) = \frac{\partial \Gamma_p(a)}{\partial a}$ be the multivariate digamma function. I believe the following function to be strictly convex at least for real values $a>2p$ (and possibly for real values $a>p$): $$a \psi_p(a)$$
Any suggestions on proving this would be helpful.

The following is a plot for $p=20$, but looks similar for other values of $p$: Image: Scaled multivariate digammma.

Things I have tried...

Several approaches below use the relationship to the standard digamma function $\psi(\cdot)$ as: $$\psi_p(a) = \sum_{i=1}^p \psi\left(a + \frac{1-i}{2}\right)$$

1. I have tried using the definition of convexity as $f( \lambda a_1 + (1-\lambda) a_2 ) \leq \lambda f(a_1) + (1-\lambda) f(a_2)$ for $2p < a_1 < a_2$ and $f(a) = a \psi_p(a)$. I believe this approach could work but the formulas become unwieldy quickly.

2. I have tried showing that the second derivative is non-negative. I wind up with a formula: $$f''(a) = 2 \sum_{i=1}^p \psi^{(1)}\left( a + \frac{1-i}{2} \right) + a \sum_{i=1}^p \psi^{(2)}\left(a + \frac{1-i}{2}\right)$$
   where $\psi^{(n)}$ is the polygamma function of order $n$. But I have been unable to show that this function is non-negative.

3. I have been able to show that it is convex in the limit as $p\rightarrow \infty$ through connection to the harmonic numbers. But I think this result should hold for any $p>1$.

Things I know that might be helpful...

1. The digamma function $\psi(\cdot)$ is strictly monotonically increasing and strictly concave on $(0,\infty)$.

2. For $n$ odd the polygamma function $\psi^{(n)}$ is strictly positive and strictly convex on $(0,\infty)$.

3. For $n$ even the polygamma function $\psi^{(n)}$ is strictly negative and strictly concave on $(0,\infty)$.

4. Let $n\leq 1$ and $c$ a real number then $f_c(x) = x^c|\psi^{(n)}(x)|$ is strictly decreasing on $(0,\infty)$ if and only if $c \leq n$, and $f_c$ is strictly increasing on $(0,\infty)$ if and only if $c \leq n+1$. (This result is due to Alzer 2004. "Sharp inequalities for the digamma and polygamma functions.")

5. With the definition of $f_c$ above it is strictly convex on $(0,\infty)$ if an only if $c \leq n$, or $c=n+1$, or $c \geq n+2$. (Also due to Alzer 2004)