I need to evaluate the expectation of $\ln(1+x)$ when $x\sim \text{Gamma}(k,\theta)$. I know that this integral
$$\mathbb{E}\left[ \ln(1+x) \right] = \int_{0}^{\infty} \ln(1+x) \frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k\Gamma(k)}dx$$
has no closed form solution unfortunately and only when $x\gg1$, we can approximate it with $\mathbb{E}\left[ \ln(1+x) \right] \approx \mathbb{E}\left[ \ln(x) \right] = \gamma(k) + \ln(\theta)$. However none of these results are convex. I wanted to know if there is any studies on finding convex upper or lower bounds for this integral $\mathbb{E}\left[ \ln(1+x) \right]$?!
Any hints and/or references would be appreciated.