This is not a full answer, but according to this explanation I think the question asks whether $\Gamma(k, k \lambda) \geq e^{-\lambda} \Gamma(k)$ when $k \in Z^+$, and $\lambda \in (0, 1)$.

Edit: Some answers to this math.se question have lower bounds for the upper incomplete gamma function.

This is not a full answer, but according to this explanation I think the question asks whether $\Gamma(k, k \lambda) \geq e^{-\lambda} \Gamma(k)$ when $k \in Z^+$, and $\lambda \in (0, 1)$.

Edit: Some answers to this math.se question have lower bounds for the upper incomplete gamma function.

This is not a full answer, but I think this question can be rewritten as asking whether

$\Gamma(k, k \lambda) \geq e^{-\lambda} (k-1)!$

where $\Gamma$ is the upper incomplete gamma function, $k \in Z^+$, and $\lambda \in (0, 1)$.

http://en.wikipedia.org/wiki/Incomplete_gamma_function#Regularized_Gamma_functions_and_Poisson_random_variables

This is not a full answer, but according to this explanation I think the question asks whether $\Gamma(k, k \lambda) \geq e^{-\lambda} \Gamma(k)$ when $k \in Z^+$, and $\lambda \in (0, 1)$.

Edit: Some answers to this math.se question have lower bounds for the upper incomplete gamma function.