I want to show the following.
$$ \phi((\lambda g_1 + (1-\lambda)g_2)^k) - (\lambda g_1 + (1-\lambda) g_2)^k \nabla \phi(f^k) \le \lambda \phi(g_1^k) + (1-\lambda)\phi(g_2^k) - (\lambda g_1^k + (1-\lambda) g_2^k) \nabla \phi(f^k).$$
where $\phi$ is convex function, $k>0$, $0 \le \lambda \le 1$, and $g_1, g_2, f$ are probability densities (non-negative).
In the second term, I understand the power function itself is convex so that $$ (\lambda g_1 + (1-\lambda) g_2)^k \le \lambda g_1^k + (1-\lambda) g_2^k $$ holds.
I think the above inequality does not hold in general. Please correct me if I am wrong. If the inequality indeed holds, what constraints are needed?
Thanks.