Gibbs' inequality is equivalent to: \begin{equation} \sum_{i} \ln q_i^{p_i}-\ln p_i^{p_i} \leq 0 \end{equation} where $p_i,q_i \in [0,1]$ and $\sum_i p_i = \sum_i q_i=1$. Now, a friend of mine suggested that assuming $p_i,q_i \in [0,1]$ and $\sum_i p_i = \sum_i q_i=1$, Gibbs' inequality implies: \begin{equation} \sum_{i} q_i^{p_i}-p_i^{p_i} \leq 0 \end{equation} Right now I doubt this is true but I can't think of a counter-example. **Update**: I have found experimental evidence for my friends' conjecture by running the following Python code: import numpy as np count = 0 for i in range(10000): # randomly create distributions: P, Q = np.random.rand(10), np.random.rand(10) p, q = P/np.sum(P), Q/np.sum(Q) M = np.sum([p[i]**p[i] for i in range(10)]) m = np.sum([q[i]**p[i] for i in range(10)]) if m <= M: count+=1 The inequality was satisfied every single time I ran this script.