I am trying to prove three inequalities that would help me solve the proof of a larger theorem.

Let $P(X,Y)$ be a discrete bivariate distribution and $$ I(X;Y) = \sum_{i,j} p(x_i, y_j) \log \frac{p(x_i, y_j)}{p(x_i)p(y_j)} $$ the mutual information between $X$ and $Y$.

Let's call $\bar{P}(X,Y)$ the function (it is not a probability distribution) obtained by $P(X,Y)$ by adding $0 \le a \le 1$ to $p(x_{\bar{i}}, y_{\bar{j}})$ for a given pair $\bar{i}, \bar{j}$ $$ \bar{p}(x_{\bar{i}}, y_{\bar{j}}) = p(x_{\bar{i}}, y_{\bar{j}}) + a \quad \Rightarrow \quad \bar{p}(x_{\bar{i}}) := \sum_j \bar{p}(x_i, y_j) = p(x_i) + a $$ and $$ I^a(X;Y) := \sum_{i,j} p(x_i, y_j) \log \frac{\bar{p}(x_i, y_j)}{\bar{p}(x_i)p(y_j)} $$

Does the relationship $$ 0 \le I^a(X:Y) \le I(X:Y) \qquad \text{(inequality 1)} $$ hold for any $a$?

Furthermore, if $0 \le a \le p(x_{\bar{i}}, y_{\bar{j}})$, do the following relationships also hold (I use the apex $-a$ to indicate that $a$ is subtracted instead than added to $p(x_{\bar{i}}, y_{\bar{j}})$)? $$ I(X;Y) \le I^{-a}(X;Y) \qquad \text{(inequality 2)} $$ $$ I^{-a}(X;Y) - I(X;Y) \le I(X;Y) - I^a(X;Y) \qquad \text{(inequality 3)} $$