$$ L=\left \{ a^n b^m c^z : m \neq z, n \geq 1 \right \} $$
Can anyone have a look at my answer? Seems correct? $$ \\S\rightarrow TD|AR \\R\rightarrow bRc|Y \\X\rightarrow Xb|b \\Y\rightarrow Yc|c \\A\rightarrow Aa| \lambda \\T\rightarrow Ta|a \\D\rightarrow bDc|\lambda $$
Your grammar is more complicated that it needs to be, if I'm not mistaken. We need something to generate a nonempty string of a's, followed by a string of b's, followed by a string of c's, where the the b and c strings must be of different lengths.
$$S\to AR\\ A\to a|aA$$
Takes care of the first part. As for $R$, we want to generate a number of b's followed by the same number of c's, and then stick a nonempty constant string in the middle.
$$R\to bRc|B|C\\ B\to b|bB\\ c|\to c|cC $$