Let $\mathbf{R_n}$ be the space of rational functions with prescribed poles. If $t_1,t_2,\dots,t_n$ are the zeros of $B(z)+\lambda$ and $s_1,s_2,\dots,s_n$ are zeros of $B(z)-\lambda$ where $B(z)$ is the Blaschke product and $\lambda\in T$, then for $z\in T$ \[ |r'(z)|eq\frac{|B'(z)|}2[(\max_{1eq keq n}|r(t_k)|)^2+(\max_{1eq keq n}|r(s_k)|)^2]. \] Let $r,s\in\mathbf{R_n}$ and assume $s$ has all its $n$ zeros in $D^-\cup T$ and $|r(z)|\leq|s(z)|$ for $z\in T$, then for any $\alpha$ with $|\alpha|\leq\frac12$ and for $z\in T$ \[ |r'(z)+lpha B'(z)r(z)|eq|s'(z)+lpha B'(z)s(z)|. \] In this paper, we consider a more general class of rational functions $rof\in\mathbf R_{m^\star n}$, defined by $(rof)(z)=r(f(z))$, where $f(z)$ is a polynomial of degree $m^\star$ and prove some generalizations of the above inequalities.