It is well known that, for all $x\in [-1,1]$ and for all $j>0$, the Chebyshev polynomials of the first and second kind satisfy $$ |T_j(x)|\leq 1 ,\qquad |U_j(x)|\leq j+1. $$
I am wondering about the values of $T_j(1+\eta)$ and $U_j(1+\eta)$ for some $\eta$ small. For instance, are there simple formulas for $\eta>0$ (in terms of $j$) such that $$ |T_j(1+\eta)|\leq 2 ,\qquad |U_j(1+\eta)|\leq 2(j+1)? $$ I have derived some bounds myself, but it was a bit tedious and I made many simplifications, so I suspect tighter bounds should be known.