Given roots of $x^2-mx+4=0$ are distinct and lies in $\left(1, 5\right),$ Following conditions must be true

$\left(i\right)$ $D>0\Rightarrow m^2-16>0$

$\Rightarrow \left(m-4\right)\left(m+4\right)>0$

$\Rightarrow m\in \left(-\infty , -4\right)\cup \left(4, \infty \right)$

$\left(ii\right)$ $f\left(1\right)>0\Rightarrow m<5$

$\left(iii\right)$ $f\left(5\right)>0\Rightarrow m<\frac{29}{5}$

$\left(iv\right)$ $1<-\frac{b}{2a}<5\Rightarrow 2<m<10$

Taking intersection of all conditions, we get $m\in \left(4, 5\right)$