From a cylinder of radius $R$, a cylinder of radius $R/2$ is removed, as shown. Current flowing in the remaining cylinder is $I$. Magnetic field strength is:

For cylinder; 

$$\begin{gathered} B=\frac{{\mu }_{0}ir}{2\pi R^2};r<a \\ =\frac{{\mu }_{0}i}{2\pi r};r>a \end{gathered}$$

We can consider the given cylinder as a combination of two cylinders. One of radius ‘$R$’ carrying current I in one direction and other of radius  $\frac{R}{2}$ carrying current  $\frac{I}{3}$  in both directions.
At point A:
$B=\frac{{\mu }_0\left(\frac{I}{3}\right)}{2\pi \left(\frac{R}{2}\right)}+0=\frac{{\mu }_{0}I}{3\pi R}$

At point B:

$B=\frac{{\mu }_0\left(\frac{4I}{3}\right)}{2\left(\pi R^2\right)}\left(\frac{R}{2}\right)+0=\frac{{\mu }_{0}I}{3\pi R}$