Uniform $\lambda$-property in $L^1\cap L^{\mathrm{\infty }}$

Here it is proved that the space $L^1\cap L^{\mathrm{\infty }}$ equipped with the standard interpolation norm ${\Vert \cdot \Vert }_{L^1\cap L^{\mathrm{\infty }}}=max\{{\Vert \cdot \Vert }_{L^1},{\Vert \cdot \Vert }_{L^{\mathrm{\infty }}}\}$ has the uniform $\lambda$-property if and only if $\mu (T)\le 1.$ Replacing the standard norm with an equivalent one ${\Vert \cdot \Vert }_{L^1\cap L^{\mathrm{\infty }}}^{\prime }=$ ${\Vert \cdot \Vert }_{L^1}+{\Vert \cdot \Vert }_{L^{\mathrm{\infty }}}$, a different result is obtained.: $(L^1\cap L^{\mathrm{\infty }},{\Vert \cdot \Vert }_{L^1\cap L^{\mathrm{\infty }}}^{\prime })$ has the uniform $\lambda$-property if and only if \(\mu (T)