For ${\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{y}}_1,{\mathrm{y}}_2 \in \mathrm{R},$ if $0<{\mathrm{x}}_1<{\mathrm{x}}_2, {\mathrm{y}}_1={\mathrm{y}}_2$ and ${\mathrm{z}}_1={\mathrm{x}}_1+{\mathrm{iy}}_1, {\mathrm{z}}_2={\mathrm{x}}_2+{\mathrm{iy}}_2$ and ${\mathrm{z}}_3=\frac{1}{2}({\mathrm{z}}_1+{\mathrm{z}}_2),$ then ${\mathrm{z}}_1,{\mathrm{z}}_2$ and ${\mathrm{z}}_3$ satisfy

$0<{\mathrm{x}}_1<{\mathrm{x}}_2, {\mathrm{y}}_1={\mathrm{y}}_2$

${\mathrm{x}}_1<{\mathrm{x}}_2, \mathrm{ }{\mathrm{x}}_1+{\mathrm{x}}_2< 2{\mathrm{x}}_2\Rightarrow \boxed{\frac{{\mathrm{x}}_1+{\mathrm{x}}_1}{2}<{\mathrm{x}}_2}$

$\Rightarrow 2{\mathrm{x}}_1<{\mathrm{x}}_1+{\mathrm{x}}_2\mathrm{ }, \Rightarrow \boxed{x_1<\frac{{\mathrm{x}}_1+{\mathrm{x}}_1}{2}}$

$\mathrm{ }\therefore \mathrm{ }{\mathrm{x}}_1<\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2}<{\mathrm{x}}_2$

${\Rightarrow {\mathrm{x}}^2}_1<{\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2}\right)}^2<{\mathrm{x}}_2^2$

${\Rightarrow {\mathrm{x}}_1}^2+{{\mathrm{y}}_1}^2 < {\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2}\right)}^2+{\left(\frac{{\mathrm{y}}_1+{\mathrm{y}}_2}{2}\right)}^2<{{\mathrm{x}}_2}^2+{{\mathrm{y}}_2}^2$

$\Rightarrow \mathrm{ }{\left|{\mathrm{z}}_1\right|}^2<{\left|{\mathrm{z}}_3\right|}^2<{\left|{\mathrm{z}}_2\right|}^2$

$\therefore \left|{\mathrm{z}}_1\right|<\left|{\mathrm{z}}_3\right|< \left|{\mathrm{z}}_2\right|$