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Notice that Z is a complex variable, and |arg z| is the length of an arc. I didn't get that what's the meaning of multiply a real number |Z| by the length of an arc |argz|. Can anyone give me a hint about what's the geometric meaning of this inequality? Thank you guys.

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    $\mbox{arg} z$ is the angle (in radians) of the complex number $z$. It would be the length of an arc of radius one and that angle, but $|z|$ isn't necessarily 1. The product $|z| | \mbox{arg}(z)|$ is the length of the circular arc from $|z|$ to $z$ in the complex plane. – 2017-01-25

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To get from $1$ to $z$, you can either travel along the purple path then the green path, or you can travel along the orange path. The orange path is a straight line so it's shorter.

pseudo triangle inequality

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    why rθ is the arc length? Suppose θ=60°, why the product of r and 60° is the arc length? – 2017-01-31
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    $\theta$ has to be in radians. The unit of radians was chosen specifically so that the length of an arc of angle $\theta$ would be $r\theta$. For example, with $\theta=2\pi$, $r\theta=2\pi r$, which is the correct formula for the circumference. An arc of smaller angle will yield a proportionally smaller arc length. – 2017-01-31