(i) a function f(z) is analytic at all points z in the upper half plane y≥0 that are exterior to a circle |z| =

(ii) denotes a semicircle , where

(iii) for all points z on , there is a positive constant such that |f(z)|≤,where:

Then,for every positive constant a,

The proof is based on a result that is known as Jordan’s inequality:

To verify this inequality, we first note from the graphs of the function and when that for all values of in that interval. Consequently, if R > 0,

e^{-Rsin theta}

and so

Hence