Geodesic
On the Monotonicity of Certain Functionals in the Theory of Analytic Functions
Canadian mathematical bulletin, Tome 10 (1967) no. 5, pp. 723-729

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Let f(z) be a function regular for | z | < R, and denote by L(r) the length of the curve Γ onto which the circle | z | = r is mapped by f(z), i. e. If Dr is the image of | z | ≤ r on the Riemann surface of f(z) then its area S(r) is given by 
Rahman, Q.I. On the Monotonicity of Certain Functionals in the Theory of Analytic Functions. Canadian mathematical bulletin, Tome 10 (1967) no. 5, pp. 723-729. doi: 10.4153/CMB-1967-074-9
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     author = {Rahman, Q.I.},
     title = {On the {Monotonicity} of {Certain} {Functionals} in the {Theory} of {Analytic} {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {723--729},
     year = {1967},
     volume = {10},
     number = {5},
     doi = {10.4153/CMB-1967-074-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1967-074-9/}
}
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