The Schwarzian derivative and the Wiman-Valiron property
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- Langley, J.K. JAMA (2016) 130: 71. doi:10.1007/s11854-016-0029-5
Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has finitely many transcendental singularities. Using the Wiman-Valiron method, it is shown that if the Schwarzian derivative Sf of f is transcendental, then f has infinitely many multiple points, the inverse function of Sf does not have a direct transcendental singularity over ∞, and ∞ is not a Borel exceptional value of Sf. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.