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This paper is a continuation of two earlier papers ([3],[4]) in the sense that in all of these papers we investigate ideas related to linear invariance and uniform local convexity, but in different geometries in the three papers. We assume the reader has some familiarity with these two papers since we sometimes omit details of proofs that are very similar to corresponding proofs in these two preceding papers; in such instances clear reference to the earlier papers is always provided. This paper deals with two notions of spherical linear invariance, one for locally schlicht meromorphic functions from the unit disk ID to the Riemann sphere IP and another for locally univalent meromorphic functions from the complex plane< L tothe Riemann sphere. Our two earlier papers deal with linear invariance and uniform local convexity for locally univalent functions from the disk ID to either the complex plane< L orthe unit disk. Here we extend the concept oflinear invariance tolocally univalent meromorphic functions/: ID—> IP or/: C-» IP by paralleling the definition of euclidean linear invariance ([11],[3]) rather than that of hyperbolic linear invariance [4]. In the case of euclidean and hyperbolic linear invariance the order of a schlicht function is less than or equal to 2; there is no analogous result for spherical linear invariance because there is no uniform upper bound for the second coefficient of normalized (f (0)= 0,/'(0)= 1) meromorphic univalent functions in the unit disk ID. Therefore, it should come as no surprise that there are differences between spherical linear invariance and euclidean or hyperbolic linear invariance.

Here is a brief outline of the paper. Preliminary …