Algebraic Geometry Santa Cruz 1995: Summer Research by David R. Morrison, Janos Kolla Summer Research Institute on

By David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry

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Additional info for Algebraic Geometry Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz (Proceedings of Symposia in Pure Mathematics) (Pt. 2)

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15) prove that, for every positive e, lim(1 — r)2P'M(r, f) = 0. 5 Suppose that f i (z) = f(z) C where C is a constant and that W(R), Wi(R) refer to f(z), f 1 (z) respectively. Prove that W(R —1c1) W1(R) W(R +10). 15) then so does f i (z). 13) without f (z) doing so. 4 Simultaneous growth near different boundary points We have seen that a function f (z) mean p-valent in lz I < 1 satisfies (z)1 = 0(1 — r) -2P (1z1 = r). However a function can be as large as this only on a single rather small arc of Izi = r.

4). 58 ... ,z q there counting multiplicity. Let y be a simple closed Jordan curve which surrounds the origin and lies in A. 8). Landau [1922] has proved a sharp version of this result, when y is a circle. The present simple argument is due to Hayman and Nicholls [1973]. 8). We define zv F(z) = f (z) / H( z 1— f v z) v=1 Then F(z) yields 0 in A and so the maximum principle applied to 1/F(z) inf If(z)1 zey inf IF(z)1 zey IF(0)1 = IaoI/ftz t, I. V =1 34 The growth of finitely mean valent functions Suppose next that if(z)i > E V =0 on y.

Her method, based on a distortion theorem of Ahlfors [1930], was extended by Spencer [1940b] to the more general case. If f(z) = ao + aiz + ... 3) = v=0 This dependence is essential. In fact any polynomial of degree p is p-valent and has at most p zeros in iz 1 < 1, but bounds for M(r) must clearly depend on all the coefficients. 1 A length-area principle 29 cannot grow too rapidly near several points of lz1 = 1 simultaneously, and in particular that, if M(r) attains the growth (1 — r)-2P, then if (r )1 attains this magnitude for a single fixed value of 0, and is quite small for other constant 0 as r 1.

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