# 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

Read or Download 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) PDF

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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.