By Bloch S. (ed.)
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Additional info for Algebraic Geometry - Bowdoin 1985, Part 1
In [PBM, p. 443, Formulas 57-65] some formulas are given for the differentiation of P Fq with respect to the parameters. 22) for r E (0, 1) are called the complete elliptic integrals of the first and second kind, respectively. They will be studied in detail in Chapter 3. 23. Methods for proving inequalities. Throughout this book we will con stantly use inequalities to study functions of a real variable. An inequality of the form a (x) � b(x) (or a (x) :::: b (x) ) is said to be sharp if equality holds for some x .
Deduce that F(a - t, a + t; c; r) ::: F(a , a ; c; r) for a , c > 0 , r E (0, 1 ) , t E [O, a] . (4) For a , b positive and n a positive integer, show that with equality iff n = I . /ab. /(a, n)(b, n) ::: (a , n) The first inequality reduces to equality iff a to equality iff a = b . + (b, n). 35. Lemma. ( I ) For a, x E (0, 1), b, c E (0, oo), F(a, b; c; x) + F(-a , b; c; x) I. The second reduces > 2. (2) Let a , b, c E (0, 00), c > a + b . Then, for x E [O, I ] , r (c)r (c - a - b) . F(a, b; c; x) ::: r (c - a)r (c - b) If also a E (0, I ) , then and r (c)r (c + a - b) ::: F(-a, b; c; x) r (c + a)r (c - b) 2 ::: F(a, b; c; x) + F(-a , b; c; x) r (c) r (c - a - b) < [ r (c - b) r (c - b) + r (c + a - b) ] r (c - b) .
37. Lemma. ( 1 ) For a E (0, I ) and b , c > 0, the function ( 1 - F(-a, b; c; r))/r is strictly increasing and con vex on (0, I ) . (2) Th e function (F(a, b; c ; r) - F(-a , b; c ; r))/r i s strictly increasing and con vex on (0, I ) and has limit 2ab I c as r tends to 0. Proof. All coefficients of the power series of the functions in (1 ) and (2) are positive, and the constant term in the second series is 2ab/c. 38. Exercises. ( 1 ) Let (a, b) be an open interval in R , let f, g : (a, b) � (0, oo) be concave functions, and let f be increasing and g decreasing.
Algebraic Geometry - Bowdoin 1985, Part 1 by Bloch S. (ed.)