By Stanislaw Saks

ISBN-10: 0444408738

ISBN-13: 9780444408730

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**Sample text**

The letter s always represents a complex scalar. The letter z may or may not represent a complex scalar. The inequalities c ≤ a ≤ d and c ≤ b ≤ d are written simultaneously as c≤ a b ≤ d. The preﬁx “non” means “not” in the words nonconstant, nonempty, nonintegral, nonnegative, nonreal, nonsingular, nonsquare, nonunique, and nonzero. ” “Increasing” and “decreasing” indicate strict change for a change in the argument. The word “strict” is superﬂuous, and thus is omitted. Nonincreasing means nowhere increasing, while nondecreasing means nowhere decreasing.

Let I ⊆ R be a ﬁnite or inﬁnite interval, let f : I → R. Then, the following statements are equivalent: i) f is convex. ii) f is continuous, and, for all x, y ∈ I, 2 1 1 1 3 (f [ 2 (x+y)]+f [ 2 (y+z)]+f [ 2 (x+z)] ≤ 13 [f (x)+f (y)+f (z)]+f [ 31 (x+y+z). (Remark: This result is Popoviciu’s inequality. See [1039, p. 4. 7. Let [a, b] ⊂ R, let f : [a, b] → R, and assume that f is convex. Then, b f [ 12 (a + b)] ≤ 1 b−a a f (x) dx ≤ 12 [f (a) + f (b)]. (Proof: See [1039, pp. 1. Let x and α be real numbers, and assume that x ≥ −1.

Iv) The symmetric hull of G is the graph sym(G) = (X, sym(R)). v) The transitive hull of G is the graph trans(G) = (X, trans(R)). vi) The equivalence hull of G is the graph equiv(G) = (X, equiv(R)). vii) G is reﬂexive if R is reﬂexive. viii) G is symmetric if R is symmetric. In this case, the arcs (x, y) and (y, x) in R are denoted by the subset {x, y} of X, called an edge. ix) G is transitive if R is transitive. x) G is an equivalence graph if R is an equivalence relation. xi) G is antisymmetric if R is antisymmetric.

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