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By Stanislaw Saks

ISBN-10: 0444408738

ISBN-13: 9780444408730

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4], and [1010, p. 65]. 26. ) (Remark: These results are Bernoulli’s inequality. 2. Let x be a nonnegative number, and let α be a real number. If α ∈ [0, 1], then α + xα ≤ 1 + αx, whereas, if either α ≤ 0 or α ≥ 1, then 1 + αx ≤ α + xα. 1. 21. ) (Remark: This result is equivalent to Bernoulli’s inequality. 3. Let x and α be real numbers, assume that either α ≤ 0 or α ≥ 1, and assume that x ∈ [0, 1]. Then, (1 + x)α ≤ 1 + (2α − 1)x. Furthermore, equality holds if and only if either α = 0, α = 1, x = 0, or x = 1.

2) The set with no elements, denoted by ∅, is the empty set. If X = ∅, then X is nonempty. A set cannot have repeated elements. For example, {x, x} = {x}. However, a multiset is a collection of elements that allows for repetition. The multiset consisting of two copies of x is written as {x, x}ms . However, we do not assume that the listed elements x, y of the conventional set {x, y} are distinct. The number of distinct elements of the set S or not-necessarily-distinct elements of the multiset S is the cardinality of S, which is denoted by card(S).

In this case, f (x) = y. 2. Let R be a relation on X. Then, the following terminology is defined: i) R is reflexive if, for all x ∈ X, it follows that (x, x) ∈ R. ii) R is symmetric if, for all (x1, x2 ) ∈ R, it follows that (x2 , x1 ) ∈ R. iii) R is transitive if, for all (x1, x2 ) ∈ R and (x2 , x3 ) ∈ R, it follows that (x1, x3 ) ∈ R. iv) R is an equivalence relation if R is reflexive, symmetric, and transitive. 3. Let R1 and R2 be relations on X. If R1 and R2 are (reflexive, symmetric) relations, then so are R1 ∩ R2 and R1 ∪ R2 .

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Analytic Functions by Stanislaw Saks

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