By Niculescu C.P., Persson L.-E.

ISBN-10: 0387243003

ISBN-13: 9780387243009

Thorough advent to a massive sector of arithmetic comprises fresh effects comprises many workouts

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**Example text**

A geometric application of the Hardy–Littlewood–P´ olya inequality; see M. S. Klamkin [127]) Let P , A and P , A denote the perimeter and area, respectively, of two convex polygons P and P inscribed in the same circle (the center of the circle lies in the interior of both polygons). If the greatest side of P is less than or equal with the smallest side of P, prove that P ≥P and A ≥ A with equality if and only if the polygons are congruent and regular. [Hint: Express the perimeter and area of a polygon via the central angles subtended by the sides.

N}. On the other hand, a moment’s reﬂection shows that we can pass immediately from these discrete inequalities to their integral analogues, corresponding to ﬁnite measure spaces. 6 It is important to notice that all numerical inequalities of the form f (x1 , . . , xn ) ≥ 0 for x1 , . . 9) where f is a continuous and positively homogeneous function of degree 1 (that is, f (λx1 , . . , λxn ) = λf (x1 , . . , xn ) for λ ≥ 0), extend to the context of Banach lattices, via a functional calculus invented by A.

Consequently, Ind is at most countable. Notice that this reasoning depends on the axiom of choice. 2 below. See also Exercise 3 at the end of this section. Simple examples such as f (x) = 0 if x ∈ (0, 1), and f (0) = f (1) = 1, show that upward jumps could appear at the endpoints of the interval of deﬁnition of a convex function. 4 If f : [a, b] → R is f (b−) exist in R and ⎧ ⎪ ⎨f (a+) f (x) = f (x) ⎪ ⎩ f (b−) a convex function, then f (a+) and if x = a if x ∈ (a, b) if x = b is convex too. 5 If f : I → R is convex, then either f is monotonic on int I, or there exists an ξ ∈ int I such that f is nonincreasing on the interval (−∞, ξ] ∩ I and nondecreasing on the interval [ξ, ∞) ∩ I.

### Convex functions and their applications by Niculescu C.P., Persson L.-E.

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