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Concerning triangle preserving functions.

Article Excerpt
Abstract. -- The concept of a triangle preserving function is introduced and it is proved that functions of the form f(x) = [cos.sup.s](x/k) belong to this class for integers k [greater than or equal to] 2 and real numbers [less than or equal to] s [less than or equal to] k.

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Conditions under which three arbitrary positive real numbers a,b, and c may serve as the length of the sides a triangle has been the subject of inquiry for some time. Arguably, the oldest of the necessary and sufficient conditions is the cyclic inequality

a + b - c > 0, b + c - a > 0, c + a - b > 0, (1)

from which further necessary and sufficient conditions have been derived. An argument that used the following reformulation of (1)

1/2([a.sup.4] + [b.sup.4] + [c.sup.4]) < [a.sup.2][b.sup.2] + [b.sup.2][c.sup.2] + [c.sup.2][a.sup.2] (2)

lead V. E. Hoggat, Jr. (1959) to proposed the following problem:

Show that if a,b,c form a triangle, then [square root of a], [square root of b], [square root of c] form a triangle.

Together with Hoggat's own proof, two other solutions extending his result were also published: R.T. Hood's (1960) proof which replaces the square roots with roots of an arbitrary order n, and J. L. Brown's proof (1960) where these roots are replaced with a nonnegative nondecreasing subadditive function f defined on (0, [infinity]).

The search for sufficient conditions implying the feasibility of triangles that are based on the measurement of the angles of a given triangle naturally leads to triangle preserving functions.

For this paper a triangle preserving function (TPF) is a continuous function f : (0,[pi]) [right arrow] [R.sup.+] which satisfies the property that for any triangle T with interior angles [alpha], [beta] and [gamma], there exists a triangle T* whose sides have lengths f([alpha]), f([beta]) and f([gamma]). Nothing is implied about triangles T* and T being the same or different.

T. P. [arc.C] erpanova in (1963; 1966) studied the functions f(x) = cos(x/2) and f(x) = [cos.sup.2](x/2). She showed that each function was a TPF. In this paper the authors expand this work by showing that functions of the form f(x) = [cos.sup.s](x/k) are TPFs for integers k [greater than or equal to] 2 and real numbers [less than or equal to] s [less than or equal to] k.

BACKGROUND

In this section a number of theorems will be presented which will serve as background for the main results of this paper. First, from elementary geometry one has that a necessary and sufficient condition for three line segments to be the sides of a triangle is that the sum of the lengths of any two of the segments must be greater than the length of the third line segment. This is the first theorem:

Theorem 1. Suppose that f : (0,[pi]) [right arrow] [R.sup.+] is a continuous function. Then f will be a TPF if and only if each of the following inequalities hold for any triple [alpha], [beta] and [gamma] of positive numbers whose sum is [pi]:

i) f([alpha]) < f([beta]) + f([gamma]);

ii) f([beta]) < f([alpha]) + f([gamma]); and

iii) f([gamma]) < f([alpha]) + f([beta]).

Note 1. Whenever convenient and without loss of generality one assumes that if f : (0,[pi]) [right arrow] [R.sup.+] is a continuous function and if [alpha], [beta] and [gamma] are the interior angles of...

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