On (k, f, l)-Chordal Polygons

T.K. Pogány, M. Radić

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract The article generalizes the concept of the k-chordal polygons and the (k, λ, l)- chordal polygons as introduced in papers [3], [4] respectively. Namely, here we describe the positive real function f where f(a_{1}), ⋯, f(a_{n}) are the side lengths of a l-chordal polygon and in the same time a_{1}, ⋯, a_{n} are the lengths of the sides of a k-chordal polygon.

Mathematics Subject Classification: 51E12 Key words: Geometrical inequality, k-chordal polygon, (k, f, l)-chordal polygon, convexity, star-likeness

1 Preliminary definitions and results

This article is an addendum to the papers [3], [4], where the concepts of the kchordal and its generalized variant the (k, λ, l)-chordal polygons are introduced and discussed. The (k, λ, l) property means the following. Let a_{1}, ⋯, a_{n} be the lengths of the sides of a k-chordal polygon \underline{A}, and a_{1}^{λ}, ⋯, a_{n}^{λ}, λ ∈ R_{+} are the side lengths of a lchordal polygon, then \underline{A} is called (k, λ, l). The properties of these general geometrical objects are exposed in detail in [4]. Here we are interested in the following natural generalization of the main results of the paper [4].

Our main goal is now to describe a positive, real function f so that it maps the side lengths a_{1}, ⋯, a_{n} of a k-chordal polygon into f(a_{1}), ⋯, f(a_{n}), being these the lengths of the sides of a l-chordal polygon.

For convenience we first repeat briefly some definitions and the results of article [3].

Definition 1.1. Let \underline{A} = A_{1}A_{2} ⋯ A_{n} be a chordal polygon and let Cn ; \underline{A} be its circumcirle. By S_{A_{i}} and Ŝ_{A_{i}} we denote the semicircles so that

S_{A_{i}} ∪ Ŝ_{A_{i}} = Cn ; \underline{A}, A_{i} ∈ S_{A_{i}} ∩ Ŝ_{A_{i}}.

The polygon \underline{A} is said to be of the first kind if it is fulfiled: (i) not all vertices A_{1}, A_{2}, ⋯, A_{n} lie on the same semicircle; (ii) for every three consecutive vertices A_{i}, A_{i+1}, A_{i+2} it is valid that A_{i} ∈ S_{A_{i+1}} ⟹ A_{i+2} ∈ Ŝ_{A_{i}}; (iii) any two consecutive vertices A_{i}, A_{i+1} do not lie on the same diameter.

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 91-96 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Definition 1.2. Let \underline{A} = A_{1}A_{2} ⋯ A_{n} be a chordal polygon and let k be a positive integer. The polygon \underline{A} is said to be k-inscribed and called k-chordal polygon if it

is of the first kind and if ∑_{i=1}^{n}^{}A̸_{i}CA_{i+1} = 2kπ, where C denotes the centre of the

circumscribed circle to the polygon \underline{A}.

In the sequel we will write β_{i} for ^{}C̸A_{i}A_{i+1} according to the notations introduced in [3],[4]. It follows that for a k-chordal polygon \underline{A} it results that

(1)  ∑_{i=1}^{n} β_{i} = (n − 2k)π.

The result that we need frequently throughout the paper and concerning the sum of the side lengths in the k-chordal polygons, is: If a_{1}, a_{2}, ⋯, a_{n} are the side lenghts of the k-chordal polygon \underline{A}, then

(2)  ∑_{i=1}^{n} a_{i} > 2ka^{⋆},

where a^{⋆} = max_{1≤j≤n} a_{j}, cf. [3], Corrolary 1.2.

Finally we are ready to introduce the concept of the (k, f, l)-chordal polygons.

Definition 1.3. When \underline{A} is k-chordal polygon with the sides of lenghts a_{1}, ⋯, a_{n} and in the same time f(a_{1}), ⋯, f(a_{n}) are the side lenghts of a l-chordal polygon, then \underline{A} is called (k, f, l)-chordal polygon. The polygon that consists of the sides of the lengths f(a_{i}), we denote as f(\underline{A}).

We also need the definition of the convexity and the definition of the star-likeness with respect to the origin. According to the references [1], [2] we give these definitions.

Definition 1.4. The real function f : [a, b] → R is said to be convex on [a, b] when for all x, y ∈ [a, b] and all t ∈ [0, 1] we have f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y).

Definition 1.5. The real function f : [0, b] → R; f(0) = 0, that satisfies f(tx) ≤ tf(x) for all x ∈ [0, b] and t ∈ [0, 1] is called star-like with respect to the origin.

In the following section we give our principal results. Namely, we are focusing to the sufficient conditions for the existence of the (k, f, l)-chordal polygons and we are looking for some sufficient conditions upon the lengths f(a_{i}); i = \overline{1, n} that ensures the existence of the chordal polygons \underline{A} and f(\underline{A}).

2 Main results

Now, according to the already given definitions we are ready to give our principal results concerning the existence of the introduced (k, f, l)-chordal polygons. At first we give the sufficient conditions for the existence of this geometrical object, when we possess the information that \underline{A} is k-chordal.

Theorem 2.1. Let \underline{A} be a k-chordal polygon with the sides of the lengths a_{1}, ⋯, a_{n}

where k ∈ N; k ≤ [\frac{n − 1}{2}]. Then f(a_{1}), ⋯, f(a_{n}) are the side lengths of a l-chordal

polygon, i.e. \underline{A} is (k, f, l)-chordal if

1. f : R_{+} → R_{+} is convex and monotonously nondecreasing,

2.

(3)  arcsin \frac{f (\frac{2k}{n} a^{⋆})}{f(a^{⋆})} > \frac{lπ}{n} .

Proof. It is clear that if we prove the existence of some angles β_{i}, i = \overline{1, n} that satisfies

(4)  β_{1} + β_{2} + ⋯ + β_{n} = (n − 2l)\frac{π}{2}

(5)  \frac{cos β_{1}}{f(a_{1})} = ⋯ = \frac{cos β_{n}}{f(a_{n})} ,

then f(\underline{A}) is obviously l-chordal, i.e. \underline{A} is (k, f, l)-chordal.

For this purpose consider

L := ∑_{i=1}^{n} arcsin \frac{f(a_{i})}{f(a^{⋆})}.

As the main part of arcsin x is convex when x ∈ (0, 1), and f is convex as well, by the assumptions of the Theorem, it follows that

(6)

L ≥ n arcsin \{\frac{1}{nf(a^{⋆})} ∑_{i=1}^{n} f(a_{i})\} ≥ n arcsin \frac{f (\frac{1}{n} ∑_{i=1}^{n} a_{i})}{f(a^{⋆})}

> n arcsin \frac{f (\frac{2k}{n} a^{⋆})}{f(a^{⋆})} > \frac{lπ}{n} .

Here we use the fact that \underline{A} is k-chordal, that means it holds (2), and the arcsine functions are always well-defined, since f is nondecreasing. Therefore (6) ensures the existence of the unique θ ∈ (0, 1) so that

(7)  ∑_{i=1}^{n} arcsin \{\frac{f(a_{i})}{f(a^{⋆})} θ\} = lπ.

But this is equivalent to

∑_{i=1}^{n} arccos \{\frac{f(a_{i})}{f(a^{⋆})} θ\} = (n − 2l)\frac{π}{2} .

There is also an unique β^{⋆} ∈ (0, \frac{π}{2} ) corresponding to the maximal side length f(a^{⋆}) that θ = cos β^{⋆}. Consecutively putting

β_{i} = arccos \{\frac{f(a_{i})}{f(a^{⋆})} cos β^{⋆}\} , i = \overline{1, n},

we get the angles β_{i} of a l-chordal polygon the sides of which are of the lengths f(a_{i}), i = \overline{1, n} so that (4) and (5) are valid. Thus, the proof is complete. Q.E.D.

In the continuation we formulate a modest improvement of the previous Theorem, but with more weeker assumptions. Namely, we have not assumed the k -chordality of the initial polygon \underline{A}, but the price of this fact has to be paid by some additional conditions upon f.

Theorem 2.2. Let a_{1}, ⋯, a_{n} be any given lengths with the property (2), i.e. so that a_{1} + ⋯ + a_{n} > 2ka^{⋆}. Then \underline{A} is (j, f, l)-chordal, j = \overline{1, k} if

1. f : R_{+} → R_{+}, is convex, monotonously increasing,

2. f(0) = 0,

3.

(8)  arcsin \frac{f (\frac{2k}{n} a^{⋆})}{f(a^{⋆})} > \frac{lπ}{n} .

Proof. The first step in proving procedure is completely the same as in the previous Theorem. Therefore f(\underline{A}) is clearly l-chordal. Now it remains to prove that \underline{A} is k-chordal under the assumed conditions. For this purpose let us consider (8). This inequality is equivalent to

(9)  f (\frac{2k}{n} a^{⋆}) > f(a^{⋆}) sin \frac{lπ}{n} .

It is not hard to see that f is star-like with respect to the origin, because it is convex and f(0) = 0 ^{1}. Therefore it is

(10)  f (\frac{2k}{n} a^{⋆}) > f(a^{⋆}) sin \frac{lπ}{n} ≥ f (a^{⋆} sin \frac{lπ}{n} ) .

Furthermore, since f is bijective too, i.e. f is convex on a compact and it is increasing, by (10) we conlude that

(11)  arcsin \frac{2k}{n} > \frac{lπ}{n} .

Now, repeating the procedure for finding the exact angles β_{i} so that (1) holds and

\frac{cos β_{i}}{a_{i}} = \frac{cos β_{j}}{a_{j}}; 1 ≤ i < j ≤ n,

1Speaking in the convexity hierarchy terminology, we assume that f ∈ K_{1}(1). The K_{3}(1) class consists of all functions being star-like with respect to the origin. So, as K_{1}(1) ⊂ K_{3}(1)consult e.g.

[2], the star-likeness of f is obvious.

exposed in the proof of the Theorem 2.1., we finish the proof. Q.E.D.

Finally, we give a result in some way reverse to the Theorem 2.1. Namely, we are looking for the broadest possible class of functions so that a condition like (2) e.g.

a_{1} + ⋯ + a_{n} > γ(k, n)a^{⋆}

suffices for the existence of the k-chordal polygon with the side lengths f(a_{i}), i = \overline{1, n}. The existence of such constant γ(k, n) is shown in [4], and it has to be γ(k, n) ≥

n sin \frac{kπ}{n}, cf. [4], Theorem 3.4.

Theorem 2.3. Let a_{1}, a_{2}, ⋯, a_{n} be any given lenghts so that

(12)  f(a_{1}) + f(a_{2}) + ⋯ + f(a_{n}) > γ(k, n)f(a^{⋆}),

where k ∈ N and k ≤ [\frac{n − 1}{2}]. Assume γ(k, n) ≥ n sin \frac{kπ}{n}. If the positive increas-

ing function f is star-like with respect to the origin, then there is a (k, f, k)-chordal polygon whose sides have lengths a_{1}, a_{2}, ⋯, a_{n}.

Proof. By the convexity of the arcsine function we conclude:

(13) ∑_{i=1}^{n} arcsin \frac{f(a_{i})}{f(a^{⋆})} ≥ n arcsin \{\frac{1}{nf(a^{⋆})} ∑_{i=1}^{n} f(a_{i})\} ≥ n arcsin \frac{γ(k, n)}{n} ≥ kπ.

Now, it follows that there is the unique θ_{1} ∈ (0, 1), so that by (13) we get

∑_{i=1}^{n} arcsin \{\frac{f(a_{i})}{f(a^{⋆})} θ_{1}\} = kπ;

thus f(\underline{A}) is k-chordal.

In the same time it is

(14)  f(a_{i}) = f (\frac{a_{i}}{a^{⋆}} · a^{⋆}) ≤ \frac{a_{i}}{a^{⋆}} f(a^{⋆}),

henceforth f is star-like with respect to the origin. That means

(15)  \begin{aligned}γ(k, n) & < & \frac{f(a_{1}) + ⋯ + f(a_{n})}{f(a^{⋆})} \\ & ≤ & \frac{\frac{a_{1}}{a^{⋆}} f(a^{⋆}) + ⋯ + \frac{a_{n}}{a^{⋆}} f(a^{⋆})}{f(a^{⋆})} \\ & = & ∑_{i=1}^{n} \frac{a_{i}}{a^{⋆}} < ∑_{i=1}^{n} arcsin \frac{a_{i}}{a^{⋆}} . \end{aligned}

Finally the k-chordality of \underline{A} immediately follows from (15), or, what is the same, \underline{A} is (k, f, k)-chordal. Q.E.D.

Remark 2.1 Taking f(x) = x^{λ} the Theorems 2.1., 2.2. and 2.3. give the main results of the articles [3], [4] respectively as λ = 1 and λ ≥ 1.

References

[1] D.S. Mitrinović, Analytic Inequalities, Grad.evinska knjiga, Belgrade, 1970 (in Serbian)

[2] D.S. Mitrinović and E.L. Pečarić, Means in Mathematics, Naučna knjiga, Belgrade, 1989 (in Serbian)

[3] M. Radić, Some inequalities and properties concerning chordal polygons, Math. Inequal. Appl. 2, No.1.(1998), pp. 141-150

[4] M. Radić and T. Pogány, Some inequalities concerning the existence of (k, λ, l)- chordal polygons, (submitted).

Tibor K. Pogány University of Rijeka Department of Maritime Studies 51000 Rijeka, Studentska 2, Croatia e-mail: poganj@brod.pfri.hr

Mirko Radić University of Rijeka Faculty of Philosophy Department of Mathematics 51000 Rijeka, Omladinska 14, Croatia