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ON BEST QUADRATIC TRIANGLE INEQUALITIES
ROBERT FRUCHT AND MURRAY S. KLAMKIN
ABSTRACT. Contrary to published results, it is shown that there do not exist 'strongest' (or 'best possible') homogeneous quadratic polynomial triangle inequalities of the form$$q(R, r) \leqslant s^{2} \leqslant Q(R, r)$$without further restrictions. Also, several best inequalities for symmetric functions of three positive variables are considered.
1. INTRODUCTION
As usual, R, r, s will denote, respectively, the radii of the circumscribed and inscribed circles, and the semi-perimeter of a triangle.
The necessary and sufficient condition for the existence of a triangle with the elements R, r, s is that$$s^{4}-2\left(2 R^{2}+10 R r-r^{2}\right) s^{2}+r(4 R+r)^{3} \leqslant 0\tag{1}\label{eq1}$$ and was given by Sondat [1, 13.8] back in 1891. Apparently, this result was not exploited until Blundon in two important papers [2], [3], on the structure of triangle inequalities, chose these variables as his canonical ones and showed that the B.I.'s (best possible homogeneous triangle inequalities) of the form$$f(R, r) \leqslant s^{2} \leqslant F(R, r)\tag{2}\label{eq2}$$are given by$$f(R, r)=2 R^{2}+10 R r-r^{2}-2(R-2 r) \sqrt{R^{2}-2 R r},\tag{3}\label{eq3}$$$$F(R, r)=2 R^{2}+10 R r-r^{2}+2(R-2 r) \sqrt{R^{2}-2 R r}\tag{4}\label{eq4}$$with simultaneous equality iff the triangle is equilateral. Incidentally, \eqref{eq3} and \eqref{eq4} follow immediately from Sondat's result.
Blundon [3] also gives the following theorem:
Let q(R, r) and Q(R, r) be quadratic forms with real coefficients. Then the strongest possible inequalities of the form$$q(R, r) \leqslant s^{2} \leqslant Q(R, r)\tag{5}\label{eq5}$$with equality only for the equilateral triangle, occur when$$q(R, r)=16 R r-5 r^{2}\tag{6}\label{eq6}$$$$Q(R, r)=4 R^{2}+4 R r+3 r^{2}\tag{7}\label{eq7}$$
Although inequalities \eqref{eq5}, which have appeared earlier in papers by Gerretsen [4] and Steinig [5], are valid, it will be shown by counter-examples that there are no 'best' ones of the form considered unless one makes some further restrictions. The error in Blundon's proof resides in the tacit assumption of the existence of B.I.'s of the required form. Blundon's results \eqref{eq5} are repeated in [1, 5.9] and by Bottema [6]. However, Bottema avoids Blundon's error by restricting the forms q and Q. We shall return to this point at the end of Section 2.
2. BEST TRIANGLE INEQUALITIES
Although Blundon does not give any explanation of what is meant by a B.I., it appears from his proof that the following interpretation should be given:An inequality $s^{2} \geqslant f^{*}(R, r)$ is defined as a B.I. of a given type iff $f^{*}(R, r)\geqslant f(R, r)$ for any other inequality $s^{2} \geqslant f(R, r)$ of the same type (and
similarly for $F^{*}(R, r) \geqslant s^{2}$).
THEOREM 1. Inequality \eqref{eq9} is valid only if either
(a) $\mu≥-5\quad \textit{and}\quad \kappa≤0$,
or
(b) $\mu<-5\quad \textit{and}\quad \kappa≤\frac{(\mu+5)^{2}}{ 4(\mu+1)}$
Inequality \eqref{eq10} is valid only if either
(c) $\mu^{\prime} \leqslant 3 \quad \textit { and } \quad \kappa^{\prime} \geqslant 4$
or
(d) $\mu^{\prime}>3 \quad \textit { and } \quad \kappa^{\prime} \geqslant\frac{\left(\mu^{\prime}+5\right)^{2}}{4\left(\mu^{\prime}+1\right)}$
Corresponding to cases (a) and (c), there is equality only for equilateral triangles. This also holds for (b) and (d) provided, additionally, $\kappa \neq\frac{(\mu+5)^{2}}{4\mu+1}$ and similarly for $\kappa^\prime$. Otherwise there is also equality for an isosceles triangle whose sides are in the ratio $$(-\mu-1):(-\mu-1):(-\mu+3) \text { or }\left(\mu^{\prime}+1\right):\left(\mu^{\prime}+1\right):\left(\mu^{\prime}-3\right)$$Since the proof of Theorem 1 given by the first author [7] is rather lengthy and not readily accessible, we sketch, in the Appendix, another proof based on the very useful method of Blundon [3].
Using Theorem 1, we can now give counter-examples to \eqref{eq5} being best possible. First, we consider \eqref{eq9} for κ=-0.1 and μ=-6, giving$$s^{2} \geqslant-0.1 R^{2}+16.7 R r-6 r^{2}\tag{11}\label{eq11}$$But here the left hand half of \eqref{eq5} is only stronger than \eqref{eq11} if R≥5r. For 2r<R<5r, \eqref{eq11} is stronger. For example, consider the right triangle of sides 6, 8, 10. Here, R=5, r=2, s=12. Then (11) gives $s^2$≥140.5 as compared with $s^2≥ 140$ for \eqref{eq5}. Analogously, it can be shown that the right hand half of \eqref{eq5} is incomparable with the inequality$$s^{2} \leqslant 4.75 R^{2}+0.5 R r+7 r^{2}\label{eq12}\tag{12}$$which is weaker than \eqref{eq5} only for $R>\frac{8r}3$, but not for $2r<R<-\frac{8r}3$.
So far, we have only given two examples of incomparable inequalities. The set of all such inequalities with respect to \eqref{eq5} is given by
THEOREM 2. An inequality of type \eqref{eq9} is incomparable with the left hand half of \eqref{eq5} iff
(e) $\mu<-5 \quad\textit { and }\quad\frac{\mu+5}4<\kappa \leqslant\frac{(\mu+5)^{2} } {4(\mu+1)}$
An inequality of type \eqref{eq10} is incomparable with the right hand half of \eqref{eq5} iff
(f) $\mu^{\prime}>3 \quad\textit { and }\quad\frac{\left(\mu^{\prime}+5\right)^{2} }{4\left(\mu^{\prime}+1\right)} \leqslant \kappa^{\prime}<\frac{\left(\mu^{\prime}+13\right)} 4$
Proof. For \eqref{eq9} to be weaker than the left hand half of \eqref{eq5}, we must have$$16 R r-5 r^{2} \geqslant \kappa R^{2}+(27-4 \kappa-\mu) \frac{R r } 2+\mu r^{2}\tag{13}\label{eq13}$$or,$$(x-2)(\mu+5-2 \kappa x) \geqslant 0$$where $x = \frac Rr \ge2$. Since $\kappa\le0$, the required condition is $4\kappa< \mu + 5$. The latter condition and Theorem 1, then imply part (e). Part (f) is proven similarly.
So far, we have disregarded the fact that Blundon only considers inequalities with equality solely for the equilateral triangle. If we wish to impose the same condition in Theorem 2 by ruling out the equality case for isosceles triangles, we merely replace the ≤ sign by < in (e) and (f).
We have previously mentioned that Bottema quotes Blundon's theorem, but instead of the two-parameter set of inequalities of type (9), he only considers the one-parameter family$$s^{2} \geqslant \mu r^{2}+(27-\mu) \frac{R r}2\label{eq14}\tag{14}$$ |
