On certain new refinements of Finsler-Hadwiger inequalities
- Omran Kouba^{1}Email author
DOI: 10.1186/s13660-017-1356-5
© The Author(s) 2017
Received: 22 December 2016
Accepted: 1 April 2017
Published: 18 April 2017
Abstract
Several refinements of the Finsler-Hadwiger inequality and its reverse in the triangle are discussed. A new one parameter family of Finsler-Hadwiger inequalities and their reverses are proved. This allows us to obtain new bounds for the sum of the squares of the side lengths of a triangle in terms of other elements in the triangle. Finally, these new bounds are compared to known ones.
Keywords
triangle inequalities Finsler-Hadwiger inequalityMSC
11B681 Introduction
This inequality was published in 1919, but the authors in [2] traced it back to 1897 where it was proposed as Problem 273 in the Romanian magazine Gazeta Matematică (III(2), p.52) by Ionescu. So they proposed to call it the ‘Ionescu-Weitzenböck inequality’. At least five distinct proofs of (1.1) could be found in [3] and [4].
Surprisingly, it was noted in [14] that the Ionescu-Weitzenböck inequality (1.1) is equivalent to Finsler-Hadwiger inequality (1.2) by showing that the second follows from the first by applying the first to another triangle.
2 Results and discussion
But what about the upper bound \(\psi^{+}\)? To the best of the knowledge of the author, it seems that no refinement better than the constant \(\psi^{+}_{0}\equiv1\) is known or has been published. In this note we will prove that \(\psi^{+}=\psi_{1}^{+}\) with \(\psi_{1}^{+}(t)=\sqrt{t}\) is a refinement of the reverse Finsler-Hadwiger inequality, and we will also provide an alternative proof (different from the one given in [16]) of the lower bound with \(\psi_{1}^{-}\).
3 Theorems and proofs
The main tool in our proofs is the fundamental inequality in the triangle. This inequality has a long history, the reader may consult [21] or [22], Chapter 1, for more information.
Theorem 3.1
The fundamental inequality
We will use also several algebraic inequalities that are gathered in the next two lemmas.
Lemma 3.2
- (i)For \(t\in[0,1]\) we have$$ 2+5t-\frac{t^{2}}{4}+2(1-t)^{3/2}\le\frac{(8+t)^{2}}{4(4-t)}. $$(3.4)
- (ii)For \(t\in[0,1]\) we have$$ 2+5t-\frac{t^{2}}{4}-2(1-t)^{3/2}\ge\frac{t ( \sqrt{128+16 t+3t^{2}}-\sqrt{3} t )^{2}}{16}. $$(3.5)
- (iii)For \(t\in[0,1]\) we have$$ 2+5t-\frac{t^{2}}{4}-2(1-t)^{3/2} \ge\frac{ t(8+t)^{2}}{4(2+t)}. $$(3.6)
Proof
Proposition 3.4
Proof
Remark 3.5
Inequalities (3.8) and (3.9) are not new, see [23], Inequality 2.15, and particularly, (3.8) is Kooi’s inequality [19].
Now we are ready to give an alternative proof of the refinement of Finsler-Hadwiger inequality given in [16] and to present our refinement of its reverse, as announced in the Introduction.
Theorem 3.6
Proof
In the next result we provide an alternative ‘reverse Finsler-Hadwiger inequality’.
Theorem 3.7
Proof
Theorem 3.8
Proof
Taking \(\mu=3/4\) we obtain the next corollary.
Corollary 3.9
In fact Theorem 3.8 allows us to give the following ‘parametric Finsler-Hadwiger inequality’.
Corollary 3.10
Proof
Remark 3.11
This inequality is the best of its kind in the sense that if \(\lambda \in (-\frac{8\sqrt{6}+17}{95},0 )\) then there are triangles \(\triangle ABC\) that satisfy (3.21) and others that violate it. Indeed, testing (3.21) with an isosceles triangle with side lengths \(a=1\), \(b=c=t\ge1/2\) for large t shows that the condition \(\lambda\ge0\) is necessary for its validity. Testing its reverse for t near \(1/2\) (but larger than \(1/2\)) shows that the condition \(\lambda\le-\frac{8\sqrt{6}+17}{95}\) is necessary for the validity of the reverse.
Remark 3.12
For \(\lambda\le-\frac{8\sqrt{6}+17}{95}\) we have two different lower bounds for \(a^{2}+b^{2}+c^{2}\) given by Theorem 3.6 and Corollary 3.10, respectively. Testing the difference with our famous isosceles triangle with side lengths \(a=1\), \(b=c=t\ge1/2\) shows that these two lower bounds are not comparable.
4 Conclusion
In this work, we considered the problem of refining the Finsler-Hadwiger inequality and its reverse in the triangle. Several refinements are proposed and compared, and an optimal parametric refinement of this inequality and its reverse is proved.
Declarations
Acknowledgements
The author would like to thank the anonymous referees for reading this article carefully, providing valuable suggestions, and for bringing references [3, 4, 8] to his attention. The author declares that he has not received any financial support to do this research.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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