Refinements of the Erdös-Mordell inequality, Barrow’s inequality, and Oppenheim’s inequality
- Jian Liu^{1}Email author
https://doi.org/10.1186/s13660-015-0947-2
© Liu 2015
Received: 18 July 2015
Accepted: 16 December 2015
Published: 4 January 2016
Abstract
In this paper, we present some new refinements of the Erdös-Mordell inequality, Barrow’s inequality, and Oppenheim’s inequality. Based on verification by computer, several related interesting conjectures are put forward.
Keywords
Erdös-Mordell inequality Barrow’s inequality Oppenheim’s inequality triangle interior pointMSC
51M161 Introduction
Inequality (1.1) was conjectured by Erdös [1] in 1935. Mordell and Barrow [2] first proved it in 1937, and since then this inequality is known as the Erdös-Mordell inequality and has attracted the attention of many mathematicians who offered various new proofs, generalizations, variations, sharpness, and conjectures (see [3–33] and the references therein).
The main purpose of this paper is to establish some new refinements of the Erdös-Mordell inequality, Barrow’s inequality (1.2) and Oppenheim’s inequality (1.3). We also propose some closely related interesting conjectures.
The remainder of this paper is organized as follows. In the next section, we first establish a new inequality involving an interior point of a triangle. Then we use this inequality, the Erdös-Mordell inequality and another well-known inequality to deduce a refinement of the Oppenheim inequality. In Section 3, we present a refinement with one parameter for Barrow’s inequality (1.2). In Sections 4 and 5, some new refinements of the Erdös-Mordell inequality are established. Finally, in Section 6, we propose some interesting related conjectures as open problems.
2 A new refinement of Oppenheim’s inequality
In this section, we first establish a new geometric inequality which may be of independent interest. We shall make use of this result and its weaker form (see inequality (4.3) below) several times in the sequel.
Lemma 2.1
Proof
It is well known that the following inequality is related to the Erdös-Mordell inequality. See [9] for instance.
Lemma 2.2
We now state and prove the following refinement of the Oppenheim inequality.
Theorem 2.1
Proof
It is clear that the equality conditions in (2.8) are the same as in (1.1) and (2.1), i.e., if and only if \(\triangle ABC\) is equilateral and P is its center. The proof of Theorem 2.1 is completed. □
3 A refinement of Barrow’s inequality
We first give the following compact inequality.
Lemma 3.1
If we let \(x=R_{1}\), \(y=R_{2}\), and \(z=R_{3}\) in (3.2), then inequality (3.1) follows immediately. To our surprise, inequality (3.1) has not been given in [4, 10] although it is compact.
Remark 3.1
Oppenheim [10] obtained a weighted generalization of Barrow’s inequality (1.2), which is equivalent with (3.2) when x, y, z are positive.
For Barrow’s inequality (1.2), we now give the following refinement with one parameter.
Theorem 3.1
Proof
The following particular case (\(k=0\)) of Theorem 3.1 is of interest.
Corollary 3.1
This double inequality can be regarded as an associated result of (2.1) and (3.1).
4 Refinements of the Erdös-Mordell inequality I
In this section and next section, we shall give some new refinements of the Erdös-Mordell inequality. First, we point out that we have the following result, which is a counterpart of Theorem 3.1.
Theorem 4.1
Proof
Since the equalities of \(w_{1}\ge r_{1}\), \(w_{2}\ge r_{2}\), and \(w_{3}\ge r_{3}\) are all valid if and only if P is the circumcenter of \(\triangle ABC\), thus by the equality conditions of (2.1) and (3.1), it is easy to conclude that both equalities of (4.3) and (4.4) hold if and only if \(\triangle ABC\) is equilateral and P is its center. Further, we know that the statement in Theorem 4.1 for the equalities is right. This completes the proof of Theorem 4.1. □
In particular, for \(k=0\) in Theorem 4.1, we obtain the following.
Corollary 4.1
Next, we apply inequality (4.6) to prove another refinement with one parameter for the Erdös-Mordell inequality.
Theorem 4.2
Proof
For \(k=0\) in Theorem 4.2, we get the following.
Corollary 4.2
Theorem 4.3
Proof
In view of the equality conditions of (4.10), (2.7), and (1.1), we immediately conclude that the equalities in (4.11) hold if and only if \(\triangle ABC\) is equilateral and P is its center. This completes the proof of Theorem 4.3. □
In the proof of the final theorem given in this section, we shall use the following well-known result (for proofs, see e.g., [5, 14, 19]).
Lemma 4.1
Theorem 4.4
Proof
5 Refinements of the Erdös-Mordell inequality II
The refinements of the Erdös-Mordell inequality, given in the last section, involve six segments \(R_{1}\), \(R_{2}\), \(R_{3}\), \(r_{1}\), \(r_{2}\), \(r_{3}\) but not the geometric elements of \(\triangle ABC\). In this section, we use an unified method based on two lemmas (Lemmas 5.1 and 5.2 below) to establish three new refinements of the Erdös-Mordell inequality, which also cover the sides, the altitudes, and the medians of the triangle \(ABC\) besides the above six segments.
Before offering the results of this kind of double inequalities, we first prove two related lemmas.
Lemma 5.1
Proof
Lemma 5.2
Proof
Lemma 5.1 and Lemma 5.2 show that if the positive real numbers \(k_{1}\), \(k_{2}\), \(k_{3}\) form a triangle and satisfy (5.2) and (5.6), then the refinement (5.1) of the Erdös-Mordell inequality holds. Next, we shall establish three refinement of the Erdös-Mordell inequality by applying this conclusion.
Theorem 5.1
Proof
In (5.10), for \(\mu=0\), we obtain the following.
Corollary 5.1
Remark 5.1
In (5.10), for \(\lambda=0\), we obtain
Corollary 5.2
Since \(r_{1}+R_{1}\ge h_{a}=r\sum a/a\), where r is the inradius of \(\triangle ABC\). Thus, by the first inequality of (5.12), we obtain a lower bound of \(\sum R_{1}\), namely
Corollary 5.3
Theorem 5.2
Proof
Summarizing, we deduce that the double inequality (5.17) holds for \(0\le k\le2\). In addition, by Lemma 5.1 and Lemma 5.2, we easily conclude that the equalities in (5.17) occur hold only when \(\triangle ABC\) is equilateral and P is its center. The proof of Theorem 5.2 is completed. □
In particular, for \(k=0\) in Theorem 5.2, we obtain
Corollary 5.4
Finally, we give a dual result of Theorem 5.2, which shows that if we replace the altitudes \(h_{a}\), \(h_{b}\), \(h_{c}\) by the corresponding medians \(m_{a}\), \(m_{b}\), \(m_{c}\) in (5.17) then the inequalities still hold, i.e., we have the following conclusion:
Theorem 5.3
Proof
The equality conditions of (5.19) follow easily from Lemma 5.1 and 5.2. This completes the proof of Theorem 5.3. □
For \(k=0\) in (5.19), we get the following counterpart of (5.18).
Corollary 5.5
6 Open problems
For the inequalities established in this paper, we can propose a lot of new problems. We next introduce some related conjectures as open problems, which have been checked by computer.
The first inequalities of (1.4) and (2.8) prompt the author to propose the following conjecture.
Conjecture 6.1
Conjecture 6.2
The following conjecture is similar to Theorem 4.1.
Conjecture 6.3
Another conjecture with one parameter, which comes from considering generalizations of the previous double inequality (5.12), is as follows.
Conjecture 6.4
Many years ago, the author found the following interesting ‘r-w’ phenomenon: If an inequality involving the segments \(r_{1}\), \(r_{2}\), \(r_{3}\) and other geometric elements holds for any interior P of \(\triangle ABC\), then the inequality from replacing \(r_{1}\), \(r_{2}\), \(r_{3}\) by \(w_{1}\), \(w_{2}\), \(w_{3}\) in the original inequality, respectively, often still holds for either any or acute triangle \(ABC\). Based on this phenomenon and with the verification by computer, we can propose some dual inequalities for the results presented in this paper. Here, we only give one example, which is a dual conjecture of Theorem 5.1 as follows.
Conjecture 6.5
Declarations
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Authors’ Affiliations
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