A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY

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1 Volume , Iue 3, Article 68, 3 pp. A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ DEPARTMENT OF APPLIED MATHEMATICS NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943, USA cfrenzen@np.edu DEPARTMENT OF MATHEMATICS COLUMBUS STATE UNIVERSITY COLUMBUS, GA 3907, USA ionacu_eugen@coltate.edu DEPARTMENT OF APPLIED MATHEMATICS NAVAL POSTGRADUATE SCHOOL MONTEREY, CA 93943, USA ptanica@np.edu Received 26 September, 2006; accepted 0 September, 2007 Communicated by S.S. Dragomir ABSTRACT. In thi paper we prove ome reult which imply two conjecture propoed by Janou on an etenion to the p-th power-mean of the Erdö Debrunner inequality relating the area of the four ub-triangle formed by connecting three arbitrary point on the ide of a given triangle. Key word and phrae: Erdö-Debrunner inequality, Area, Etrema, p-th power mean Mathematic Subject Claification. 26D07, 5M6, 52A40.. MOTIVATION Given a triangle ABC and three arbitrary point on the ide AB, AC, BC, the Erdö- Debrunner inequality [] tate that. F 0 minf, F 2, F 3, where F 0 i the area of the middle formed triangle DEF and F, F 2, F 3 are the area of the urrounding triangle ee Figure.. The author would like to acknowledge the thorough, very contructive comment of the referee who helped to correct the calculation in an earlier argument and who untintingly trived to improve the paper. The third author acknowledge partial upport by a Reearch Initiation Program grant from Naval Potgraduate School

2 2 C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ Figure.: Triangle ABC The p-th power-mean i defined for p on the etended real line by min,..., n, if p =, M p, 2,..., n = n i= p i n p, if p 0, M 0 = n n i= i, if p = 0, ma,..., n, if p =. It i known ee [2, Chapter 3] that M p i a nondecreaing function of p. Thu, it i natural to ak whether. can be improved to:.2 F 0 M p F, F 2, F 3. The author of [4] invetigated the maimum value of p, denoted here by p ma, for which.2 i true, howing that p ma ln 3 and diproving a previouly publihed claim. ln 2 Since p ma < 0, by etting = BD AC, y = EC AB, z = AF BC, and q = p, it i hown in AE BC F B AC DC AB [4] that.2 i equivalent to.3 f, y, z := g, y q + gy, z q + gz, q 3, where g, y := + y, q min, the analogue of p ma, atifie ln 3 ln 2 q min, and the variable are uch that g, y 0, gy, z 0, gz, 0 and, y, z > 0. Let u introduce the natural domain of f, ay D, to be the et of all triple, y, z R 3 with, y, z > 0 and g, y 0, gy, z 0 and gz, 0. Since f, y, z 0, the function f ha an infimum on D. Let u denote thi infimum by m. To complete the analyi begun in [4], the author propoed the following two conjecture. Conjecture.. For any q q 0 = ln 3, if f, y, z = m, then yz =. ln 2 Conjecture.2. If q q 0, then m = 3. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

3 A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY 3 In thi paper we prove Theorem 2. that for every q > 0, the function f ha a minimum m, and if thi infimum i attained for, y, z D, then yz =. Moreover, we how Theorem 3. that for every q > 0 we have m = min{3, 2 q+ }. Our reult are more general than Conjecture. and.2 above, and imply them. After the initial ubmiion of our paper, we learned that the initial conjecture of Janou were alo proved by Macioni [5]. However, our method are different and Macioni Theorem can be obtained from our Theorem 3.2 for q = q 0. In other word, we etend the Erdö-Debrunner inequality to the range p < 0, and p = q 0 = ln3/2 i jut a particular value of p for which C ln2 p = in Theorem 3.2. Thi range can be etended for p > 0 only in the trivial way, i.e., F 0 0 M p F, F 2, F 3, ince F 0 = 0 and M p F, F 2, F 3 0 if, for intance, the point F coincide with the point B and the point E coincide with the point C. A hown net, becaue the minimum of f i attained at the ame point for every p > q 0, we cannot have an inequality of the type F 0 C 3 F F 2 F 3 with C > 0 either. So, our Theorem 3.2 i, in a ene, jut a far a one can go along thee line in generalizing the Erdö-Debrunner inequality. Of coure, one may try to how that there i a contant c p > 0 uch that however, that i beyond the cope of thi paper. F 0 c p M p F, F 2, F 3, p 0, 2. PROOF OF CONJECTURE. We are going to prove the following more general theorem from which Conjecture. follow. Theorem 2.. For every q > 0, the function f defined by.3 ha a minimum m and if f, y, z = m for ome, y, z D then yz =. Proof. Since f,, = 3 and f2, /2, = 2 q+ we ee that 0 m min{3, 2 q+ }. Since g, y > y, we ee that if y > +3 q =: a then f, y, z > 3. Similarly, f, y, z > 3 if or z i greater than a. On the other hand, if < then g, y > / > a = a 3/q, which implie that f, y, z > 3, again. Clearly, if y or z are le than /a we alo have f, y, z > 3. Hence, we can introduce the compact domain { C :=, y, z } a, y, z a, g, y 0, gy, z 0 and gz, 0, which ha the property that 2. m = inf{f, y, z, y, z C}. Since any continuou function defined on a compact et attain it infimum, we infer that m i a minimum for f. Moreover, every point at which f take the value m mut be in C. Let u aume now that we have uch a point, y, z a in the tatement of Theorem 2.: f, y, z = m. We will conider firt the cae in which, y, z i in the interior of C. By the firt derivative tet ometime called Fermat principle for local etrema, thi point mut be a critical point. So, f,y,z = 0, which i equivalent to 2 g, yq = gz,. q Hence the ytem 2.2 f, y, z = 0, 0, 0 J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

4 4 C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ i equivalent to = g, yq gz, q ; y2 = gy, zq g, y q ; z2 = gz, q gy, z q. Multiplying the equalitie in 2.3 give yz =, and thi prove the theorem when the infimum occur at an interior point of C. Now let u aume that the minimum of f i attained at a point, y, z on the boundary of C. Clearly the boundary of C i {, y, z C {, y, z} {a, /a} or g, ygy, zgz, = 0}. We ditinguih everal cae. Cae : Firt, if = a, ince /z > 0, we have q f, y, z z + > a q = 3 m. Thu, we cannot have f, y, z = m in thi ituation. Similarly, we eclude the poibility that y or z i equal to a. Cae 2: If = /a, becaue y > 0, it follow that q f, y, z + y > a q = 3 m. Again, thi implie that f, y, z = m i not poible. Likewie, we can eclude the cae in which y, or z i /a. Cae 3: Let u conider now the cae in which g, y = 0, that i y = oberve that we need >. Therefore, f, y, z = f,, z become the following function of two variable q q k, z = + z + z + = z + q q + z +. Hence, uing the arithmetic-geometric inequality, we obtain z + q q q q z z z + = 2 [ 2 + z + 2 q+, z where we have ued X + /X 2 for X > 0. We oberve that if m = 2 q+ thi i equivalent to q q 0, ince f, y, z = m, we mut have equality in 2.4, which, in particular, implie that z =, that i, yz =. If m < 2q+, then 2.4 how that we cannot have f, y, z = m. Either way, the conjecture i alo true in thi ituation. The other cae are treated in a imilar way. ] q J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

5 A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY 5 3. RESULTS IMPLYING CONJECTURE.2 We are going to prove a reult lightly more general than Conjecture.2: Theorem 3.. Aume the notation of Section 2. Then, for every q > 0 we have m = min{3, 2 q+ }. In [4], Theorem 3. wa hown to be true for ln 3 q. So we are going to aume ln 2 without lo of generality that q < throughout. Baed on what we have hown in Section 2, we can let z = and tudy the minimum of the function h, y = f, y, on the trace of y y the domain C in the pace of the firt two variable: { H =, y, y [/a, a] and + } y. Before we continue with the analyi of the critical point inide the domain H we want to epedite the boundary analyi. We define A := / + y, B := /y + /y and C := y +. It i a imple matter to how 3. ABC + AB + AC + BC = 4. If, y i on the boundary of H, then either y = +, or y =. The firt poibility i equivalent to B = 0, and the econd i equivalent to A = 0 if >, or C = 0 if <. Now, if C = 0 then AB = 4. Hence f, y, z A q + B q + C q = A q + B q 2 AB q = 2 +q. Similar argument can be ued for the cae A = 0 or B = 0. Hence, ince h, 2 = 2 q+ we obtain the following reult. Lemma 3.2. The minimum of h on the boundary of H, ay H, i 3.2 min{h, y, y H} = 2 q+. Net, we analyze critical point inide H. By Fermat principle, thee critical point will atify h h = 0, = 0, that i, y and 2 qaq 2 y qbq + y + qc q = 0, qa q + y 2 qbq + qc q = 0. We remove the common factor q in both of thee equation to obtain Aq 2 y Bq + y + C q = 0, A q + y 2 Bq + C q = 0. Solving for A q in 3.4 and ubtituting in 3.3 we get or + 3 y 2 Bq + Cq 2 y Bq + y + C q = 0 y + + C q = + + y B q. 3 y 2 J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

6 6 C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ Since y + + > 0, > 0, by implifying the previou equation we obtain 3.5 C q = Bq 2 y 2. Moreover, replacing 3.5 in 3.4, ay, we get which implie A q + y 2 Bq + Bq 2 y = 0, 2 A q 3.6 = Bq 2 2 y. 2 Therefore, if we put 3.5 and 3.6 together, we obtain 3.7 The equality Aq 2 A q 2 = C q i equivalent to 2 q + y = Bq 2 y 2 = Cq. = y +. If we introduce the new variable = +q q >, the lat equality can be written a y = +. Similarly, the equality Aq 2 = Bq can be manipulated in the ame way to obtain 2 y 2 + y = y 2 q y + y, or y = y + y. So, the two equation in 3.7 give the critical point inide the domain H, which can be claified in the the following way: C :,; C 2 : {, : atifie = + }; C 3 : {, y : y atifie y = y + y }; { } C 4 :, y : y = + y and =,, y. y y + Let t +t if t > 0, tt φt = 2 if t =, which i continuou for all t > 0. Since it i going to be ueful later, we note that φ atifie 3.8 φ = tφt, for all t > 0. t Thu C 2 i the et of all, with φ = ; C 3 i the et of all, y y with φ/y = ; and C 4 i the et of all, y, y with y = φ 3.9 = φ/y. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

7 A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY 7 Remark 3.3. Due to 3.8, the cla C 3 i in fact the et of all point, y, where y = / and, i in C 2. To determine the nature of the critical point, we compute the econd partial derivative, and analyze the Heian of h at thee critical point. Uing relation 3.7 we obtain: 2 h = qq 2 4 Aq y 2 Bq 2 + y + 2 C q q 3 Aq y Bq 2q + ycq = q qc q 2 A + y B C = qcq 2 + y q A + BC + 2 y + 2 AB ABC qq + = ABCC + 4, 2 ABC 2 q uing the fact that y + = C +. Similarly, we get 2 h = qq A q B q C q q y2 2 y 4 y 3 2q + Cq = q qc q y A y 2 B + 2 C 3. = qcq 2 + q 2 y 2 A + CB AC y yabc qq + 2 = ABCB + 4, ABC 2 q uing yb + = +. Further, the mied econd derivative i h y = qq 2 Aq y 3 Bq 2 + y + C q 2 + q 2 y 2 Bq + C q = 2qC q q qc q A + + yb = qc q + q q = qq + ABC 2 q ABC 2 2 BC, y + + C ACB + + AB BC ABC uing the identitie yb + = +, and y + = C +. The dicriminant determinant of the Heian D := 2 h 2 h y h 2 y B q J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

8 8 C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ can be calculated uing 3.0, 3. and 3.2 to obtain D = q2 q + 2 A 2 B 2 C 2 B + C + A 2 B 2 C 4 2q 4 ABC B + C BC BC 2 = q2 q + 2 A 2 B 2 C 2 BC + B + C A 2 B 2 C 4 2q 4 ABCBC + B + C + 4 4BC 2 B2 C 2. Now, by 3. we have 4BC B 2 C 2 = BC4 BC = BCABC + AB + AC = ABCBC + B + C and o we have the factor ABCBC + B + C in all the term above. Thi implie that the dicriminant of h at the critical point, that i, auming relation 3.7 can be implified to 3.3 D = q2 q + 2 BC + B + C ABC + 4 ABC3 2q 4. 2 Our net lemma claifie the critical point,. Lemma 3.4. For q /3, the point, i a local minimum. For q < /3 the critical point, i not a point of local minimum. Proof. If q = /3, h, = 3, o, ince h, y = f, y, 3 by inequality.3, we y etablih that, i a local minimum point of h. Aume q /3. For = and y = the formulae etablihed above become 2 h, = 2 h, = 2q3q > 0, 2 y2 2 h, = q3q > 0 y and D = 3q 2 3q 2. Hence, the Heian i poitive definite and o we have a local minimum at thi point cf. [3, Theorem 2.9.7, p. 74]. For the econd part, oberve that D, > 0, but 2 h, < 0 if 2 q < /3, and o, i not a local minimum if q < /3. Theorem 3.5. If q /3, there eit only one olution 0 of φ =, 0 <, uch that a 0, if q > /3 > 2; 2 2 b 0 [2, 2 2 if q < /3 < < 2. 2 Furthermore, there i only one olution y 0 = / 0 to φ/y =, 0 < y. If q = /3 = 2, there are no poitive olution for φ =, 0 <, or φ/y =, 0 < y. Proof. Firt, aume q = /3. Then = 2. It i traightforward to how that, i in C 2 implie =. However, = i not allowed. Similarly,, y i in C 3 implie y = 0, or, which are not allowed. Thu, if q = /3, there are no poitive olution for φ =, 0 <, or φ/y =, 0 < y. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

9 A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY t Figure 3.: The graph of ψ Now we hall aume throughout that q /3. Let u oberve that the equation φ = can be written equivalently a ψ = 0, where ψt := t 2t +, t 0. We firt aume that q > /3, which i equivalent to > 2. The derivative of ψ i ψ 2 which ha only one critical point t 0 = 2/. Since > 2, we obtain that t0 <. We have ψ0 =, ψ = 0 and then automatically ψt 0 = 2/ 22/ + = 2/ < 0. The econd derivative of ψ i: ψ 2. Thi how that ψ i a conve function and o it graph lie above any of it tangent line and below any ecant line paing through it graph, a in Figure 3.. We conclude that 0 i between the interection of the tangent line at 0, with the -ai and the interection between the ecant line connecting 0, and t 0, ψt 0 with the -ai. Since ψ 0 = 2, the equation of the tangent line i y = 2 and o it interection with the -ai i /2, 0. The equation of the ecant line through 0, and t 0, ψt 0 i y = ψt 0 t 0, or y = 2. Thi give the interection with the -ai:, 0. 2 Therefore the firt part of our theorem i proved. The lat claim i hown imilarly. Remark 3.6. A q approache from below, become large and the interval around 0 part a in Theorem 3.5 i very mall. Theorem 3.7. The critical point in C 2 and C 3 are not point of local minimum for h. Proof. We how that the Heian of h i not poitive emi-definite by howing that the dicriminant D i le than zero. We will treat only the critical point of type C 2, ince the cae C 3 i imilar. We get A = A 0, = / 0, B = B 0, = / 0 and C = C 0, = 2 0. The condition D < 0 i the ame a < 0, J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

10 0 C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ t Figure 3.2: The graph of γ, γ 2 which i equivalent to or = < 0 0, 2 0, 2, if q /3 < 2 and,, if q > /3 > By Theorem 3.5 part a and b, and the inequality 2 checked, we ee that D < 0, which complete the proof. Net, we define the two function 3.5 γ t := t + t t > 2 2 that can be eaily t + t and γ 2 t :=, t > 0, t. t t Thee function are etended by continuity at t = 0 and t =. We ketch the graph of thee two function for = 6 in Figure 3.2. The following two lemma will be crucial for our final argument. Lemma 3.8. For every >, the function γ i conve and the function γ 2 i concave. Proof. For γ, one can readily check that where γ t = 2t 2 t 3 β t β t = t + + t t. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

11 A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY Net we oberve that β t = + β 2 t, where β 2 t = t t + and oberve that β 2 t 2 = t 2 t. The ign of β 2 i then eaily determined, howing that β 2 ha a point of global minimum at t =. Hence β 2 t β 2 = 0. Thi implie that β i trictly increaing. Since β = 0 we ee that the ign of β i the ame a the ign of t 3. Thi mean that γ t > 0 for all t > 0. At t = the limit i 2 > 0 alo. 3 In order to deal with γ 2, we rewrite it a t r t r+ γ 2 t = = θt t r r+, where r := > 0. Since γ 2 t = r + θt 2r+ r+ θtθ t r r + θ 2, we have to how that δt := θtθ t r r + θ 2 < 0 for all t > 0. The firt and econd derivative of θ are given by and θ t = t2r r + t r + rt r t r 2, θ t = r[r t2r r + t 2r 2 + r + t r r t r 2 ] t r 3. Thee two epreion ubtituted into δt yield where the ign of δ i determined by δt = rt2r 2 r + δ t, δ t := t 2r+2 t r+2 + t r r t r+ 2r 2 + 4r +. However, δ = 0 and δ r δ 2 t, where δ 2 t = 2t r+2 r + 2t 2 + rr + + 2r 2 + 4rt. Again, oberve that δ 2 = 0 and δ 2t = 2r + 2δ 3 t, where δ 3 t = t r+ r + t + r. Finally, δ 3 = 0 and δ 3 r. Now δ 3 ha only a ingle critical point at t = which i a global minimum. Thu δ 3 t δ 3 = 0. Thi how that δ 2 i trictly increaing on 0, and i zero at t =. Therefore, δ t ha a minimum at t = implying that δ t 0 with it only zero at t =. Hence δt < 0 for all t. Thi, and lim t δt = r+r+2 how that γ 2r 2 2 i a concave function and complete the proof. We hall need the following well-known reult which may be formulated with weaker hypothee. For convenience, we include it here. Lemma 3.9. The graph of two function f and g twice differentiable on [a, b], f conve f > 0 and g concave g < 0 cannot have more than two point of interection. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

12 2 C. L. FRENZEN, E. J. IONASCU, AND P. STĂNICĂ Proof. Suppoe by way of contradiction that they have at leat three point of interection. We thu aume, f =, g, 2, f 2 = 2, g 2, 3, f 3 = 3, g 3, with a < 2 < 3 b are uch point. Net, we look at the epreion E = f 2 f f 3 f 2 = g 2 g g 3 g By the Mean Value Theorem applied twice to f and f the epreion E i equal to E = f c f c 2 = f cc c 2 < 0, c, 2, c 2 2, 3, c c, c 2 and applied to g and g give E = g ξ g ξ 2 = g ξξ ξ 2 > 0, ξ, 2, ξ 2 2, 3, ξ ξ, ξ 2 which i a contradiction. Let u oberve that if 0 i a olution of the equation φ 0 = then / 0, 0 i a olution of the ytem 3.9. Theorem 3.0. If q /3, then the only critical point of h are,, 0,,, 0, 0, 0, where 0 i a in Theorem 3.5. If q = /3,, i the only critical point. Proof. Start with q = /3. Then Lemma 3.4 and Theorem 3.5 imply the claim that, i the only critical point. Net, for q /3, we conider the following ytem in the variable and y: = y + y y 3.6 y + = y. y y In what follow net we how that every olution of C 4 i a olution of 3.6. Indeed, if, y i in C 4, then it atifie Thi implie that = +, = y + y = y y + y. y y + y, or + y + y + y + y = y y. Now, ue y y + = y to implify the firt term of the previou equality and derive + y + y + y y y = 0. Finally, we olve for to obtain y y + y + + y y + + y = y + y + y + y y + y +, which i equivalent to 2y 2y + = 2y 2y. So, if y thi implie = y y which implie that = y + y /. y + y y y We oberve that, / 0, / 0, 0 are olution of 3.6. By Lemma 3.8 and 3.9, thee two point are the only olution of thi ytem, which prove our theorem. Uing Lemma 3.8 and Theorem 3.0 we infer the net reult. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.

13 A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY 3 Theorem 3.. The point in / 0, 0 in C 4 i not a minimum point. Proof. Since at thi point, A = 2 0, B = / 0, C = / 0 we ee that ABC = 2 0 and 2 0 the dicriminant D take the ame form a in Theorem 3.7. Hence the proof here follow in the ame way a in Theorem 3.7. Putting together Lemma 3.2, 3.4, and Theorem 3.7, 3.0, and 3., we infer the truth of Theorem 3.. In term of our original problem, we have obtained the following theorem. Theorem 3.2. Given the point D, E, F on the ide of a triangle ABC, and F 0, F, F 2, F 3 the area a in Figure., then where C p = min, F 0 C p M p F, F 2, F 3, /p, for all p < 0. Proof. We know from [4] that F 0 F = z + = gz,, F 0 F 0 = + z = gy, z. F 3 y = + y = g, y, F 2 We howed that f, y, z = g, y q +gy, z q +gz, q ha the minimum m = min3, 2 q+. Hence F q 0 F q + F q 2 + F q 3 min3, 2 q+. Thi i equivalent to Raiing thi to power p Thi give C p = min ma3, 2 q F q + F q 2 + F q 3 F q < 0 p = q, we get min3 /p, 2 /p F p + F p 2 + F p 3 p F0. /p., REFERENCES [] O. BOTTEMA, R.Z. DJORDJEVIĆ, R.R. JANIC, D.S. MITRINOVIĆ, P.M. VASIC, Geometric Inequalitie, Groningen, Wolter Noordhoff, 969. [2] P.S. BULLEN, Handbook of Mean and Their Inequalitie, Kluwer, Dordrecht, [3] J.J. DUISTERMAT AND A.C. KOLK, Multidimenional Real Analyi I Differentiation, Cambridge Studie in Advanced Mathematic Vol. 86, Cambridge Univerity Pre, New York, [4] W. JANOUS, A hort note on the Erdö-Debrunner inequality, Elem. Math., , [5] V. MASCIONI, On the Erdö-Debrunner inequality, J. Ineq. Pure Appl. Math., , Art. 32. [ONLINE: 0. J. Inequal. Pure and Appl. Math., , Art. 68, 3 pp.