IWAMOTO, Seiichi

KIRA, Akifumi

Kyoto University Research Information Repository

Title On Golden inequalities(The Development of Information andDecision Processes)Author(s)IWAMOTO, Seiichi; KIRA, AkifumiCitation数理解析研究所講究録 (2006), 1504: 168-176Issue Date2006-07URL http://hdl.handle.net/2433/58489RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn Golden inequalitiesSeiichi IWAMOTO and Akifumi KIRADepartment of Economic EngineeringGraduate School of Economics, Kyushu UniversityFukuoka 812-8581, Japantel&fax. +81(92)642-2488, email: iwamoto@en.kyushu-u.ac.jpAbstractWe consider an inequality with equality condition where one side is greater (resp.less) than or equal to a multiple of the other side and an equality holds if and onlyif one value is a multiple of the other variable. When the two multiples constitutea Golden ratio, the inequality is called Golden. This paper presents six Goldeninequalities both among one-variable functions and among two-variable functions.We show a cross-duality between four pairs of Golden inequalities for one-variablefunctions. Similar results for twovariable functions are stated. Further a graphicrepresentation for Golden inequalities is shown.1 IntroductionBoth historically and practically, it is well known that the Golden raio has been takingan interesting part in many fields in science, technology, art, architecture, biology and soon $[3, 13]$ . In mathematical science, the Golden ratio has recently been incorporated intooptimization field [8-10, 12]. This paper is motivated by trying to introduce and discussthe Golden ratio in the fields of inequalities [1, 2, 4-7, 11]. In fact, there exists a mutualrelationship between optimizaion and inequality [1, 2,4, 5, 7, 10].In this paper we consider a class of Golden inequalities between two given functions.A pair of two real values is called Golden if it constitutes the Golden ratio. We consideran inequality of the following form. One side consists of one function only. The other sideconsists of a multiple of the other function. One side is greater (resp. less) than or equalto the other side. Further the sign of equality holds if and only if one value is a multipleof the other value. If the pair of two multiples is Golden, the inequality is called Golden.We present six Golden inequalities for one-variable quadratic functions. We show across-duality between four pairs of Golden inequalities for one-variable functions. Similarresults for two-variable quadratic functions are stated. Further a graphic representationfor Golden inequalities is shown.1504 2006 168-176 1682 The Golden RatioWe take a basic standard real number$\phi=\frac{1+\sqrt{5}}{2}\approx$ 1.61803The number $\phi$ is called the Golden number. It is defined as the positive solution toquadratic equation$x^{2}-x-1=0$ .A Fibonacci sequence $\{a_{n}\}$ is defined by second-order linear difference equation$a_{n+2}-a_{n+1}-a_{n}=0$ .Then we have a famous relation.Lemma 2.1$\phi^{n}=a_{n}\phi+a_{n-1}$$(\phi-1)^{n}=a_{-n}\phi+a_{-n-1}$$n=\cdots,$ $-2,$ $-1,0,1,2,$ $\cdots$where $\{a_{n}\}$ is the Fibonacci sequence urith $a_{0}=0,$ $a_{1}=1$ .The Fibonacci sequence is tabulated in Table 1:Table 1 Fibonacci sequence $\{a_{n}\}$On the other hand, the Fibonacci sequence has the analytic formLemma 2.2$a_{n}= \frac{1}{2\phi-1}\{\phi^{n}-(1-\phi)^{n}\}$Lemma 2.3$n=\cdots,$ $-2,$ $-1,0,1,2,$ $\cdots$(i) $\frac{\phi}{1}=\frac{1+\emptyset}{\phi}=\frac{1+2\phi}{1+\emptyset}=\frac{2+3\phi}{1+2\emptyset}=\frac{3+5\phi}{2+3\phi}=\cdots=\frac{a_{n}+a_{n+1}\phi}{a_{n-1}+a_{n}\phi}\approx 1.61803$(\"u) $\frac{a_{\mathrm{n}}+a_{n+1}\phi}{a_{n-1}+a_{n}\phi}=\cdots=\frac{2\phi-3}{5-3\phi}=\frac{2-\emptyset}{2\phi-3}=\frac{\phi 1}{2\phi}==\frac{1}{\phi-1}=\frac{\phi}{1}$(\"ui) $\frac{0236}{0146}:\approx:\frac{0382}{0236}\approx:\frac{0618}{0382}\approx\frac{1}{0.618}\approx\frac{1.618}{1}\approx:\frac{2618}{1618}\approx:\frac{4236}{2618}\approx\cdots\approx 1.618$where $\{a_{n}\}$ is the Fibonacci sequence.1693 One-variable FunctionsLet us consider an inequality between two one-variable functions $f,$ $g:R^{1}arrow R^{1}$ with anequality condition. We assume that inequality$f(u)\leq(\geq)\alpha g(u)$ on $R^{1}$ (1)holds. The sign of equality holds if and only if $u=\beta$ , where $\alpha$ and $\beta$ are real constants.Definition 3.1 We say that the pair $(\alpha, \beta)$ constitutes the Golden ratio if$| \frac{\beta}{\alpha}|=\emptyset$ or $| \frac{\alpha}{\beta}|=\emptyset$ .Deflnition 3.2 When the pair constitutes the Golden ratio, the inequality (1) is calledGolden.For instance we see that inequality$1+u^{2}\leq(1+\phi)\{1+(u-1)^{2}\}$ on $R^{1}$ (2)holds. The sign of equality holds if and only if $u=\emptyset$ . Further $(\phi, 1+\emptyset)$ constitutes theGolden ratio. Thus (2) is a Golden inequality.First we consider six Golden inequalities between one-variable quadratic functions.The inequalities (3) and (4) are pairs of Golden inequalities. The inequalities (5) and (6)are Golden. Thus we have six Golden inequahities in the following.Lemma 3.1 (i) It holds that$(2-\phi)\{1+(u-1)^{2}\}\leq 1+u^{2}\leq(1+\phi)\{1+(u-1)^{2}\}$ on $R^{1}$ . (3)The sign of left equality holds if and only if $u=1-\emptyset$ and the sign of right equality holdsif and only if $u=\phi([\mathit{8}-\mathit{1}\mathit{0}J)$.(ii) It holds that$(1-\phi)\{1+(v-1)^{2}\}\leq-1+v^{2}\leq\phi\{1+(v-1)^{2}\}$ on $R^{1}$ . (4)The sign of left equality holds if and only if $v=2-\phi$ and the sign of right equality holdsif and only if $v=1+\phi$ (Figure 2).(iii) The middle-right inequality (resp. left-middle) in (3) is equivalent to the left-middle(resp. middle-right) inequality in (4).Lemma 3.2 (i) It holds that$(-1+\phi)(1-u^{2})\leq u^{2}+(u-1)^{2}$ on $R^{1}$ (5)The sign of equality holds if and only if $u=2-\emptyset$ (Figure 3).It holds that$-u^{2}-(u-1)^{2}\leq\phi(1-u^{2})$ on $R^{1}$ (6)The sign of equality holds if and only if $u=1+\phi$ (Figure 4).(ii) The inequality (5) $\dot{w}$ equivalent to the middle-right inequality in (4). The inequality(6) is equivalent to the left-middle inequality in (4).1703.1 A pair of Golden inequalities between $1+(1-v)^{2}\mathrm{a}\mathrm{n}\mathrm{d}-1+v^{2}$Figure 2 A pair of Golden Inequalities$(1-\phi)\{1+(1-v)^{2}\}\leq-1+v^{2}\leq\phi\{1+(1-v)^{2}\}$ .The left and right equaiities attain at $\hat{v}=2-\emptyset$ and $v^{*}=1+\phi$ , respectively.1713.2 One Golden inequality between $1-u^{2}$ and $u^{2}+(1-u)^{2}$$x$$u$Figure 3 Golden Inequality $(-1+\phi)(1-u^{2})\leq u^{2}+(1-u)^{2}$ .The equality attains at $u^{*}=2-\emptyset$ .1723.3 The other Golden inequality between $1-u^{2}$ and $u^{2}+(1-u)^{2}$$.\tau$Figure 4 Golden Inequality $-\{u^{2}+(1-u)^{2}\}\leq\phi(1-u^{2})$ .The equality attains at $\text{\^{u}}=1+\emptyset$.1734 Two-variable ffinctionsLet us take two two-variable functions $f,$ $g:R^{2}arrow R^{1}$ . We assume that inequality$f(x,y)\leq(\geq)\alpha g(x,y)$ on $R^{2}$ (7)holds and that the sign of equality holds if and only if $y=\beta x$ .Deflnition 4.1 When the pair $(\alpha, \beta)$ constitutes the Golden ratio, the inequality (7) iscalled Golden.For instance, the inquality$x^{2}+y^{2}\geq(2-\phi)\{x^{2}+(y-x)^{2}\}$ on $R^{2}$ (8)holds. The sign of equality holds if and only if $y=(1-\phi)x$ . Thus inequality (8) is alsoGolden.4.1 Cauchy-SchwarzLet us take $f(x,y)=(ax+by)^{2},$ $g(x,y)=x^{2}+y^{2}$ , where $a(\neq 0),$ $b$ are real constants.Then it holds that$(ax+by)^{2}\leq(a^{2}+b^{2})(x^{2}+y^{2})$ on $R^{2}$ . (9)$\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}\alpha=\frac{b\mathrm{e}}{\mathrm{o}1a}\mathrm{a}\mathrm{n}.\mathrm{d}\beta=a^{2}+b^{2}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{G}\mathrm{o}1\mathrm{d}\mathrm{b}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\text{\’{e}} \mathrm{G}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{q}\mathrm{u}\mathrm{a}1\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{h}\mathrm{o}1\mathrm{d}\mathrm{s}\mathrm{i}\mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{n}1\mathrm{y}\mathrm{i}\mathrm{f}ay=bx$en ratio, the Cauchy-Schwart inequality4.2 Golden inequalitiesWe specify six Golden inequalities between two-variable quadratic functions. Each of (10)and (11) yields a pair of Golden inequalities. Both (12) and (13) are Golden inequalities.Thus we have also six Golden inequalities in the following.Theorem 4.1 (i) It holds that$(2-\phi)\{x^{2}+(y-x)^{2}\}\leq x^{2}+y^{2}\leq(1+\phi)\{x^{2}+(y-x)^{2}\}$ on $R^{2}$ . (10)The sign of left equality holds if and only if $y=(1-\phi)x$ and the sign of right equalityholds if and only if $y=\phi x$ .(ii) It holds that$(1-\phi)\{x^{2}+(y-x)^{2}\}\leq-x^{2}+y^{2}\leq\phi\{x^{2}+(y-x)^{2}\}$ on $R^{2}$ . (11)The sign of lefl equality holds if and only if $y=(2-\phi)x$ and the sign of right equalityhol& $if$ and only if $y=(1+\phi)x$ .(iii) The middle-right inequality (resp. left-middle) in (10) is equivalent to the left-middle(resp. middle-ri9ht) inequality in (11).174Theorem 4.2 (i) It holds that$(-1+\phi)(x^{2}-y^{2})\leq y^{2}+(y-x)^{2}$ on $R^{2}$ . (12)The sign of left equality holds if and only if $y=(2-\phi)x$ . It holds that$-y^{2}-(y-x)^{2}\geq\phi(x^{2}-y^{2})$ on $R^{2}$ . (13)The sign of right equality holds if and only if $y=(1+\phi)x$ .(ii) The inequality (12) is equivalent to the middle-right inequality in (11). The inequality(13) is equivalent to the left-middle inequality in (11).References[1] E.F. Beckenbach and R.E. BeUman, Inequalities, Springer-Verlag, Ergebnisse 30,1961.[2] R.E. Bellman, Introduction to Matriv Analy8is, $\mathrm{M}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{w}$-Hill, New York, NY, 1970(Second Edition is a SIAM edition 1997).[3] A. Beutekpacher and B. Petri, $\Leftrightarrow*j*\S\rfloor$–eas $k\mathfrak{B}\mathrm{E}\ _{\overline{\mathrm{R}}}^{-}\mathrm{f}\mathrm{f}\mathrm{l}\ -(\mathfrak{M}\#\mathrm{B}\mathfrak{F})$ , $\#\mathrm{E}$$\mathrm{f}\mathrm{f}\mathrm{l}\mathfrak{B}$ , 2005; (Original) Der Goldene Schnitt 2., \"uberarbeitete und erweitene Auflange,ELSEVIER GmbH, Spectrum Akademischer Verlag, Heidelberg, 1996.[4] S. Iwamoto, Inverse theorem in dynamic programming I, II, III, J. Math. Anal. Appl.58(1977), 113-134, 247-279, 439-448.[5] S. Iwamoto, Dynamic programming approach to inequalities, J. Math. Anal. Appl.58(1977), 687-704.[6] S. Iwamoto, Reverse function, reverse program and reverse theorem in mathematicalprogramming, J. Math. Anal. Appl. 95(1983), 1-19.[7] $\pi^{\mathrm{J}}\mathrm{B}\mu$ ru–, r#\S - $\varpi \mathrm{U}\epsilon*\Leftrightarrow \mathfrak{X}\}_{\mathrm{c}’}^{\vee}\supset \mathrm{v}\backslash \tau,$ $\Re*k_{\mathrm{r}}^{<}\lceil\{T\mathrm{f}\mathrm{f}\mathrm{l}\mathfrak{F}_{\mathrm{r}\mathrm{W}}^{\mathfrak{B}}\ *\sigma)\ovalbox{\tt\small REJECT}_{\grave{\mathrm{J}}}H\mathrm{J}$ , $\overline{R}\star\ \mathfrak{B}$$\Re\not\in\Re 8653$ , $1988*4$ fl, pp. 109-129.[8] S. Iwamoto, \yen 4BEffi\emptyset ae\not\equiv A\star f, $\Re\Re\#^{\underline{<}}\lceil\ovalbox{\tt\small REJECT}\#\emptyset\ovalbox{\tt\small REJECT} \mathrm{E}ffl\Re\rfloor$ , $\overline{R}\star \mathrm{a}\mathrm{e}\not\in\Re\Re\Re 8$1443, $2005\not\in 7$ fl, pp. 27-43.[9] S. Iwamoto, The Golden optimum solution in quadratic programming, Ed. W. Taka-hashi and T. Tanaka, Proceedings of The Fourth International Conference on Non-$1_{\dot{\mathrm{i}}}$ear Analysis and Convex Analysis (NACA05), under submission.[10] S. Iwamoto, The Golden trinity –optimility, inequality, identity –, $\lceil\Phi\#\emptyset\Re \mathfrak{B}$$\mathrm{f}\mathrm{f}\mathrm{l}\Re\rfloor$ , fi $\lambda\#\not\in\Re \mathfrak{F}\# 8$ , $2006\not\in$ , to appear.175[11] S. Iwamoto, R.J. Tomkins and C.-L. Wang, Some theorems on reverse inequalities,J. Math. Anal. Appl. 119(1986), 282-299.[12] S. Iwamoto and M. Yasuda, “Dynamic programming creates the Golden Ratio, too,”Proc. of the Sixth Intl Conference on Optimization: Techniques and Applications(ICOTA 2004), Ballarat, Australia, December 9-11, 2004.[13] H. Walser, $\mathrm{g}*_{J}^{j}+\Leftrightarrow \mathrm{j}(\mathfrak{B}^{\grave{\backslash }}l\mathrm{I}\doteqdot \mathbb{R}\ovalbox{\tt\small REJECT})$ , $\mathrm{B}\mathrm{X}^{-\Supset \mathrm{A}}--H_{\beta \mathrm{f}\mathrm{f}\mathrm{l}}\dagger\pm$, 2002; (Original) DER GOLDENESCHNITT, B.G. Teubner, Leibzig, 1996.176

On Golden inequalities(The Development of Information and Decision Processes)

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On Golden inequalities(The Development of Information and Decision Processes)

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