Vol. 10 No 3 - Pi Mu Epsilon

A WEIGHTED AM-GM-HM INEQUALITYDan Kalman was originally attracted to college mathematics by its lackof laboratories. He was an applied mathematician at the AerospaceCorporation and now teaches at the American University. John Mathewsearned his doctorate at Michigan State University and has written texts oncomplex variables and numerical methods.Ayoub B. AyoubPennsylvania State U., Ogontz CampusThe familiar arithmetic mean-geometric mean inequality,Chapter ReportsThe NEW YORK OMEGA Chapter (St. Bonaventure University) had asits major activity, Professor Francis Leary reports, its popular MathematicsForum. Fifteen talks were presented last year, mostly by students, includingone on "The mathematics of coyotes, roadrunners, and ants". TheChapter's graduating vice-president, Heather Lecceardone, won thedepartment's Mathematics Medal.The MICHIGAN ZETA chapter (University of Michigan-Dearborn)cosponsored a student-faculty mixer which was attended by most of thefaculty, nine alumni, and more than fifty students. Professor JohnFrederick Fink says that is the best attendance ever at such an event. TheChapter inducted eighteen new members last year.The CONNECTICUT GAMMA Chaper (Fairfield University) sponsored itsannual High School Math Bowl, similar to the College Bowl. ProfessorJoan Wyzkoski Weiss reports that eight teams from local high schoolsparticipated. At the spring initiation ceremony, nineteen new members wereinitiated and Carole Lacapagne of the U. S. Department of Education spokeon "The prime number connection: bow number theory helps secure vitaldata. "holds with more general weights [I]:mla + q b 2 a *Ibrn2where a, b > 0, ml + q = 1, and mi, 3 2 0. We will modify thisinequality to permit negative weights, then extend it to include the harmonicmean. To that end, we will first prove the weighted AM-GM inequalityusing the natural logarithm function. If we consider the points A: (a, In a)and B: (b , ln b) on the graph of y = lnx , then the point C that divides ABin the ratio 3 : mi will bemla +-b mllna +m^\nb')"I +m^. m~ +"'2However, if ml + q = 1, then C will have the coordinatesIf, in addition, we assume that ml, 3 > 0, then C divides AB internally(see Figure 1). Since the graph of y = lnx is concave down (y" =- llx 2 < 0) , the point D: (ml a + + ln(mI a + q b)) lies verticallyabove C. It is obvious that C will coincide with A or B if q = 0 or ml= 0, respectively. Thus, ml , q Si 0 implies thatSince the function y = lnx is increasing (y' = llx > O), then