Continuous Solutions of a Functional Equation Involving the Harmonic and Arithmetic Means

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1 Continuous Solutions o a Functional Equation Involving the Harmonic and Arithmetic Means Rebecca Whitehead and Bruce Ebanks aculty advisor) Department o Mathematics and Statistics Mississippi State University PO Drawer MA Mississippi State, MS 3976 ebanks@math.msstate.edu May 17, 01 Abstract Let : R + R + R be a continuous solution to the unctional equation ), ab = a, b) or all a, b > 0. The aim o this paper will be to prove that is o the orm a, b) = φab) where φ : R + R is an arbitrary continuous unction. The procedure used to prove the theorem will be to develop a sequence related to the arithmetic and harmonic means and show that it converges to a limit. Then the continuous unction o the convergent sequence will also converge. It will be shown that ta, t a) is actually a unction o the variable t alone. I, y > 0 such that ta = and t a = y, then a and t are ound to be a = y and t = y. Thus the unction, y) is equal to a continuous unction o the single variable y. 1 Introduction A unctional equation is an equation in which one or more unctions appearing there are unknown. These unctions are implicitly dened by the equation, and the job o the solver is to determine their eplicit orms. Functional equations can involve unctions o one or more variables. The unctional equation discussed in this paper is related to the arithmetic and harmonic means Aa, b) = 1

2 and Ha, b) = ab, respectively, where a, b > 0. In particular, the unctional equation discussed in this paper is related to the Gaussian arithmetic-geometric mean iteration. Starting with a = a 0 > 0 and b = b 0 > 0, the recursive relationship or terms a n and b n o the Gaussian arithmetic-geometric mean iteration are dened by and a n+1 = a n + b n b n+1 = a n b n. The limits have been shown to have the property lim b n = lim a n as n. The limit Ma, b) = lim b n = lim a n as n is called the Gaussian arithmetic-geometric mean o a and b. This limit has been shown see [], pages -3) to satisy the unctional equation Ma, b) = M, ) ab. The unctional equation discussed in this paper is a variant in which we replace the geometric mean by the harmonic mean. Problem o Haruki and Rassias In 1995, Haruki and Rassias [1] posed the ollowing problem. They observed that any unction o the orm, y) = φy) satises the unctional equation ), ab = a, b),.1) and they asked whether all continuous solutions o this unctional equation must be o this orm. This question was again posed in the 1998 book o Sahoo and Riedel [] see pages 30-31). We answer this question in the armative. That is, we will prove the ollowing. Theorem.1. A unction : R + R + R is a continuous solution o the unctional equation.1), i and only i a, b) = φab) or an arbitrary continuous unction φ : R + R.

3 3 A Key Lemma The proo o our theorem hinges on the convergence o a certain sequence o real numbers. To motivate our consideration o this sequence, suppose t and are two positive real numbers and let a = t and b = t. Then equation.1) becomes t + t t, t + t = t, t ) which simplies to ) t + t, t = t, t ). 3.1) + 1 Let h : R + R + R be the continuous unction dened by h, t) := t, t ) 3.) or all, t > 0. Then rom 3.1) we obtain ) + 1 h, t = h, t) 3.3) or all, t > 0. Our plan is to iterate this unctional equation, so we consider the unction s : R + R + dened by s) := + 1, where > 0. Now consider the sequence {s n) )} = {, s), ss)), s 3) ),... } or n 0. That is, we dene s 0) ) :=, s 1) ) := s), and s n+1) ) = ss n) )) or each positive integer n. Lemma 3.1. For each > 0 and or all positive integers n, we have 1. s n) ) 1, and. { s n) ) } is a monotone non-increasing sequence. Proo. We prove part 1 o the lemma by induction. For the case n = 1, we note that 1) 0 or any > 0. Epanding, we have + 1 0, and so or all > 0. s) =

4 Now let > 0 and suppose that s m) ) 1 or some positive integer m. Then, mimicking the argument above, rom s m) ) 1) 0 it ollows that s m) )) s m) ) + 1 0, and thereore s m+1) ) = sm) )) + 1 s m) ) 1. Thus, by the Principal o Mathematical Induction, s n) ) 1 or all > 0 and or all positive integers n. We also prove part by induction. For the case n = 1, ) s) s) s ) + 1 ) = s) s) = s)) 1 s) s) 1)s) + 1) = s) By the proo o part 1, we know that s) 1 or all > 0. Thus the displayed quantity is nonnegative, and this yields s) s ) ) or > 0. Now let > 0 and suppose that s m) ) s m+1) ) or some positive integer m. Then we calculate ) s s m+1) ) s m+) ) = s m+1) m+1) )) + 1 ) s m+1) ) = sm+1) )) 1 s m+1) ) = sm+1) ) 1)s m+1) ) + 1). s m+1) ) Since or > 0 we have s n) ) 1 or all n 1 by part 1, the displayed quantity is nonnegative. Hence s m+1) ) s m+) ) or all > 0. By the Principal o Mathematical Induction, s n) ) s n+1) ) or all > 0 and or all positive integers n. This completes the proo o the lemma. 4 Proo o our Theorem Since the sequence { s n) ) } is monotone non-increasing and bounded below by 1, it converges to a limit L 1. Letting n in the equation s n+1) ) = sn) )) + 1 s n) ) 4

5 we obtain L = L + 1 L Thereore L = 1, and the series { s n) ) } converges to 1. Now we rewrite equation 3.3) as hs), t) = h, t) and iterate: and so on. Thus by iteration we have hs), t) = h, t), hs ) ), t) = hs), t), hs 3) ), t) = hs ) ), t), h, t) = hs n) ), t), or all, t > 0 and or all positive integers n. Since the unction h is continuous, letting n in this last equation gives h, t) = h1, t), a unction o only one variable. Let us dene the unction φ : R + R by φt) := h1, t), 4.1) or all t > 0. The unction φ inherits continuity rom the unction h. The equation 3.) can now be written as t, t ) = h, t) = φt ), or all, t > 0. Now given a, b > 0, we can nd t, > 0 such that t = a and t = b. a Namely, we choose = b and t = ab. Then we have a, b) = t, t ) = φt ) = φab), where φ is an arbitrary continuous unction. This completes the proo o our theorem. As a nal observation, we note that all values o solution unctions are determined by their values on the diagonal. By equations 3.) and 4.1), we see that a, b) = φab) = h1, ab) = ab, ab). Reerences [1] H. Haruki and Th. Rassias, A new analogue o Gauss's unctional equation, Int. J. Math. Math. Sci ), [] P.K. Sahoo and T. Riedel, Mean Value Theorems and Functional Equations, World Scientic Publishing Co. Pte. Ltd., Singapore,