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Asymptotic Behaviour of the Iterative Pythagorean Means

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Asymptotic Behaviour of the Iterative Pythagorean Means RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 19 = 523 (2015): 117-127 ASYMPTOTIC BEHAVIOUR OF THE ITERATIVE PYTHAGOREAN MEANS Tomislav Burić Abstract. Asymptotic expansion and behaviour of the iterative com- binations of the Pythagorean means (arithmetic, geometric and harmonic mean) is obtained and analyzed. Results are used for asymptotic compar- ison of means. 1. Introduction The well-known Pythagorean means are three classical means: arithmetic, geometric and harmonic mean defined by s + t 2 √ (1.1) A(s,t)= , G(s,t)= st, H(s,t)= 1 1 , 2 s + t where s and t are real positive numbers. These means have following ordering (1.2) min(s,t) H(s,t) G(s,t) A(s,t) max(s,t), ≤ ≤ ≤ ≤ and there are many results and papers concerning their properties, inequalities and comparison to other classical means. In recent papers [5–7] authors studied asymptotic expansions of means and developed new technique of deriving and calculating coefficients in this expansions. They succesfully used new method to establish various relations between classical means, see cited papers for details. The asymptotic expansion of the mean M(s,t) is representation of this mean in the form ∞ n (1.3) M(x + s, x + t)= x c (s,t)x− , x , n → ∞ n=0 2010 Mathematics Subject Classification. 26E60, 41A60. Key words and phrases. Pythagorean means, iterative means, arithmetic-geometric mean, asymptotic expansions. 117 118 T. BURIĆ where cn(s,t) are homogeneous polynomials of the degree n in variables s and t. It is shown that simpler form of these coefficients is obtained under substitution (1.4) t = α + β, s = α β, − so variables α and β will be used in the rest of the paper. Obviously, expansion of the arithmetic mean is (1.5) A(x + s, x + t)= x + α and it has only these two terms. Asymptotic expansion of the geometric and harmonic mean is derived in [6] and it reads as β2 1 αβ2 2 β2 2 2 3 (1.6) G(x + s, x + t) x + α x− + x− (4α + β )x− + . ∼ − 2 2 − 8 2 1 2 2 2 2 3 (1.7) H(x + s, x + t) x + α β x− + αβ x− α β x− + . ∼ − − Recall that arithmetic-geometric mean is an example of an interesting mean obtained by the iterative combination of arithmetic and geometric mean in the following way. Define a0 = s, g0 = t and a + b (1.8) a = k k , g = a b , k 0. k+1 2 k+1 k k ≥ Then both of this sequences converge to the same limit AG(s,t) which is called arithmetic-geometric mean. Same idea can be used to the other combinations of Pythagorean means as well. For an example, iteration of geometric and harmonic mean 2 (1.9) gk+1 = gkhk, hk+1 = , 1 + 1 gk hk leads to same limit geometric-harmonic mean and iteration of arithmetic and harmonic mean ak + hk 2 (1.10) ak+1 = , hk+1 = , 2 1 + 1 ak hk defines arithmetic-harmonic mean. It is not hard to see that this mean is exactly equal to the geometric mean. In a recent paper [4], authors studied arithmetic-geometric mean and derived its asymptotic expansion: β2 1 αβ2 2 β2 2 2 3 (1.11) AG(x + t, x + s) x + α x− + x− (16α +5β )x− + . ∼ − 4 4 − 64 By comparing coefficients in this expansion with expansion of A and G mean, one can see that AG clearly lies somewhere in the middle, but is a little bit closer to the geometric mean. This iterative process also showed interesting convergence and stationary properties of the coefficients in the asymptotic expansion of AG mean, for details see [4]. ASYMPTOTIC BEHAVIOUR OF THE ITERATIVE PYTHAGOREAN MEANS 119 The main aim of this paper is to derive coefficients in asymptotic expan- sion of the arithmetic-harmonic and harmonic-geometric mean by analyzing their iterative proccesses in a similar way as it has been done for AG mean. These coefficients will be used in the third section where we give relations and inequalities between this means. We shall need the following fundamental lemma for the transformation of the asymptotic series. The proof is easy, see e.g. [6]. Lemma 1.1. Let function f(x) have asymptotic expansion (a0 =1) ∞ n f(x) a x− , x . ∼ n → ∞ n=0 Then for all real p it holds p ∞ n [f(x)] c (p)x− , ∼ n n=0 where c0 =1 and n 1 (1.12) cn = [k(1 + p) n]akcn k. n − − k=1 We shall also need the following standard result: Lemma 1.2. Let f(x) and g(x) have asymptotic expansions (a0 = b0 =1): ∞ ∞ n n f(x) a x− , g(x) b x− . ∼ n ∼ n n=0 n=0 Then their product f(x)g(x) and quotient f(x)/g(x) have asymptotic expan- sions ∞ ∞ n f(x) n f(x)g(x) c x− , d x− , ∼ n g(x) ∼ n n=0 n=0 where c0 = d0 =1 and n n (1.13) cn = akbn k, dn = an bkdn k. − − − k=0 k=1 120 T. BURIĆ 2. Asymptotic expansions of iterative Pythagorean means To derive asymptotic expansion of the combination of two Pythagorean means, we shall follow its iterative process. Let us start with arithmetic-harmonic mean. Let sequence (an) be lim- iting sequence which converges above to the limit AH(s,t) and let An(s,t) be the value of n-th iteration. Similarly, let Hn(s,t) be defined through the sequence (hn). An and Hn are also means because they are obtained as the composition of arithmetic and harmonic mean to the previous members of these sequences. Then, let ∞ (n) k (2.1) An(s,t)= x ak (s,t)x− , k=0 and ∞ (n) k (2.2) Hn(s,t)= x hk (s,t)x− , k=0 be asymptotic expansions of the n-th iteration. (n) (n) We shall show that functions ak and hk converge to the same limit c (t,s) when n , and it holds k → ∞ ∞ k (2.3) AH(s,t)= x ck(s,t)x− . k=0 Theorem 2.1. Let n be arbitrary natural number. Then we have (n) (n) (2.4) ak = hk for all k 2n. ≤ (n) In other words, for fixed k, the sequence ak is stationary sequence which defines the limiting value ck. Proof of the theorem follows from the next lemma by mathematical induction. (n) (n) Lemma 2.2. Suppose that coefficients ak and hk , for arbitary n 1, satisfy ≥ (n) (n) ak = hk , k =0, 1,...,K. Then it holds (2.5) a(j) = h(j) = a(n), j n +1, k =0,...,K, k k k ≥ and (n+1) (n+1) (n+1) (n+1) (2.6) aK+1 = hK+1 , aK+2 = hK+2 . Therefore, in each following step, at least two new coefficients coincide and remain equal in the future. ASYMPTOTIC BEHAVIOUR OF THE ITERATIVE PYTHAGOREAN MEANS 121 Proof. For the arithmetic series the statement (2.5) is obvious. Let us examine (hn). Denote k ∞ (n) j+1 (n) j+1 An,k = aj x− , ARn,k = aj x− , j=0 j=k+1 and similarly for Hn,k, HRn,k. Then 2 2AnHn 2(An,k + ARn,k)(Hn,k + HRn,k) Hn+1 = 1 1 = = . An− + Hn− An + Hn An,k + ARn,k + Hn,k + HRn,k Since An,k = Hn,k, this can be written as AR HR 1+ n,k 1+ n,k A A (2.7) H = A n,k n,k n+1 n,k AR + HR 1+ n,k n,k 2An,k and clearly (2.5) follows. Now, to prove (2.6), let us find the first two coefficients in the expansion of the quotient ARn,k/An,k. Using Lemma 1.2 (recall that a0 = g0 = 1), we have (n) (n) (n) (n) AR a a a a n,k k+1 k+2 − 1 k+1 = k+1 + k+2 + . An,k x x In a same way, (n) (n) (n) (n) HR h h a h n,k k+1 k+2 − 1 k+1 = k+1 + k+2 + . An,k x x Now, from (2.7) it follows (n) (n) (n) (n) a(n) a a a a 1 k+1 k+2 − 1 k+1 Hn+1 = x 1+ + . 1+ k+1 + k+2 + ... x x x · 1 h(n) h(n) a(n)h(n) u(n) u(n) a(n)u(n) − k+1 k+2 − 1 k+1 k+1 k+2 − 1 k+1 1+ k+1 + k+2 + ... 1+ k+1 + k+2 + ... , · x x x x where a(n) + h(n) u(n) = k+i k+i , k+i 2 and therefore h(n+1) = a(n) + h(n) u(n) = a(n+1), k+1 k+1 k+1 − k+1 k+1 h(n+1) = a(n)a(n) + a(n)h(n) a(n)u(n) + a(n) a(n)a(n) + k+2 1 k+1 1 k+1 − 1 k+1 k+2 − 1 k+1 + h(n) a(n)h(n) u(n) a(n)u(n) = a(n+1). k+2 − 1 k+1 − k+2 − 1 k+1 k+2 122 T. BURIĆ Now we can calculate coefficients ck of the asymptotic expansion (2.3). In (n) each iteration, coefficients ak are easily obtained and for calculating sequence (n 1) (n 1) (n) 2a − h − h = (n 1) (n 1) , a − + h − we use Lemma 1.2 for multiplication and quotient of asymptotic series.
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