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The Weighted Arithmetic Mean-Geometric Mean Inequality Is The Weighted Arithmetic Mean-Geometric Mean Inequality is Equivalent to the H¨older Inequality Yongtao Lia, Xian-Ming Gub,c,∗, Jianxing Zhaod aSchool of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China bSchool of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China cBernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK Groningen, The Netherlands dCollege of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China Abstract In the current note, we investigate the mathematical relations among the weighted arithmetic mean– geometric mean (AM–GM) inequality, the H¨older inequality and the weighted power-mean inequality. Mean- while, the proofs of mathematical equivalence among the weighted AM–GM inequality, the weighted power- mean inequality and the H¨older inequality are fully achieved. The new results are more generalized than those of previous studies. Key words: weighted AM-GM inequality; H¨older inequality; weighted power-mean inequality; L’Hospital’s rule 1. Introduction In the field of classical analysis, the weighted arithmetic mean–geometric mean (AM–GM) inequality (see e.g., [1], pp. 74–75) is often inferred from Jensen’s inequality, which is a more generalized inequality compared to the AM–GM inequality; refer to, e.g., [1, 2]. In addition, the H¨older inequality [2] found by Leonard James Rogers (1888) and discovered independently by Otto H¨older (1889) is a basic and indispensable inequality for studying integrals and Lp spaces, and is also an extension of the Cauchy–Bunyakovsky–Schwarz (CBS) inequality [3]. The H¨older inequality is used to prove the Minkowski inequality, which is the triangle p m inequality in the space L (µ) [4, 5]. The weighted power mean (also known as the generalized mean) Mr (a) m r r r 1 for a sequence a = (a1,a2,...,an) is defined as Mr (a) = (m1a1 + m2a2 + ··· + mnan) r , which is a family of functions for aggregating sets of numbers, and plays a vital role in analytical inequalities; see [2, 6] for instance. In recent years, many researchers have been interested in studying the mathematical equivalence among arXiv:1504.02718v4 [math.FA] 15 Mar 2021 some famous analytical inequalities, such as the Cauchy–Schwarz inequality, the Bernoulli inequality, the Wielandt inequality, and the Minkowski inequality; see [7–13] for details. Additionally, these studies note the relations among the weighted AM–GM inequality, the H¨older inequality, and the weighted power-mean inequality are still less clear, although one inequality is often helpful to prove another inequality [1, 12]. Motivated by these aforementioned studies, in the present note, the mathematical equivalence among three such well-known inequalities is proved in detail; the result introduced in [14] is also extended. ∗Corresponding author Email addresses: [email protected] (Yongtao Li), [email protected] (Xian-Ming Gu), [email protected] (Jianxing Zhao) The rest of the present note is organized as follows. In the next section, we will present the detailed proofs of mathematical equivalence among three celebrated mathematical inequalities. Finally, the paper ends with several concluding remarks in Section 2. The weighted AM–GM inequality, the H¨older inequality, and the weighted power-mean inequality [1] (pp. 111–112, Theorem 10.5) are first reviewed and they are often related to each other. Then the results of mathematical equivalence among three such inequalities will be shown. Weighted AM–GM Inequality. If 0 ≤ ci ∈ R (i = 1,...,n) and 0 ≤ λi ∈ R (i = 1,...,n) such that n λi = 1, then i=1 P n n λk ck ≤ λkck. (1) k=1 k=1 Y X + −1 −1 H¨older Inequality. If 0 ≤ ai,bi ∈ R (i =1,...,n) and p, q ∈ R such that p + q = 1, then 1 1 n n p n q p q akbk ≤ ak bk . (2) k=1 k=1 ! k=1 ! X X X n + Weighted Power-Mean Inequality. If 0 ≤ ci, λi ∈ R (i =1,...,n) such that λi = 1, and r, s ∈ R i=1 such that r ≤ s, then P 1 1 n r n s r s λkck ≤ λkck . (3) k=1 ! k=1 ! X X The word “equivalence” between two statements A and B, by convention, is understood as follows: A implies B and B implies A. Two equivalent sentences have the same truth value. Thus, this note reveals a connection (in the sense of art) between these two well-known facts. Theorem 1.1. The H¨older inequality is equivalent to the weighted AM–GM inequality. 1 1 Proof. To show that (2) implies (1), let ak = (λkck) p ,bk = (λk) q in (2) for all k; then 1 1 n n p n q 1 1 (λkck) p (λk) q ≤ λkck λk . (4) k=1 k=1 ! k=1 ! X h i X X −1 −1 Since p + q = 1 and λ1 + λ2 + ··· + λn = 1, the inequality (4) can be rewritten as: p n n 1 p λkck ≥ λkck . (5) k=1 k=1 ! X X Now using the inequality (5) successively, it follows that p p2 n n 1 n 1 p p2 λkck ≥ λkck ≥ λkck ! ! k=1 k=1 k=1 (6) X X Xpm n 1 pm ≥···≥ λkck ≥ · · · . k=1 ! X By L’Hospital’s rule, it is easy to see that n n x lim ln λkck x = λk ln ck. x→0+ k=1 ! k=1 X . X 2 Thus, 1 n x n x λk lim λkck = ck . x→0+ k=1 ! k=1 X Y n n λk Thus, in (6), we can pass to the limit by m → +∞, giving λkck ≥ ck ; hence, (2) implies (1). k=1 k=1 To show the converse, all that is needed is a special case ofP (1), Q λ1 λ2 λ1c1 + λ2c2 ≥ c1 c2 . (7) Since p−1 + q−1=1, and by (7), thus 1 1 n n n p n q 1 p q 1 q p q p a b + b a ≥ akbk b a . p k k q k k k k k=1 k=1 k=1 ! k=1 ! X X X X Summing over k =1, 2,...,n, it follows that 1 1 n n n n p n q p q q p ak bk ≥ akbk bk ak . k=1 k=1 k=1 k=1 ! k=1 ! X X X X X Thus, (1) implies (2). Remark 1.1. The CBS inequality (the AM–GM inequality) is the special case of the H¨older inequality (the weighted AM–GM inequality); therefore, Theorem 1.1 is a generalization of the result established in [14]. Theorem 1.2. The H¨older inequality is equivalent to the weighted power-mean inequality. s s Proof. For r ≤ s, let p = r ≥ 1 and ck = dk in (5) for k =1,...,n, then s n n r s r λkdk ≥ λkdk . (8) k=1 k=1 ! X X 1 1 n s n r s r Thus, rewriting (8) as λkdk ≥ λkdk . Now the task is to prove that (3) implies (2); let k=1 k=1 n q q −1 Pq−1 P λk = bk bk , dk = ak/bk , r =1,s = p in (3), then k=1 P n n n q q bk ak akbk = bk n · q−1 q k=1 k=1 ! k=1 b bk X X X k k=1 1 P p n n q p q bk ak ≤ bk n · q−1 q k=1 ! k=1 b bk ! X X k k=1 1 1 n p n P q p q = ak bk . k=1 ! k=1 ! X X Remark 1.2. Here, we can give another proof that (3) implies (2). Repeatedly using inequality (3), it follows that 3 n n 2 n 3 n n 1 1 1 2 3 n λkck ≥ λkck ≥ λkck ≥···≥ λkck ≥ · · · . k=1 k=1 ! k=1 ! k=1 ! X X X X By L’Hospital’s rule, it is easy to see that n n λk λkck ≥ ck . k=1 k=1 X Y By using analogous methods from Theorem 1.1, it can be proved that 1 1 n n n n p n q p q q p ak bk ≥ akbk bk ak . k=1 k=1 k=1 k=1 ! k=1 ! X X X X X Theorem 1.3. The weighted power-mean inequality is equivalent to the weighted AM–GM inequality. Proof. To show that (1) implies (3), we merely exploit a special case of (1), λ1 λ2 a1 a2 ≤ λ1a1 + λ2a2. (9) s s s s −1 r r Here, we define Un(a)= λ1a1 +λ2a2 +···+λnan; let a1 = λkak(Un(a)) ,a2 = λk and λ1 = s , λ2 =1− s in (9), then r − r r s −1 r λka (Un(a)) s ≤ · λka (Un(a)) + 1 − · λk. k s k s Summing over k =1, 2,...,n, then n n r − r r s −1 r λka (Un(a)) s ≤ · λka (Un(a)) + 1 − · λk =1. k s k s k=1 k=1 X X h i Therefore, 1 1 n r n s r s λkck ≤ λkck . k=1 ! k=1 ! X X The converse is trivial from Remark 1.2. 2. Concluding Remarks In this note, the mathematical equivalence among the weighted AM–GM inequality, the H¨older inequality, and the weighted power-mean inequality is investigated in detail. Moreover, the interesting conclusions of Lin’s paper [14] are also extended. At the end of the present study, for convenience, the results on the equivalence of some well-known analytical inequalities can be summarized as follows: • Equivalence of the H¨older’s inequality and the Minkowski inequality; see [9].
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