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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]. • Equivalence of the Cauchy–Schwarz inequality and the H¨older’s inequality; see [8]. • Equivalence of the Cauchy–Schwarz inequality and the – inequality; see [7]. • Equivalence of the Kantorovich inequality and the Wielandt inequality; see e.g., [11] • Equivalence of the AM–GM inequality and the Bernoulli inequality; see e.g., [10] 4 • Equivalence of the H¨older inequality and Artin’s theorem; see e.g., [12] (pp. 657–663) for details. • Equivalence of the H¨older inequality and the weighted AM–GM inequality; refer to Theorem 1.1. • Equivalence of the H¨older inequality and the weighted power-mean inequality; see Theorem 1.2. • Equivalence of the weighted power-mean inequality and the weighted AM–GM inequality; refer to Theorem 1.3.

Author Contributions

All the authors inferred the main conclusions and approved the current version of this manuscript.

Acknowledgments

The authors would like to thank Dr. Minghua Lin for his valuable discussions, which considerably im- proved the presentation of our manuscript. This work was funded by the National Natural Science Foundation of China (Grant No. 11501141), Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant No. QJHKYZ [2016]066).

Abbreviations

The following abbreviations are used in this manuscript:

AM–GM – CBS Cauchy–Bunyakovsky–Schwarz

References

References

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