A note on mean inequalities

Introduction

Throughout this note, we consider mean of x=(x1,,xn), where n is an arbitrary positive integer and real number xi>0 for 1in. Carvalho [1] discusses Kolmogorov’s general definition of mean: μ(x;g)=g1(1ni=1ng(xi)), where g() is a continuous monotone function and g1() is its inverse. Arithmetic, geometric, harmonic, and quadratic means are special cases of the above, because we can let g(x)=x, log(x), 1/x, and x2, respectively.

Next, we introduce the so-called power mean. We define a family of functions P(x;α)=xα, if α0, otherwise=logx, where the domain is x(0,+). The power mean of x=(x1,,xn) is μ(x;P(x;α)).

The following facts are well known:

  • [F1] limαμ(x;P(x;α))=minx1,,xn.

  • [F2] limα+μ(x;P(x;α))=maxx1,,xn.

  • [F3] As a function of α, μ(x;P(x;α)) is increasing when α increases.

The above fact [F3] implies that μ(x;P(x;1))μ(x;P(x;0))μ(x;P(x;1))μ(x;P(x;2)), that is, H-meanG-meanA-meanQ-mean, where ‘H’, ‘G’, ‘A’ and ‘Q’ stand for ‘harmonic’, ‘geometric’, ‘arithmatic’ and ‘quadratic’, respectively.

In this note, based on the above, an inequality is derived in Theorem 1, which, I believe, deserves to be widely known. Two examples are shown as special cases of Theorem 1.

Main results

Theorem 1: If μ(x;g)μ(x;h), where both g() and h() are continuous monotone functions and their definite integrals on any interval [c1,c2] (0<c1<c2<+) exist, then for any 0<a<b<+, g1(1baabg(x)dx)h1(1baabh(x)dx).

Proof: Take an arbitrary positive integer n, and let xi=a+i(ba)/n, for i=1,,n. Let xn=(x1,,xn). According to the assumption,

μ(xn;g)μ(xn;h),

thus taking limit on the both sides leads to the desired result.

Example 1: Using Theorem 1 and noticing μ(x;P(x;2))μ(x;P(x;1)) together with the result H-meanG-meanA-meanQ-mean, we have the following inequality chain. For any 0<a<b<+, abbalogblogaexp(blogbalogaba1)

b+a2b2+ab+a23.

Example 2: Since μ(x;P(x;p))μ(x;P(x;q)), if p<q, by Theorem 1, we derive bp+1ap+1(ba)(p+1)pbq+1aq+1(ba)(q+1)q, where 0<a<b<+, p<q, p1 and q1.

References

[1] Miguel de Carvalho, Mean, What do You Mean? The American Statistician 70 (2016) 270–274.

Lingyun Zhang (张凌云)
Lingyun Zhang (张凌云)
Design Analyst

I have research interests in Statistics, applied probability and computation.