We present the best possible power mean bounds for the product Mpα(a,b)M-p1-α(a,b) for any p>0, α∈(0,1), and all a,b>0 with a≠b. Here, Mp(a,b) is the pth power mean of two positive numbers a and b.
1. Introduction
For p∈ℝ, the pth power mean Mp(a,b) of two positive numbers a and b is defined by Mp(a,b)={(ap+bp2)1/p,p≠0,ab,p=0.
It is well known that Mp(a,b) is continuous and strictly increasing with respect to p∈ℝ for fixed a,b>0 with a≠b. Many classical means are special cases of the power mean, for example, M-1(a,b)=H(a,b)=2ab/(a+b), M0(a,b)=G(a,b)=ab and M1(a,b)=A(a,b)=(a+b)/2 are the harmonic, geometric and arithmetic means of a and b, respectively. Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities and properties for the power mean can be found in literature [1–22].
Let L(a,b)=(a-b)/(loga-logb), P(a,b)=(a-b)/[4arctan(a/b)-π] and I(a,b)=1/e(aa/bb)1/(a-b) be the logarithmic, Seiffert and identric means of two positive numbers a and b with a≠b, respectively. Then it is well known that min{a,b}<H(a,b)<G(a,b)<L(a,b)<P(a,b)<I(a,b)<A(a,b)<max{a,b},
for all a,b>0 with a≠b.
In [23–29], the authors presented the sharp power mean bounds for L, I, (IL)1/2 and (L+I)/2 as follows: M0(a,b)<L(a,b)<M1/3(a,b),M2/3(a,b)<I(a,b)<Mlog2(a,b),M0(a,b)<L(a,b)I(a,b)<M1/2(a,b),12(L(a,b)+I(a,b))<M1/2(a,b),
for all a,b>0 with a≠b.
Alzer and Qiu [12] proved that the inequality 12(L(a,b)+I(a,b))>Mp(a,b)
holds for all a,b>0 with a≠b if and only if p≤(log2)/(1+log2)=0.40938….
The following sharp bounds for the sum αA(a,b)+(1-α)L(a,b), and the products Aα(a,b)L1-α(a,b) and Gα(a,b)L1-α(a,b) in terms of power means were proved in [5, 8]: Mlog2/(log2-logα)(a,b)<αA(a,b)+(1-α)L(a,b)<M(1+2α)/3(a,b),M0(a,b)<Aα(a,b)L1-α(a,b)<M(1+2α)/3(a,b),M0(a,b)<Gα(a,b)L1-α(a,b)<M(1-α)/3(a,b),
for any α∈(0,1) and all a,b>0 with a≠b.
In [2, 7] the authors answered the questions: for any α∈(0,1), what are the greatest values p1=p1(α), p2=p2(α), p3=p3(α), and p4=p4(α), and the least values q1=q1(α), q2=q2(α), q3=q3(α), and q4=q4(α), such that the inequalities Mp1(a,b)<Pα(a,b)L1-α(a,b)<Mq1(a,b),Mp2(a,b)<Aα(a,b)G1-α(a,b)<Mq2(a,b),Mp3(a,b)<Gα(a,b)H1-α(a,b)<Mq3(a,b),Mp4(a,b)<Aα(a,b)H1-α(a,b)<Mq4(a,b),
hold for all a,b>0 with a≠b?
It is the aim of this paper to present the best possible power mean bounds for the product Mpα(a,b)M-p1-α(a,b) for any p>0, α∈(0,1) and all a,b>0 with a≠b.
2. Main ResultTheorem 2.1.
Let p>0, α∈(0,1) and a,b>0 with a≠b. Then
M(2α-1)p(a,b)=Mpα(a,b)M-p1-α(a,b)=M0(a,b) for α=1/2,
M(2α-1)p(a,b)>Mpα(a,b)M-p1-α(a,b)>M0(a,b) for α>1/2 and M(2α-1)p(a,b)<Mpα(a,b)M-p1-α(a,b)<M0(a,b) for α<1/2, and the bounds M(2α-1)p(a,b) and M0(a,b) for the product Mpα(a,b)M-p1-α(a,b) in either case are best possible.
Proof.
From (1.1) we clearly see that Mp(a,b) is symmetric and homogenous of degree 1. Without loss of generality, we assume that b=1, a=x>1.
Letf(x)=αplog1+xp2-1-αplog1+x-p2-1(2α-1)plog1+x(2α-1)p2,
then simple computations lead to
f(1)=0,f′(x)=g(x)x(1+xp)(1+x(2α-1)p),
where
g(x)=(α-1)x2αp+αxp-αx(2α-1)p+1-α,g(1)=0,g′(x)=αpxp-1h(x),
where
h(x)=2(α-1)x(2α-1)p-(2α-1)x2(α-1)p+1,h(1)=0,h′(x)=-2p(1-α)(2α-1)x2(α-1)p-1(xp-1).
If α∈(1/2,1), then (2.9) implies that h(x) is strictly decreasing in [1,+∞). Therefore, M(2α-1)p(x,1)>Mpα(x,1)M-p1-α(x,1) follows easily from (2.2)–(2.8) and the monotonicity of h(x).
If α∈(0,1/2), then (2.9) leads to the conclusion that h(x) is strictly increasing in [1,+∞). Therefore, M(2α-1)p(x,1)<Mpα(x,1)M-p1-α(x,1) follows easily from (2.2)–(2.8) and the monotonicity of h(x).
Secondly, we compare the value of M0(x,1) to the value of Mpα(x,1)M-p1-α(x,1). It follows from (1.1) thatlog[Mpα(x,1)M-p1-α(x,1)]-logM0(x,1)=αplog1+xp2-1-αplog1+x-p2-12logx.
LetF(x)=αplog1+xp2-1-αplog1+x-p2-12logx,
then simple computations lead to
F(1)=0,F′(x)=(2α-1)(xp-1)x(1+xp)(1+x(2α-1)p).
If α∈(1/2,1), then (2.13) implies that F(x) is strictly increasing in [1,+∞). Therefore, Mpα(x,1)M-p1-α(x,1)>M0(x,1) follows easily from (2.10)–(2.12) and the monotonicity of F(x).
If α∈(0,1/2), then (2.13) leads to the conclusion that F(x) is strictly decreasing in [1,+∞). Therefore, Mpα(x,1)M-p1-α(x,1)<M0(x,1) follows easily from (2.10)–(2.12) and the monotonicity of F(x).
Next, we prove that the bound M(2α-1)p(a,b) for the product Mpα(a,b)M-p1-α(a,b) in either case is best possible.
If α∈(0,1/2), then for any ϵ∈(0,(1-2α)p) and x>0 we haveMpα(1+x,1)M-p1-α(1+x,1)-M(2α-1)p+ϵ(1+x,1)=[1+(1+x)p2]α/p[1+(1+x)-p2](α-1)/p-[1+(1+x)(2α-1)p+ϵ2]1/[(2α-1)p+ϵ].
Letting x→0 and making use of Taylor’s expansion, one has[1+(1+x)p2]α/p[1+(1+x)-p2](α-1)/p-[1+(1+x)(2α-1)p+ϵ2]1/[(2α-1)p+ϵ]=[1+α2x+α(p+α-2)8x2+o(x2)]×[1+1-α2x-(1-α)(p+α+1)8x2+o(x2)]-[1+12x+(2α-1)p+ϵ-18x2+o(x2)]=[1+12x+(2α-1)p-18x2+o(x2)]-[1+12x+(2α-1)p+ϵ-18x2+o(x2)]=-ϵ8x2+o(x2).
Equations (2.14) and (2.15) imply that for any α∈(0,1/2) and ϵ∈(0,(1-2α)p) there exists δ1=δ1(ϵ)>0, such that Mpα(1+x,1)M-p1-α(1+x,1)<M(2α-1)p+ϵ(1+x,1) for x∈(0,δ1).
If α∈(1/2,1), then for any ϵ∈(0,(2α-1)p) and x>0 we haveMpα(1+x,1)M-p1-α(1+x,1)-M(2α-1)p-ϵ(1+x,1)=[1+(1+x)p2]α/p[1+(1+x)-p2](α-1)/p-[1+(1+x)(2α-1)p-ϵ2]1/[(2α-1)p-ϵ].
Letting x→0 and making use of Taylor’s expansion, one has[1+(1+x)p2]α/p[1+(1+x)-p2](α-1)/p-[1+(1+x)(2α-1)p-ϵ2]1/[(2α-1)p-ϵ]=[1+α2x+α(p+α-2)8x2+o(x2)]×[1+1-α2x-(1-α)(p+α+1)8x2+o(x2)]-[1+12x+(2α-1)p-ϵ-18x2+o(x2)]=[1+12x+(2α-1)p-18x2+o(x2)]-[1+12x+(2α-1)p-ϵ-18x2+o(x2)]=ϵ8x2+o(x2).
Equations (2.16) and (2.17) imply that for any α∈(1/2,1) and ϵ∈(0,(2α-1)p) there exists δ2=δ2(ϵ)>0, such that Mpα(1+x,1)M-p1-α(1+x,1)>M(2α-1)p-ϵ(1+x,1) for x∈(0,δ2).
Finally, we prove that the bound M0(a,b) for the product Mpα(a,b)M-p1-α(a,b) in either case is best possible.
If α∈(0,1/2), then for any ϵ>0 we clearly see thatlimx→+∞Mpα(x,1)M-p1-α(x,1)M-ϵ(x,1)=+∞.
Equation (2.18) implies that for any α∈(0,1/2) and ϵ>0 there exists T1=T1(ϵ)>1, such that Mpα(x,1)M-p1-α(x,1)>M-ϵ(x,1) for x∈(T1,+∞).
If α∈(1/2,1), then for any ϵ>0 we havelimx→+∞Mpα(x,1)M-p1-α(x,1)Mϵ(x,1)=0.
Equation (2.19) implies that for any α∈(1/2,1) and ϵ>0 there exists T2=T2(ϵ)>1, such that Mpα(x,1)M-p1-α(x,1)<Mϵ(x,1) for x∈(T2,+∞).
Acknowledgments
This paper was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.
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