# Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

Authors: Hongya Gao; Jianling Guo; Wanguo Yu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

For1<r<+∞, we find the least value α and the greatest value β such
that the inequality Hα(a,b)<Ar(a,b)<Hβ(a,b) holds for all a,b>0 with a≠b. Here, Hω(a,b) and Ar(a,b) are the generalized Heronian
and the power means of two positive numbers a and b, respectively.


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## Body

## 1. Introduction and Statement of Result

Fora,b>0 with a≠b, the generalized Heronian mean of a and b is defined by Janous [1] as (1.1)Hω(a,b)={a+ωab+bω+2,0≤ω<+∞,ab,ω=+∞.
If we take ω=1 in (1.1), then we arrive at the classical Heronian mean (1.2)He(a,b)=a+ab+b3.
The domain of definition for the function ω↦Hω(a,b) can be extended to all ω with ω∈(-2,+∞), that is, (1.3)Hω(a,b)={a+ωab+bω+2,-2<ω<+∞,ab,ω=+∞.
For all fixed a,b>0, it is easy to derive that ω↦Hω(a,b), -2<ω<+∞ is monotonically decreasing, and (1.4)lim⁡ω→-2+Hω(a,b)=+∞.Let(1.5)Ar(a,b)={(ar+br2)1/r,r≠0,ab,r=0,max⁡{a,b},r=+∞,min⁡{a,b},r=-∞,
denote the power mean of order r. In particular, the harmonic, geometric, square-root, arithmetic, and root-square means of a and b are (1.6)H(a,b)=A-1(a,b)=2aa+b,G(a,b)=A0(a,b)=ab,N1(a,b)=A1/2(a,b)=(a+b2)2,A(a,b)=A1(a,b)=a+b2,S(a,b)=A2(a,b)=a2+b22.
It is well known that the power mean of order r given in (1.5) is monotonically increasing in r, then we can write (1.7)min⁡{a,b}<H(a,b)<G(a,b)<N1(a,b)<A(a,b)<S(a,b)<max⁡{a,b}.Recently, the inequalities for means have been the subject of intensive research [1–15]. In particular, many remarkable inequalities for the generalized Heronian and power means can be found in the literature [4–9].In [4], the authors established two sharp inequalities (1.8)23G(a,b)+13H(a,b)≥A-1/3(a,b),13G(a,b)+23H(a,b)≥A-2/3(a,b).In [5], Long and Chu found the greatest value p and the least value q such that the double inequality (1.9)Ap(a,b)≤A(a,b)αG(a,b)βH(a,b)1-α-β≤Aq(a,b)
holds for all a,b>0 and α,β>0 with α+β<1.In [6], Shi et al. gave two optimal inequalities (1.10)Aα(a,b)L1-α(a,b)≤A(1+2α)/3(a,b),Gα(a,b)L1-α(a,b)≤A(1-α)/3(a,b),
for 0<α<1, where (1.11)L(a,b)=a-blog⁡a-log⁡b,a≠b,
is the logarithmic mean for a,b>0.In [7], Guan and Zhu obtained sharp bounds for the generalized Heronian mean in terms of the power mean with ω>0. The optimal values α and β such that (1.12)Aα(a,b)≤Hω(a,b)≤Aβ(a,b)
holds in general are (1)
in case ofω∈(0,2],αmax=log2/log(ω+2) and βmin=2/(ω+2),(2)
in case ofω∈[2,+∞),αmax=2/(ω+2) and βmin=log2/log(ω+2).In this paper, we find the least valueα and the greatest value β, such that for any fixed 1<r<+∞, the inequality (1.13)Hα(a,b)<Ar(a,b)<Hβ(a,b)
holds for all a,b>0 with a≠b.Theorem 1.1.
For1<r<+∞, the optimal numbers α and β such that
(1.14)Hα(a,b)<Ar(a,b)<Hβ(a,b)
is valid for all a,b>0 with a≠b, are αmin=21/r-2 and βmax=2(1-r)/r.Notice that in our caser>1; the two numbers αmin and βmax are all negative see Corollary 2.2 below. Thus, the result in this paper is different from [7, Theorem A].


## 2. Preliminary Lemmas

The following lemma will be repeatedly used in the proof of Theorem1.1.Lemma 2.1.
For1<r<+∞, one has
(2.1)r21/r-1>1.Proof.
We show that(2.2)m(r)=(1-r)log⁡2+rlog⁡r>0,
which is clearly equivalent to the claim. Equation (2.2) follows from the facts
(2.3)lim⁡r→1+m(r)=0,m′(r)=-log⁡2+log⁡r+1>0.Corollary 2.2.
If1<r<+∞, then
(2.4)-2<2(1-r)r<21/r-2<0.Proof.
Since for1<r<+∞, the two functions
(2.5)φ1(r)=2(1-r)r,φ2(r)=21/r-2
are strictly decreasing, then one has
(2.6)-2=lim⁡r→+∞φ1(r)<φ1(r),φ2(r)<lim⁡r→1+φ2(r)=0.
It suffices to show that
(2.7)2-2r<r21/r-2r,
which is equivalent to (2.1).Lemma 2.3.
Forx>1 and r>1, let
(2.8)l(x)=(x2r+1)1/r-2x2(r-1)(x2r+2r-1).
Then, ℓ(x) is strictly decreasing for x>1, and
(2.9)lim⁡x→1+l(x)=r21/r-1,lim⁡x→+∞l(x)=1.Proof.
The factℓ(x)>0 for x>1 and r>1 is obvious, which allows us to take the logarithmic function of ℓ(x),
(2.10)log⁡l(x)=(1r-2)log⁡(x2r+1)+2(r-1)log⁡x+log⁡(x2r+2r-1).
Some tedious, but not difficult calculations lead to
(2.11)[log⁡l(x)]′=(1r-2)2rx2r-1x2r+1+2(r-1)x+2rx2r-1x2r+2r-1=m(x)x(x2r+1)(x2r+2r-1),
where
(2.12)m(x)=2(1-2r)x2r(x2r+2r-1)+(2r-1)(x2r+1)(x2r+2r-1)+2rx2r(x2r+1)=2(r-1)(2r-1)(1-x2r).
It is easy to see that
(2.13)lim⁡x→1+m(x)=0,(2.14)m′(x)=-4r(r-1)(2r-1)x2r-1<0.
Equation (2.14) implies that m(x) is strictly decreasing for x>1, which together with (2.13) implies m(x)<0 for x>1. Thus, by (2.11),
(2.15)[log⁡l(x)]′<0,
which implies
(2.16)l′(x)=[log⁡l(x)]′l(x)<0.
Hence, ℓ(x) is strictly decreasing.
It remains to show (2.9). The first equality in (2.9) is obvious. The second one follows from
(2.17)lim⁡x→+∞l(x)=lim⁡x→+∞(x2r+1)1/r-2x2(r-1)(x2r+2r-1)=lim⁡t→0+(2r-1)t2r+1(1+t2r)(2r-1)/r=1.
This ends the proof of Lemma 2.3.Lemma 2.4.
Forx>1, r>1, and ω=21/r-2, let
(2.18)fr(x)=21/r(x2+ωx+1)-(ω+2)(x2r+1)1/r.
Then,
(2.19)lim⁡x→+∞fr(x)=-∞,lim⁡x→+∞fr′(x)=21/r(21/r-2).Proof.
Simple calculations lead to(2.20)lim⁡x→+∞fr(x)=lim⁡x→+∞21/r(x2+ωx+1)-(ω+2)(x2r+1)1/r=lim⁡t→0+21/r(t2+ωt+1)-(ω+2)(t2r+1)1/rt2=-∞,lim⁡x→+∞fr′(x)=lim⁡x→+∞21/r(2x+ω)-2(ω+2)(x2r+1)1/r=lim⁡t→0+21/r(2+ωt)-2(ω+2)(1+t2r)(1-r)/rt=lim⁡t→0+21/r(2+ωt)(1+t2r)(r-1)/r-2(ω+2)t(1+t2r)(r-1)/r=lim⁡t→0+21/rω(1+t2r)(r-1)/r+2(1/r)+1(r-1)(2+ωt)(1+t2r)-1/rt2r-1(1+t2r)(r-1)/r+2(r-1)(1+t2r)-1/rt2r=21/rω=21/r(ω1/r-2)<0,
where we have used L'Hospital's law. This ends the proof of Lemma 2.4.


## 3. Proof of Theorem1.1

Proof.
Firstly, we prove that for1<r<+∞,
(3.1)H2(1-r)/r(a,b)>Ar(a,b),(3.2)H21/r-2(a,b)<Ar(a,b)
hold true for all a,b>0 with a≠b. It is no loss of generality to assume that a>b>0. Let x=b/a>1 and ω∈{2(1-r)/r,21/r-2}. In view of Corollary 2.2, -2<ω<0. Equations (1.3) and (1.5) lead to
(3.3)1a[Hω(a,b)-Ar(a,b)]=Hω(x2,1)-Ar(x2,1)=x2+ωx+1ω+2-(x2r+12)1/r=21/r(x2+ωx+1)-(ω+2)(x2r+1)1/r21/r(ω+2)=fr(x)21/r(ω+2),
where fr(x) is defined by (2.18). It is easy to see that
(3.4)lim⁡x→1+fr(x)=0,(3.5)fr′(x)=21/r(2x+ω)-2(ω+2)(x2r+1)1/r-1x2r-1,(3.6)lim⁡x→1+fr′(x)=0.
By Lemma 2.3,
(3.7)fr′′(x)=2{21/r-(ω+2)[2(1-r)(x2r+1)1/r-2x4r-2+(2r-1)(x2r+1)1/r-1x2(r-1)]}=2[21/r-(ω+2)l(x)]>2[21/r-(ω+2)r21/r-1]=21/r[2-(ω+2)r],(3.8)lim⁡x→1+fr′′(x)=2{21/r-(ω+2)[2(1-r)21/r-2+(2r-1)21/r-1]}=21/r[2-(ω+2)r].
We now distinguish between two cases.
Case 1 (ω=2(1-r)/r).
Since2-(ω+2)r=0, then by (3.7), fr′′(x)>0. Thus, fr′(x) is strictly increasing for x>1, which together with (3.6) implies fr′(x)>0. Hence, fr(x) is strictly increasing for x>1. Since (3.4), then fr(x)>0. Equation (3.1) follows from (3.3).Case 2 (ω=21/r-2).
By (3.5) and (2.11),
(3.9)fr′′′(x)=-2(ω+2)l′(x)=-2(ω+2)[log⁡l(x)]′l(x)=-2(ω+2)m(x)l(x)x(x2r+1)(xr+2r-1)>0.
Thus, fr′′(x) is strictly increasing. Equations (3.8) and (2.1) imply
(3.10)lim⁡x→1+fr′′(x)=21/r[2-(ω+2)r]=21/r+1(1-r21/r-1)<0.
Equations (3.7) and (2.9) imply
(3.11)lim⁡x→+∞fr′′(x)=lim⁡x→+∞2[21/r-(ω+2)l(x)]=2(21/r-1)>0.
Combining (3.10) with (3.11), we obviously know that there exists λ1>1 such that fr′′(x)<0 for x∈(1,λ1) and fr′′(x)>0 for x∈(λ1,+∞). This implies that fr′(x) is strictly decreasing for x∈(1,λ1) and strictly increasing for x∈(λ1,+∞). By (3.6) and Lemma 2.4, we know that fr′(x)<0 for x>1. Therefore, fr(x) is strictly decreasing. By (3.4) and Lemma 2.4 again, we derive that fr(x)<0 for x>1. Equation (3.2) follows from (3.3).Secondly, we prove that H21/r-2(a,b) is the best lower bound for the power mean Ar(a,b) for 1<r<+∞. For any α<21/r-2,
(3.12)lim⁡x→+∞Hα(x,1)Ar(x,1)=lim⁡x→+∞21/r(x+αx+1)(α+2)(xr+1)r=21/rα+2>1.
Hence, there exists X=X(α)>1 such that Hα(x,1)>Ar(x,1) for x∈(X,+∞).
Finally, we prove thatH2(1-r)/r(a,b) is the best upper bound for the power mean Ar(a,b) for 1<r<+∞. For any β>2(1-r)/r, by (3.7) (with β in place of ω), we have
(3.13)lim⁡x→1+fr′′(x)=21/r[2-(β+2)r]<0.
Hence, by the continuity of fr′′(x), there exists δ=δ(β)>0 such that fr′′(x)<0 for x∈(1,1+δ). Thus fr(x) is strictly decreasing for x∈(1,1+δ). From (3.6),  fr′(x)<0 for x∈(1,1+δ). This result together with (3.4) implies that fr(x)<0 for x∈(1,1+δ). Hence, by (3.3),
(3.14)Hβ(x2,1)<Ar(x2,1),
for x∈(1,1+δ).



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*Source: 679201-2011-06-08.xml*