# Sharp Power Mean Bounds for Sándor Mean

Authors: Yu-Ming Chu; Zhen-Hang Yang; Li-Min Wu
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

We prove that the double inequalityMp(a,b)<X(a,b)<Mq(a,b) holds for all a,b>0 with a≠b if and only if p≤1/3 and q≥log2/(1+log2)=0.4093…, where X(a,b) and Mr(a,b) are the Sándor and rth power means of a and b, respectively.


---

## Body

## 1. Introduction

Letp∈R and a,b>0 with a≠b. Then the pth power mean Mp(a,b) of a and b is given by
(1)Mp(a,b)=ap+bp21/p(p≠0),M0(a,b)=ab.The main properties for the power mean are given in [1]. It is well known that Mp(a,b) is strictly increasing with respect to p∈R for fixed a,b>0 with a≠b. Many classical means are the special cases of the power mean; for example, M-1(a,b)=2ab/(a+b)=H(a,b) is the harmonic mean, M0(a,b)=ab=G(a,b) is the geometric mean, M1(a,b)=(a+b)/2=A(a,b) is the arithmetic mean, and M2(a,b)=(a2+b2)/2=Q(a,b) is the quadratic mean.LetL(a,b)=(a-b)/(log⁡a-log⁡b), P(a,b)=(a-b)/[2arcsin((a-b)/(a+b))], I(a,b)=(aa/bb)1/(a-b)/e, M(a,b)=(a-b)/[2sinh⁡-1((a-b)/(a+b))], and T(a,b)=(a-b)/[2arctan((a-b)/(a+b))] be the logarithmic, first Seiffert, identric, Neuman-Sándor, and second Seiffert means of two distinct positive real numbers a and b, respectively. Then it is well known that the inequalities
(2)Ha,b<Ga,b<La,b<Pa,b<Ia,b<Aa,b<Ma,b<T(a,b)<Q(a,b)
hold for all a,b>0 with a≠b.Recently, the bounds for certain bivariate means in terms of the power mean have been the subject of intensive research. Seiffert [2] proved that the inequalities
(3)2πM1a,b<Pa,b<M1a,b<T(a,b)<M2(a,b)
hold for all a,b>0 with a≠b.Jagers [3] proved that the double inequality
(4)M1/2(a,b)<P(a,b)<M2/3(a,b)
holds for all a,b>0 with a≠b.In [4, 5], Hästö established that
(5)Pa,b>Mlog⁡2/log⁡πa,b,P(a,b)>22πM2/3(a,b)
for all a,b>0 with a≠b.Witkowski [6] proved that the double inequality
(6)22πM2(a,b)<T(a,b)<4πM1(a,b)
holds for all a,b>0 with a≠b.In [7], Costin and Toader presented that
(7)Mlog⁡2/(log⁡π-log⁡2)a,b<Ta,b<M5/3(a,b)
for all a,b>0 with a≠b.Chu and Long [8] proved that the double inequality
(8)Mp(a,b)<M(a,b)<Mq(a,b)
holds for all a,b>0 with a≠b if and only if p≤log⁡2/log⁡[2log⁡(1+2)]=1.224… and q≥4/3.The following sharp bounds for the logarithmic and identric means in terms of the power means can be found in the literature [9–16]:
(9)M0a,b<La,b<M1/3a,b,M2/3a,b<Ia,b<Mlog⁡2a,b,M0a,b<L1/2a,bI1/2a,b<M1/2a,b,Mlog⁡2/(1+log⁡2)(a,b)<L(a,b)+I(a,b)2<M1/2(a,b)
for all a,b>0 with a≠b.Recently, Sándor [17] introduced the Sándor mean X(a,b) of two positive real numbers a and b, which is given by
(10)X(a,b)=A(a,b)e(Ga,b/Pa,b)-1.In [18], Sándor proved that
(11)Xa,b<P2a,bAa,b,Aa,bGa,bPa,b<Xa,b<Aa,bPa,b2Pa,b-Ga,b,Xa,b>Aa,bLa,bPa,be(Ga,b/La,b)-1,Xa,b>Aa,bPa,b+Ga,b3Pa,b-Ga,b,A2a,bGa,bPa,bLa,be(La,b/Aa,b)-1<Xa,b<Aa,b1e+1-1eGa,bPa,b,Aa,b+Ga,b-Pa,b<Xa,b<A-1/3a,bAa,b+Ga,b24/3,P1/log⁡π-log⁡2a,bA1-1/log⁡π-log⁡2a,b<Xa,b<P-1a,bAa,b+Ga,b22
for all a,b>0 with a≠b.In the Introduction we cite only a minor part of the existing literature on the considered means. For example, an important paper on the first Seiffert meanP(a,b) is again due to Sándor [19].The main purpose of this paper is to present the best possible parametersp and q such that the double inequality Mp(a,b)<X(a,b)<Mq(a,b) holds for all a,b>0 with a≠b.


## 2. Lemmas

In order to prove our main results we need several lemmas, which we present in this section.Lemma 1.
Letg1:(0,1)×R→R be defined by
(12)g1(x,p)=x(x-1)xp-1+1(x+1)xp+1-arcsinx-1x+1.
Then(1)
g1(x,p) is strictly decreasing with respect to x on (0,1) if and only if p≥1/2;(2)
g1(x,p) is strictly increasing with respect to x on (0,1) if and only if p≤1/3.Proof.
It follows from (12) that
(13)∂g1(x,p)∂x=(1-x)xp-3/22x+12xp+12g2(x,p),
where
(14)g2x,p=-3x1-p-x2-p+xp+3xp+1+2p-1x2-2p+1.
(1) Ifg1(x,p) is strictly decreasing with respect to x on (0,1), then (13) leads to the conclusion that g2(x,p)<0 for all x∈(0,1). In particular, we have g20+,p≤0. We assert that p≥1/2. Indeed, from (14) we clearly see that g20+,0=2, g20+,p=+∞ if p<0, and g20+,p=1-2p>0 if 0<p<1/2.
Ifp≥1/2, then it follows from (14) that
(15)∂g2x,p∂p=3xp+1+3x1-p+x2-p+xplog⁡x-21-x2<0
for all x∈(0,1).
Equation (14) and inequality (15) lead to the conclusion that
(16)g2(x,p)≤g2x,12=-2x(1-x)<0
for all x∈(0,1).
Therefore,g1(x,p) is strictly decreasing with respect to x on (0,1) which follows from (13) and (16).
(2) Ifg1(x,p) is strictly increasing with respect to x on (0,1), then (13) leads to the conclusion that g2(x,p)>0 for all x∈(0,1). In particular, we have
(17)lim⁡x→1-g2(x,p)1-x=4-12p≥0
and p≤1/3.
Ifp≤1/3, then (14) and (15) lead to the conclusion that
(18)g2x,p≥g2x,13=131+x1/31+5x1/3+x2/3×1-x1/33>0
for all x∈(0,1).
Therefore,g1(x,p) is strictly increasing with respect to x on (0,1) which follows from (13) and (18).Lemma 2.
Letg1:(0,1)×R→R be defined by (12). Then there exists x0∈(0,1) such that g1(x,p) is strictly increasing with respect to x on (0,x0] and strictly decreasing with respect to x on [x0,1) if 1/3<p<1/2.Proof.
Letp∈(1/3,1/2) and g2(x,p) be defined by (14). Then (14) leads to
(19)g2(0,p)=1-2p>0,g2(1,p)=0,(20)x1-p∂g2x,p∂x=3p-1x1-2p+p-2x2-2p+2(2p-1)x2-p+3(p+1)x+p∶=g3(x,p),(21)g3(0,p)=p>0,g3(1,p)=12p-4>0,(22)x2p∂g3(x,p)∂x=3(p+1)x2p-2(2p-1)(p-2)x1+p-2p-1p-2x-32p-1p-1∶=g4(x,p),(23)g40,p=-31-p1-2p<0,g4(1,p)=4(3p-1)(2-p)>0,(24)∂2g4(x,p)∂x2=-2p(1-2p)(p+1)×3+(2-p)x1-px2p-2<0
for x∈(0,1).
Inequality (24) implies that g4(x,p) is strictly convex with respect to x on (0,1). From (22) and (23) together with the strict convexity of g4(x,p) with respect to x on (0,1) we clearly see that there exists x1∈(0,1) such that g3(x,p) is strictly decreasing with respect to x on (0,x1] and strictly increasing with respect to x on [x1,1). We assert that
(25)g3(x1,p)<0.
Indeed, ifg3(x1,p)≥0, then it follows from (20) and the piecewise monotonicity of g3(x,p) with respect to x on (0,1) that g2(x,p) is strictly increasing with respect to x on (0,1). Hence, we get g2(x,p)<g2(1,p)=0 for all x∈(0,1). This conjunction with Lemma 1 and (13) leads to the conclusion that p≥1/2, which contradicts with 1/3<p<1/2.
From (20) and (21) together with (25) and the piecewise monotonicity of g3(x,p) with respect to x on (0,1) we clearly see that there exist x11∈(0,x1) and x12∈(x1,1) such that g2(x,p) is strictly increasing with respect to x on (0,x11]∪[x12,1) and strictly decreasing with respect to x on [x11,x12].
Therefore, Lemma2 follows easily from (13) and (19) together with the piecewise monotonicity of g2(x,p) with respect to x on (0,1).Lemma 3.
Letg1:(0,1)×R→R be defined by (12). Then the following statements are true:(1)
g1(x,p)>0 for all x∈(0,1) if and only if p≥1/2;(2)
g1(x,p)<0 for all x∈(0,1) if and only if p≤1/3;(3)
if1/3<p<1/2, then there exists μ0∈(0,1) such that g1(μ0,p)=0, g1(x,p)<0 for x∈(0,μ0), and g1(x,p)>0 for x∈(μ0,1).Proof.
(1) Ifg1(x,p)>0 for all x∈(0,1), then g1(0+,p)≥0. Therefore, p≥1/2 follows from g1(0+,p)=-∞ for p<1/2.
Ifp≥1/2, then Lemma 1 (1) leads to the conclusion that g1(x,p)>g1(1,p)=0 for all x∈(0,1).
(2) Ifg1(x,p)<0 for all x∈(0,1), then by making use of L’Höspital’s rules and (12) we get
(26)lim⁡x→1-g1(x,p)1-x3=18p-13≤0
and p≤1/3.
Ifp≤1/3, then Lemma 1 (2) leads to the conclusion that g1(x,p)<g1(1,p)=0 for all x∈(0,1).
(3) If1/3<p<1/2, then it follows from (12) that
(27)g1(0+,p)=-∞,g1(1,p)=0.
Therefore, Lemma3 (3) follows from Lemma 2 and (27).Lemma 4.
Letg:(0,1)×(0,∞)→R be defined by
(28)gx,p=log⁡X1,xMp1,x=log⁡x+12+2x1-xarcsin1-x1+x-1plog⁡xp+12-1.
Then(1)
g(x,p) is strictly increasing with respect to x on (0,1) if and only if p≥1/2;(2)
g(x,p) is strictly decreasing with respect to x on (0,1) if and only if p≤1/3;(3)
if1/3<p<1/2, there exists μ0∈(0,1) such that g(x,p) is strictly decreasing with respect to x on (0,μ0] and strictly increasing with respect to x on [μ0,1).Proof.
It follows from (28) that
(29)∂g(x,p)∂x=1+x1-x2xg1(x,p),
where g1(x,p) is defined by (12).
Therefore, Lemma4 follows from Lemma 3 and (29).


## 3. Main Results

Theorem 5.
The double inequality(30)Mp(a,b)<X(a,b)<Mq(a,b)
holds for all a,b>0 with a≠b if and only if p≤1/3 and q≥log⁡2/(1+log⁡2)=0.4093….Proof.
Since both the Sándor meanX(a,b) and rth power mean Mr(a,b) are symmetric and homogeneous of degree 1, without loss of generality, we assume that a=1 and b=x∈(0,1).
We first prove that the inequalityX(1,x)>Mp(1,x) holds for all x∈(0,1) if and only if p≤1/3.
Ifp=1/3, then from (28) and Lemma 4 (2) we get
(31)log⁡X(1,x)M1/3(1,x)=gx,13>g1-,13=0
for all x∈(0,1).
Therefore,X(1,x)>Mp(1,x) for all x∈(0,1) and p≤1/3 follows from (31) and the monotonicity of the function p→Mp(1,x).
IfX(1,x)>Mp(1,x), then (28) leads to g(x,p)>0 for all x∈(0,1). In particular, we have
(32)lim⁡x→1-g(x,p)1-x2=1813-p≥0
and p≤1/3.
Next, we prove that the inequalityX(1,x)<Mq(1,x) holds for all x∈(0,1) if and only if q≥log⁡2/(1+log⁡2).
IfX(1,x)<Mq(1,x) holds for all x∈(0,1), then (28) leads to g(x,q)<0 for all x∈(0,1). In particular, we have
(33)g(0,q)=1q-1log⁡2-1≤0
and q≥log⁡2/(1+log⁡2).
Ifq=log⁡2/(1+log⁡2)∈(1/3,1/2), then (28) leads to
(34)g0,log⁡21+log⁡2=g1,log⁡21+log⁡2=0.
It follows from (28) and (34) together with Lemma 4 (3) that
(35)log⁡X(1,x)Mlog⁡2/(1+log⁡2)(1,x)=gx,log⁡21+log⁡2<0
for all x∈(0,1).
Therefore,X(1,x)<Mq(1,x) for all x∈(0,1) and q≥log⁡2/(1+log⁡2) follows from (35) and the monotonicity of the function q→Mq(1,x).Theorem 6.
Leta,b>0 with a≠b. Then the double inequality
(36)2eM1/2(a,b)<X(a,b)<4eM1/3(a,b)
holds with the best possible constants 2/e and 4/e.Proof.
Since both the Sándor meanX(a,b) and rth power mean Mr(a,b) are symmetric and homogeneous of degree 1, without loss of generality, we assume that a=1 and b=x∈(0,1). It follows from Lemma 4 (1) and (2) together with (28) that
(37)log⁡X(1,x)M1/2(1,x)=gx,12>g0,12=log⁡2e,log⁡X(1,x)M1/3(1,x)=gx,13<g0,13=log⁡4e
for all x∈(0,1).
Therefore,2/eM1/2(1,x)<X(1,x)<4/eM1/3(1,x) for all x∈(0,1) follows from (37), and the optimality of the parameters 2/e and 4/e follows from the monotonicity of the functions g(x,1/2) and g(x,1/3).Remark 7.
For alla1,a2,b1,b2>0 with a1/b1<a2/b2<1. Then from Lemma 4 (1) and (2) together with (28) we clearly see that the Ky Fan type inequalities
(38)Mp(a2,b2)Mp(a1,b1)<X(a2,b2)X(a1,b1)<Mq(a2,b2)Mq(a1,b1)
hold if and only if p≥1/2 and q≤1/3.Letp∈R and Lp(a,b)=(ap+1+bp+1)/(ap+bp) be the pth Lehmer mean of two positive real numbers a and b. Then the function g1(x,p) defined by (12) can be rewritten as
(39)g1(x,p)=12(1-x)1P(1,x)-G(1,x)A(1,x)Lp-1(1,x).From Lemma3 and (39) we get Remark 8 as follows.Remark 8.
The double inequality(40)A(a,b)G(a,b)Lp-1(a,b)<P(a,b)<A(a,b)G(a,b)Lq-1(a,b)
holds for all a,b>0 with a≠b if and only if p≤1/3 and q≥1/2.From (5) and (9) together with Theorem 5 one has the following.Remark 9.
The inequalities(41)La,b<M1/3(a,b)<X(a,b)<Mlog⁡2/(1+log⁡2)(a,b)<Mlog⁡2/log⁡π(a,b)<P(a,b)
hold for all a,b>0 with a≠b.



---

*Source: 172867-2015-01-15.xml*