Note on certain inequalities for Neuman means

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R E S E A R C H

Open Access

Note on certain inequalities for Neuman

means

Shu-Bo Chen

1

_{, Zai-Yin He}

2

_{, Yu-Ming Chu}

1*

_{, Ying-Qing Song}

1

_{and Xiao-Jing Tao}

3

*_{Correspondence:}

chuyuming2005@126.com

1_{School of Mathematics and}

Computation Science, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article

Abstract

In this paper, we give the explicit formulas for the Neuman meansNAH,NHA,NAC, and NCA, and present the best possible upper and lower bounds for these means in terms

of the combinations of harmonic meanH, arithmetic meanA, and contraharmonic meanC.

MSC: 26E60

Keywords: Schwab-Borchardt mean; Neuman mean; harmonic mean; arithmetic mean; contraharmonic mean

1 Introduction

Leta,b,c≥ withab+ac+bc= . Then the symmetric integralRF(a,b,c) [] of the ﬁrst

kind is deﬁned as

RF(a,b,c) =

 

∞



(t+a)(t+b)(t+c)–/dt.

The degenerate case ofRF, denoted byRC, plays an important role in the theory of special

functions [, ], which is given by

RC(a,b) =RF(a,b,b).

Fora,b>  witha=b, the Schwab-Borchardt meanSB(a,b) [–] ofaandbis given by

SB(a,b) =

⎧ ⎨ ⎩

√ b_–_a

cos–₍_a_/_b₎, a<b,

√ a_–_b

cosh–₍_a_/_b₎, a>b,

wherecos–₍_x_{) and}_cosh–₍_x_{) =}_log₍_x₊√_x_{– ) are the inverse cosine and inverse} hyper-bolic cosine functions, respectively.

Carlson [] (see also [, (.)]) proved that

SB(a,b) =RC

a,b –.

Recently, the Schwab-Borchardt mean has been the subject of intensive research. In par-ticular, many remarkable inequalities for the Schwab-Borchardt mean and its generated means can be found in the literature [–, –].

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Leta>b> ,v= (a–b)/(a+b)∈(, ),p∈(,∞),q∈(,π/),r∈(,log( +√)), ands∈(,π/) be the parameters such that /cosh(p) =cos(q) =  –v_,_cosh₍_r_{) =}_sec₍_s_{) =}  +v_,_H₍_a_,_b_{) = }_ab_/(_a₊_b_),_G₍_a_,_b_{) =}√_ab_, _A₍_a_,_b_{) = (}_a₊_b_)/, _Q₍_a_,_b_{) =}₍_a₊_b_)/, and C(a,b) = (a ₊_b_)/(_a₊_b_{) be, respectively, the harmonic, geometric, arithmetic,} quadratic, and contraharmonic means of a and b, SAH(a,b) = SB[A(a,b),H(a,b)],

SHA(a,b) = SB[H(a,b),A(a,b)], SAC(a,b) =SB[A(a,b),C(a,b)], SCA(a,b) = SB[C(a,b),

A(a,b)]. Then Neuman [] gave the explicit formulas

SAH(a,b) =A(a,b) tanh(p)

p , SHA(a,b) =A(a,b)

sinq

q , (.)

SCA(a,b) =A(a,b) sinh(r)

r , SAC(a,b) =A(a,b)

tans

s . (.)

Very recently, Neuman [] found a new mean N(a,b) derived from the Schwab-Borchardt mean as follows:

N(a,b) =  

a+ b



SB(a,b)

. (.)

LetNAH(a,b) =N[A(a,b),H(a,b)],NHA(a,b) =N[H(a,b),A(a,b)],NAG(a,b) =N[A(a,b),

G(a,b)], NGA(a,b) = N[G(a,b),A(a,b)], NAC(a,b) = N[A(a,b),C(a,b)], NCA(a,b) =

N[C(a,b),A(a,b)],NAQ(a,b) =N[A(a,b),Q(a,b)], and NQA(a,b) =N[Q(a,b),A(a,b)] be

the Neuman means. Then Neuman [] proved that

G(a,b) <NAG(a,b) <NGA(a,b) <A(a,b) <NQA(a,b) <NAQ(a,b) <Q(a,b)

for alla,b>  witha=b, and the double inequalities

αA(a,b) + ( –α)G(a,b) <NGA(a,b) <βA(a,b) + ( –β)G(a,b),

αQ(a,b) + ( –α)A(a,b) <NAQ(a,b) <βQ(a,b) + ( –β)A(a,b),

αA(a,b) + ( –α)G(a,b) <NAG(a,b) <βA(a,b) + ( –β)G(a,b),

αQ(a,b) + ( –α)A(a,b) <NQA(a,b) <βQ(a,b) + ( –β)A(a,b)

hold for all a,b >  with a=b if and only if α ≤/, β ≥π/, α ≤/, β ≥ (π – )/[(√ – )] = . . . . , α ≤/,β ≥/,α≤/, andβ≥[log( +

√ ) + √

 – ]/[(√ – )] = . . . . .

Zhang et al. [] presented the best possible parameters α,α,β,β ∈[, /] and α,α,β,β∈[/, ] such that the double inequalities

Gαa+ ( –α)b,αb+ ( –α)a <NAG(a,b) <G

βa+ ( –β)b,βb+ ( –β)a ,

Gαa+ ( –α)b,αb+ ( –α)a <NGA(a,b) <G

βa+ ( –β)b,βb+ ( –β)a ,

Qαa+ ( –α)b,αb+ ( –α)a <NQA(a,b) <Q

βa+ ( –β)b,βb+ ( –β)a ,

Qαa+ ( –α)b,αb+ ( –α)a <NAQ(a,b) <Q

βa+ ( –β)b,βb+ ( –β)a

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In [], the authors found the greatest valuesα,α,α,α,α,α,α,α, and the least valuesβ,β,β,β,β,β,β,βsuch that the double inequalities

Aα₍_a_,_b₎_G–α₍_a_,_b_{) <}_N

GA(a,b) <Aβ(a,b)G–β(a,b), α

G(a,b)+  –α

A(a,b)< 

NGA(a,b)

< β

G(a,b)+  –β

A(a,b),

Aα₍_a_,_b₎_G–α₍_a_,_b_{) <}_N

AG(a,b) <Aβ(a,b)G–β(a,b), α

G(a,b)+  –α

A(a,b)< 

NAG(a,b)

< β

G(a,b)+  –β

A(a,b),

Qα₍_a_,_b₎_A–α₍_a_,_b_{) <}_N

AQ(a,b) <Qβ(a,b)A–β(a,b), α

A(a,b)+  –α

Q(a,b)< 

NAQ(a,b)

< β

A(a,b)+  –β

Q(a,b),

Qα₍_a_,_b₎_A–α₍_a_,_b_{) <}_N

QA(a,b) <Qβ(a,b)A–β(a,b), α

A(a,b)+  –α

Q(a,b)< 

NQA(a,b)

< β

A(a,b)+  –β

Q(a,b)

hold for alla,b>  witha=b.

The main purpose of this paper is to give the explicit formulas for the Neuman means

NAH,NHA,NAC, andNCA, and to present the best possible upper and lower bounds for

these means in terms of the combinations of harmonic, arithmetic, and contraharmonic means. Our main results are Theorems .-..

Theorem . Let a>b > , v= (a–b)/(a+b)∈ (, ), p∈(,∞), q∈ (,π/), r∈

(,log( +√)),and s∈(,π/)be the parameters such that/cosh(p) =cos(q) =  –v_, cosh(r) =sec(s) =  +v_._{Then we have}

NAH(a,b) =

 A(a,b)

 + p sinh(p)

, (.)

NHA(a,b) =

 A(a,b)

cos(q) + q sin(q)

, (.)

NCA(a,b) =

 A(a,b)

cosh(r) + r sinh(r)

, (.)

NAC(a,b) =

 A(a,b)

 + s sin(s)

, (.)

and

H(a,b) <NAH(a,b) <NHA(a,b) <A(a,b)

<NCA(a,b) <NAC(a,b) <C(a,b). (.)

Theorem . The double inequalities

αA(a,b) + ( –α)H(a,b) <NAH(a,b) <βA(a,b) + ( –β)H(a,b), (.)

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αC(a,b) + ( –α)A(a,b) <NCA(a,b) <βC(a,b) + ( –β)A(a,b), (.)

αC(a,b) + ( –α)A(a,b) <NAC(a,b) <βC(a,b) + ( –β)A(a,b) (.)

hold for all a,b >  with a =b if and only if α ≤ /, β ≥ /, α ≤ /, β ≥ π/ = . . . . , α ≤ /, β ≥

√

log( +√)/ = . . . . , α ≤/, and β ≥ (√π– )/ = . . . . .

Theorem . The double inequalities

α

H(a,b)+  –α

A(a,b)< 

NAH(a,b)

< β

H(a,b)+  –β

A(a,b), (.)

α

H(a,b)+  –α

A(a,b)< 

NHA(a,b)

< β

H(a,b)+  –β

A(a,b), (.)

α

A(a,b)+  –α

C(a,b)< 

NCA(a,b)

< β

A(a,b)+  –β

C(a,b), (.)

α

A(a,b)+  –α

C(a,b)< 

NAC(a,b)

< β

A(a,b)+  –β

C(a,b), (.)

hold for all a,b> with a=b if and only ifα≤,β≥/,α≤,β≥/,α≤[ √

 – log( +√)]/[√ +log( +√)] = . . . . ,β≥/,α≤(

√

 – π)/(√ + π) = . . . . ,andβ≥/.

2 Lemmas

In order to prove our main results we need several lemmas, which we present in this sec-tion.

Lemma .(See [, Theorem .]) For–∞<a<b<∞,let f,g: [a,b]→Rbe continu-ous on[a,b],and be diﬀerentiable on(a,b),let g(x)= on(a,b).If f (x)/g(x)is increasing

(decreasing)on(a,b),then so are

f(x) –f(a)

g(x) –g(a) and

f(x) –f(b)

g(x) –g(b).

If f (x)/g(x)is strictly monotone,then the monotonicity in the conclusion is also strict.

Lemma .(See [, Lemma .]) Suppose that the power series f(x) =∞_n_=anxnand

g(x) =∞_n_=bnxnhave the radius of convergence r> and an,bn> for all n≥.If the

sequence{an/bn}is(strictly)increasing(decreasing)for all n≥,then the function f(x)/g(x)

is also(strictly)increasing(decreasing)on(,r).

Lemma .(See [, Theorem .]) If a>b,then

N(b,a) >N(a,b).

Lemma . The function

ϕ(t) =

sinh(t) – sinh(t) + t

sinh(t) – sinh(t)

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Proof Making use of power series expansion we get

ϕ(t) =

∞ n=

n+_–

(n+)!tn+

∞ n=

n+_–

(n+)!tn+ =

∞ n=

n+_–

(n+)!tn

∞ n=

n+_–

(n+)!tn

. (.)

Let

an=

n+_{– }

(n+ )!, bn=

n+_{– }

(n+ )!. (.)

Then

an> , bn> , (.)

andan/bn=  – /(n+– ) is strictly increasing for alln≥.

Note that

ϕ

+ =a

b =

, ϕ(∞) =nlim→∞

an

bn

= . (.)

Therefore, Lemma . follows easily from Lemma . and (.)-(.) together with the

monotonicity of the sequence{an/bn}.

Lemma . The function

ϕ(t) =

t–sin(t) sint( –cost)

is strictly increasing from(,π/)onto(/,π).

Proof Letf(t) = t–sin(t) andg(t) =sint( –cost). Then simple computations lead to

ϕ(t) =

f(t) –f()

g(t) –g()

(.)

andf_(t)/g_(t) = [ – /( + /cost)] is strictly increasing on (,π/). Note that

ϕ+ = lim t→+

f_(t)

g(t) =

, ϕ(π/) =π. (.)

Therefore, Lemma . follows from Lemma ., (.), (.), and the monotonicity of

f_(t)/g_(t).

Lemma . The function

ϕ(t) = sinh(t)cosh(t) –t [sinh(t)cosh(t) +t](cosh(t) – )

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Proof Simple computations lead to

ϕ(t) =

sinh(t) – t

sinh(t) + tcosh(t) +sinh(t) – sinh(t) – t

=

∞ n= 

n+

(n+)!t n

∞ n=

n+_–n+_+_n_+

(n+)! tn

. (.)

Let

an=

n+

(n+ )!, bn=

n+– n++ n+ 

(n+ )! . (.)

Then

an> , bn> , (.)

and

an+

bn+ –an

bn

= – 

n+_(_×_n+_{– }_n_{– )}

(n+_{– }n+_{+ }_n_{+ )(}n+_{– }n+_{+ }_n_{+ )}<  (.)

for alln≥. Note that

ϕ

+ =a

b =

, ϕ(∞) =nlim→∞

an

bn

= . (.)

Therefore, Lemma . follows easily from (.)-(.) and Lemma ..

Lemma . The function

f(t) = cost+ t sint

is strictly decreasing on the interval(,π/).

Proof Letf(t) = sintcost+tandg(t) =sint. Then simple computations lead to

f(t) = f(t) –f()

g(t) –g() ,

f_(t)

g_(t)=

cos_t_{– }

cost ,

(.)

and

f_(t)

g_(t)

= –sint(cos _t_{+ )}

cos_t <  (.)

fort∈(,π/).

Therefore, Lemma . follows easily from (.) and (.) together with Lemma ..

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Lemma . The function

ϕ(t) =

sintcost–t

(t+sintcost)( –cost)

is strictly decreasing from(,π/)onto(–, –/).

Proof Letf(t) =sintcost–tandg(t) = (t+sintcost)( –cost). Then simple computa-tions lead to

ϕ(t) =

f(t)

g(t)

= f(t) –f()

g(t) –g()

, (.)

f_(t)

g_(t)=

f_(t) –f_()

g_(t) –g_(), (.)

and

f_ (t)

g_ (t)=



 – (cost+_sint_t). (.)

Note that

ϕ

+ = lim t→+

f (t)

g_ (t)= – 

, ϕ

π 

= –. (.)

Therefore, Lemma . follows from Lemma . and Lemma . together with

(.)-(.).

3 Proofs of Theorems 1.1-1.3

Proof of Theorem. It follows from (.)-(.) as we clearly see that

NAH(a,b) =

 

A(a,b) + H ₍_a_,_b₎

SAH(a,b)

= A(a,b)

 + –v  p

tanh(p)

=  A(a,b)

 + p

tanh(p)cosh(p)

= A(a,b)

 + p sinh(p)

,

NHA(a,b) =

 

H(a,b) + A ₍_a_,_b₎

SHA(a,b)

= A(a,b)

 –v + q sinq

= A(a,b)

cosq+ q sinq

,

NCA(a,b) =

 

C(a,b) + A ₍_a_,_b₎

SCA(a,b)

= A(a,b)

 +v + r sinh(r)

= A(a,b)

cosh(r) + r sinh(r)

,

NAC(a,b) =

 

A(a,b) + C ₍_a_,_b₎

SAC(a,b)

= A(a,b)

 + +v  s

tan(s)

= A(a,b)

 + s

tan(s)cos_s

=  A(a,b)

 + s sin(s)

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Inequalities (.) follow easily from H(a,b) <A(a,b) <C(a,b) and Lemma . to-gether with the fact thatNKL(a,b) is a mean of K(a,b) andL(a,b) forK(a,b),L(a,b)∈

{H(a,b),A(a,b),C(a,b)}.

Proof of Theorem. Without loss of generality, we assume thata>b. Letv= (a–b)/ (a+b)∈(, ),p∈(,∞),q∈(,π/),r∈(,log( +√)), ands∈(,π/) be the param-eters such that /cosh(p) =cos(q) =  –v,cosh(r) =sec(s) =  +v. Then from (.)-(.) we have

NAH(a,b) –H(a,b)

A(a,b) –H(a,b) =

[ + p/sinh(p)]/ – ( –v)

v

=[ + p/sinh(p)]/ – /cosh(p)

 – /cosh(p) =ϕ(p), (.)

NHA(a,b) –H(a,b)

A(a,b) –H(a,b) =

[cosq+q/sinq]/ – ( –v₎

v

=[cosq+q/sinq]/ –cosq

 –cosq =



ϕ(q), (.)

NCA(a,b) –A(a,b)

C(a,b) –A(a,b) =

[cosh(r) +r/sinh(r)]/ – 

v

=[cosh(r) +r/sinh(r)]/ –  cosh(r) –  =



ϕ(r), (.)

NAC(a,b) –A(a,b)

C(a,b) –A(a,b) =

[ + s/sin(s)]/ – 

v

=[ + s/sin(s)]/ –  sec(s) –  =



ϕ(s), (.)

where the functionsϕandϕare deﬁned as in Lemmas . and ., respectively. Note that

ϕ

log( +√)=√log( +√)/ (.)

and

ϕ

π 

= √

π– 

 . (.)

Therefore, inequality (.) holds for all a,b>  witha=bif and only ifα≤/ and β≥/ follows from (.) and Lemma ., inequality (.) holds for alla,b>  with

a=bif and only ifα≤/ andβ≥π/ follows from (.) and Lemma ., inequality (.) holds for alla,b>  witha=bif and only ifα≤/ andβ≥

√

log( +√)/ follows from (.) and (.) together with Lemma ., and inequality (.) holds for all

a,b>  witha=bif and only ifα≤/ andβ≥(√π– )/ follows from (.) and

(.) together with Lemma ..

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we have

/NAH(a,b) – /A(a,b)

/H(a,b) – /A(a,b) =  +p/sinh(p)– 

 –v– 

=

sinh(p) p+sinh(p)– 

cosh(p) –  =ϕ(p), (.)

/NHA(a,b) – /A(a,b)

/H(a,b) – /A(a,b) = 

cos(q)+q/sin(q)–  

–v– 

=

sin(q)–sin(q)cos(q)–q sin(q)cos(q)+q

–cos(q) cos(q)

=  +ϕ(q), (.)

/NCA(a,b) – /C(a,b)

/A(a,b) – /C(a,b) =  cosh(r)+r/sinh(r)–

 +v

 –_+_v

=ϕ(r), (.)

and

/NAC(a,b) – /C(a,b)

/A(a,b) – /C(a,b) =  +ϕ(s), (.)

where the functionsϕandϕare deﬁned as in Lemmas . and ., respectively. Note that

ϕ

log( +√)= √

 –log( +√)

√ +log( +√) (.)

and

ϕ

π 

= –π–  √



π+ √. (.)

Therefore, Theorem . follows easily from (.)-(.) together with Lemmas .

and ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China.}2_{Department of}

Mathematics, Huzhou Teachers College, Huzhou, 313000, China.3_{School of Mathematics and Econometrics, Hunan}

University, Changsha, 410082, China.

Acknowledgements

This research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401192, the Natural Science Foundation of Hunan Province under Grant 12C0577, and the Research Foundation of Education Bureau of Hunan Province under Grant 14A026.

Received: 21 May 2014 Accepted: 3 September 2014 Published:25 Sep 2014

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