R E S E A R C H
Open Access
Sharp bounds for Sándor mean in terms of
arithmetic, geometric and harmonic means
Wei-Mao Qian
1, Yu-Ming Chu
2*and Xiao-Hui Zhang
2*Correspondence:
[email protected] 2School of Mathematics and
Computation Sciences, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article
Abstract
In the article, we present the best possible parameters
α
1,α
2,β
1,β
2∈(0, 1) andα
3,α
4,β
3,β
4∈(0, 1/2) such that the double inequalitiesα
1A(a,b) + (1 –α
1)H(a,b) <X(a,b) <β
1A(a,b) + (1 –β
1)H(a,b),α
2A(a,b) + (1 –α
2)G(a,b) <X(a,b) <β
2A(a,b) + (1 –β
2)G(a,b),H
α
3a+ (1 –α
3)b,α3b+ (1 –α
3)a<X(a,b) <Hβ
3a+ (1 –β
3)b,β
3b+ (1 –β
3)a,G
α
4a+ (1 –α
4)b,α4b+ (1 –α
4)a<X(a,b) <Gβ
4a+ (1 –β
4)b,β
4b+ (1 –β
4)ahold for alla,b> 0 witha=b. Here,X(a,b),A(a,b),G(a,b) andH(a,b) are the Sándor, arithmetic, geometric and harmonic means ofaandb, respectively.
MSC: 26E60
Keywords: Sándor mean; arithmetic mean; geometric mean; harmonic mean
1 Introduction
Letr∈Rand a,b> with a=b. Then the harmonic mean H(a,b), geometric mean
G(a,b), logarithmic meanL(a,b), Seiffert meanP(a,b), arithmetic meanA(a,b), Sándor meanX(a,b) [] andrth power meanMr(a,b) ofaandbare, respectively, defined by
H(a,b) = ab
a+b, G(a,b) = √
ab, L(a,b) = a–b
loga–logb, (.) P(a,b) = a–b
arcsin(aa+–bb), A(a,b) =
a+b
, X(a,b) =A(a,b)e
G(a,b)
P(a,b)– (.)
and
Mr(a,b) =
ar+br
/r
(r= ), M(a,b) =
√
ab. (.)
It is well known thatMr(a,b) is continuous and strictly increasing with respect tor∈R
for fixeda,b> witha=b, and the inequalities
H(a,b) <G(a,b) <L(a,b) <P(a,b) <A(a,b) (.) hold for alla,b> witha=b.
Recently, the Sándor mean has attracted the attention of several researchers. In [], Sándor established the inequalities
X(a,b) <P
(a,b)
A(a,b),
A(a,b)G(a,b)
P(a,b) <X(a,b) <
A(a,b)P(a,b) P(a,b) –G(a,b),
X(a,b) >A(a,b)L(a,b)
P(a,b) e
G(a,b)
L(a,b)–, X(a,b) >A(a,b)[P(a,b) +G(a,b)]
P(a,b) –G(a,b) ,
A(a,b)G(a,b)
P(a,b)L(a,b) e
L(a,b)
A(a,b)–<X(a,b) <A(a,b)
e+
–
e
G(a,b)
P(a,b)
,
A(a,b) +G(a,b) –P(a,b) <X(a,b) <A–/(a,b)
A(a,b) +G(a,b)
/
,
P/(logπ–log)(a,b)A–/(logπ–log)(a,b) <X(a,b) <P–(a,b)
A(a,b) +G(a,b)
for alla,b> witha=b.
Yanget al.[] proved that the double inequality
Mp(a,b) <X(a,b) <Mq(a,b) (.)
holds for alla,b> witha=bif and only ifp≤/ andq≥log/( +log) = . . . . . In [], Zhouet al.proved that the double inequality
Hα(a,b) <X(a,b) <Hβ(a,b) (.)
holds for alla,b> witha=bif and only ifα≤/ andβ≥log/( +log) = . . . . , whereHp(a,b) = [(ap+ (ab)p/+bp)/]/p(p= ) andH(p) =
√
abis thepth power-type Heronian mean ofaandb.
Inequalities (.) and (.) together with the identities H(a,b) =M–(a,b), G(a,b) =
M(a,b) andA(a,b) =M(a,b) lead to the inequalities
H(a,b) <G(a,b) <X(a,b) <A(a,b) (.)
for alla,b> witha=b.
Leta,b> witha=b,x∈[, /],f(x) =H[xa+ ( –x)b,xb+ ( –x)a] andg(x) =G[xa+ ( –x)b,xb+ ( –x)a]. Then both functionsf andgare continuous and strictly increasing on [, /]. Note that
f() =H(a,b) <X(a,b) <f(/) =A(a,b) (.)
and
Motivated by inequalities (.)-(.), we naturally ask: what are the best possible param-etersα,α,β,β∈(, ) andα,α,β,β∈(, /) such that the double inequalities
αA(a,b) + ( –α)H(a,b) <X(a,b) <βA(a,b) + ( –β)H(a,b),
αA(a,b) + ( –α)G(a,b) <X(a,b) <βA(a,b) + ( –β)G(a,b),
Hαa+ ( –α)b,αb+ ( –α)a
<X(a,b) <Hβa+ ( –β)b,βb+ ( –β)a
,
Gαa+ ( –α)b,αb+ ( –α)a
<X(a,b) <Gβa+ ( –β)b,βb+ ( –β)a
hold for alla,b> witha=b? The purpose of this paper is to answer this question.
2 Lemmas
In order to prove our main results, we need four lemmas, which we present in this section.
Lemma . Let p∈(, )and
f(x) =x
√
–x[( –p)x+ ]
p+ ( –p)( –x) –arcsin(x). (.)
Then the following statements are true:
() Ifp= /,thenf(x) < for allx∈(, ).
() Ifp= /e,then there existsλ∈(, )such thatf(x) > forx∈(,λ)andf(x) < for
x∈(λ, ).
Proof Simple computations lead to
f() = , f() = –π
, (.)
f(x) = x
√
–x[p+ ( –p)( –x)]f(x), (.)
where
f(x) = ( –p)x– ( –p)( –p)x+ – p. (.)
() Ifp= /, then (.) leads to
f(x) = –
x
–x < (.) forx∈(, ).
Therefore,f(x) < forx∈(, ) follows easily from (.), (.) and (.). () Ifp= /e, then (.) leads to
f() =
e–
e > , f() = –
From (.) and (.) we clearly see that there existsλ∈(, ) such thatf(x) > for
x∈(,λ) andf(x) < forx∈(λ, ).
We divide the proof into two cases.
Case .x∈(,λ]. Thenf(x) > follows easily from (.) and (.) together withf(x) >
on the interval (,λ).
Case .x∈(λ, ). Then (.) andf(x) < on the interval (λ, ) lead to the conclusion
thatf(x) is strictly decreasing on [λ, ).
From (.) and f(λ) > together with the monotonicity off(x) on [λ, ) we clearly
see that there existsλ∈(λ, )⊂(, ) such thatf(x) > forx∈(λ,λ) andf(x) < for
x∈(λ, ).
Lemma . Let p∈(, )and
g(x) =px
√
–x+ ( –p)x
( –p)√ –x+p –arcsin(x). (.)
Then the following statements are true:
() Ifp= /,theng(x) > for allx∈(, ).
() Ifp= /e,then there existsμ∈(, )such thatg(x) < forx∈(,μ)andg(x) > for
x∈(μ, ).
Proof Simple computations lead to
g() = , g() =
p– – π
, (.)
g(x) = x
√
–x[p+ ( –p)√ –x]g(x), (.)
where
g(x) =p(p– )
√
–x+ – p–p. (.)
() Ifp= /, then (.) leads to
g(x) =
–√ –x > (.)
forx∈(, ).
Therefore,g(x) > for allx∈(, ) follows easily from (.), (.) and (.). () Ifp= /e, then (.) leads to
g() =
e–
e < , g() =
e– e–
e > , (.)
g(x) = p√( –p)x
–x > (.)
for allx∈(, ).
From (.) and (.) we clearly see that there existsμ∈(, ) such thatg(x) < for
We divide the proof into two cases.
Case .x∈(,μ]. Theng(x) < forx∈(,μ] follows easily from (.) and (.)
to-gether withg(x) < on the interval (,μ).
Case .x∈(μ, ). Then (.) andg(x) > on the interval (μ, ) lead to the conclusion
thatg(x) is strictly increasing on [μ, ). Note that
g(μ) < , g() =e– –
π
> . (.) From (.) and the monotonicity ofg(x) on the interval [μ, ) we clearly see that there
existsμ∈(μ, )⊂(, ) such thatg(x) < forx∈(μ,μ) andg(x) > forx∈(μ, ). Lemma . Let p∈(, /)and
h(x) =arcsin(x) –x
√
–x[ + ( – p)x]
– ( – p)x . (.)
Then the following statements are true:
() Ifp= / –√/ = . . . .,thenh(x) > for allx∈(, ).
() Ifp= / –√ – /e/ = . . . .,then there existsσ∈(, )such thath(x) < for
x∈(,σ)andh(x) > forx∈(σ, ).
Proof Simple computations lead to
h() = , h() =π
, (.)
h(x) = – x
√
–x[ – ( – p)x]h(x), (.)
where
h(x) =
p– p+ p– p+ x
+–p+ p– p+ p– x+ p– p+ . (.)
() Ifp= / –√/, then (.) leads to
h(x) = –
x
–x < (.)
forx∈(, ).
Therefore,h(x) > for allx∈(, ) follows easily from (.) and (.) together with (.).
() Ifp= / –√ – /e/, then
h() =
p– p+ > , h() = –p( –p) < , (.)
h(x) = p– p+ p– p+ x
Note that
p– p+ p– p+ = . . . . > , (.) p– p+ p– = –. . . . < . (.) It follows from (.)-(.) that
h(x) < p– p+ p– p+ x+ –p+ p– p+ p– x
= p– p+ p– x< (.) forx∈(, ).
From (.) and (.) we clearly see that there existsσ∈(, ) such thath(x) > for
x∈(,σ) andh(x) < forx∈(σ, ).
We divide the proof into two cases.
Case .x∈(,σ]. Thenh(x) < forx∈(,σ] follows easily from (.) and (.)
to-gether withh(x) > on the interval (,σ).
Case .x∈(σ, ). Then (.) andh(x) < on the interval (σ, ) lead to the conclusion
thath(x) is strictly increasing on (σ, ). Therefore, there existsσ∈(σ, )⊂(, ) such
thath(x) < forx∈(σ,σ) andh(x) > forx∈(σ, ) follows from (.) andh(σ) <
together with the monotonicity ofh(x) on the interval (σ, ). Lemma . Let p∈(, /)and
J(x) =arcsin(x) – x
√
–x
– ( – p)x. (.)
Then the following statements are true:
() Ifp= / –√/ = . . . .,thenJ(x) > for allx∈(, ). () Ifp= / –√ – /e/ = . . . .,then there existsτ
∈(, )such thatJ(x) < for
x∈(,τ)andh(x) > forx∈(τ, ).
Proof Simple computations lead to
J() = , J() =π
, (.)
J(x) = x
√
–x[ – ( – p)x]J(x), (.)
where
J(x) =
p– p+ p– p+ x–p– p+ . (.) () Ifp= / –√/, then (.) leads to
J(x) =
x
> (.)
Therefore,J(x) > for allx∈(, ) follows easily from (.) and (.) together with (.).
() Ifp= / –√ – /e/, then (.) leads to
J() = –
p– p+ < , J() = p
p– p+ p+ > , (.)
J(x) = p– p+ p– p+ x> (.) forx∈(, ).
It follows from (.) and (.) that there existsτ∈(, ) such thatJ(x) < forx∈
(,τ) andJ(x) > forx∈(τ, ).
We divide the proof into two cases.
Case .x∈(,τ]. ThenJ(x) < forx∈(,τ] follows easily from (.) and (.)
to-gether withJ(x) < on the interval (,τ).
Case .x∈(τ, ). Then (.) andJ(x) > on the interval (τ, ) lead to the conclusion
thatJ(x) is strictly increasing on (τ, ).
Therefore, there existsτ∈(τ, )⊂(, ) such thatJ(x) < forx∈(τ,τ) andJ(x) >
forx∈(τ, ) follows from (.) andJ(τ) < together with the monotonicity ofJ(x) on
the interval (τ, ).
3 Main results
Theorem . The double inequality
αA(a,b) + ( –α)H(a,b) <X(a,b) <βA(a,b) + ( –β)H(a,b)
holds for all a,b> with a=b if and only ifα≤/e= . . . .andβ≥/.
Proof SinceH(a,b),X(a,b) andA(a,b) are symmetric and homogenous of degree one, we assume thata>b> . Letx= (a–b)/(a+b)∈(, ) andp∈(, ). Then (.) and (.) lead to
X(a,b) –H(a,b)
A(a,b) –H(a,b)=
e
√
–xarcsin(x)
x –– ( –x)
x , (.)
log X(a,b)
pA(a,b) + ( –p)H(a,b)=
√
–xarcsin(x)
x – –log
p+ ( –p) –x . (.)
Let
F(x) =
√
–xarcsin(x)
x – –log
p+ ( –p) –x . (.) Then simple computations lead to
F+ = , (.)
F() = –logp– , (.)
F(x) =
x√ –xf(x), (.)
We divide the proof into two cases.
Case .p= /. Then (.)-(.) and (.) together with Lemma .() lead to the con-clusion that
X(a,b) <
A(a,b) +
H(a,b). (.) Case .p= /e. Then (.) and Lemma .() lead to the conclusion that there exists
λ∈(, ) such thatF(x) is strictly increasing on (,λ] and strictly decreasing on [λ, ).
Note that (.) becomes
F() = . (.)
It follows from (.)-(.) and (.) together with the piecewise monotonicity ofF(x) that
X(a,b) >
eA(a,b) +
–
e
H(a,b). (.)
Note that
lim x→+
e
√
–xarcsin(x)
x –– ( –x)
x =
, (.)
lim x→–
e
√
–xarcsin(x)
x –– ( –x)
x =
e. (.)
Therefore, Theorem . follows from (.) and (.) in conjunction with the following statements.
• Ifα> /, then equations (.) and (.) lead to the conclusion that there existsδ∈
(, )such thatX(a,b) <αA(a,b) + ( –α)H(a,b)for alla>b> with(a–b)/(a+b)∈
(,δ).
•• Ifβ< /e, then equations (.) and (.) lead to the conclusion that there existsδ∈
(, )such thatX(a,b) >βA(a,b) + ( –β)H(a,b)for alla>b> with(a–b)/(a+b)∈
( –δ, ).
Theorem . The double inequality
αA(a,b) + ( –α)G(a,b) <X(a,b) <βA(a,b) + ( –β)G(a,b)
holds for all a,b> with a=b if and only ifα≤/andβ≥/e= . . . . .
Proof SinceA(a,b),G(a,b) andX(a,b) are symmetric and homogenous of degree one, we assume thata>b> . Letx= (a–b)/(a+b)∈(, ) andp∈(, ). Then (.) and (.) lead to
X(a,b) –G(a,b)
A(a,b) –G(a,b)=
e
√
–xarcsin(x)
x ––√ –x
log X(a,b)
pA(a,b) + ( –p)G(a,b)=
√
–xarcsin(x)
x – –log
p+ ( –p)√ –x. (.)
Let
G(x) =
√
–xarcsin(x)
x – –log
p+ ( –p)√ –x. (.)
Then simple computations lead to
G+ = , (.)
G() = –logp– , (.)
G(x) =
x√ –xg(x), (.)
whereg(x) is defined by (.). We divide the proof into two cases.
Case .p= /. Then (.)-(.) and (.) together with Lemma .() lead to the conclusion that
X(a,b) >
A(a,b) +
G(a,b). (.) Case .p= /e. Then from Lemma .() and (.) we know that there existsμ∈(, )
such thatG(x) is strictly decreasing on (,μ] and strictly increasing on [μ, ). Note that
(.) becomes
G() = . (.)
It follows from (.)-(.) and (.) together with the piecewise monotonicity ofG(x) that
X(a,b) <
eA(a,b) +
–
e
G(a,b). (.)
Note that
lim x→+
e
√
–xarcsin(x)
x ––√ –x
–√ –x =
, (.)
lim x→–
e
√
–xarcsin(x)
x ––√ –x
–√ –x =
e. (.)
Therefore, Theorem . follows easily from (.) and (.) together with
(.)-(.).
Theorem . Letα,β∈(, /).Then the double inequality
Hαa+ ( –α)b,αb+ ( –α)a
<X(a,b) <Hβa+ ( –β)b,βb+ ( –β)a
holds for all a,b> with a=b if and only ifα≤/ –
√
– /e/ = . . . .andβ≥
/ –√/ = . . . . .
Proof SinceH(a,b) andX(a,b) are symmetric and homogenous of degree one, we assume thata>b> . Letx= (a–b)/(a+b)∈(, ) andp∈(, /). Then (.) and (.) lead to
logH[pa+ ( –p)b,pb+ ( –p)a] X(a,b) =log
– ( – p)x–
√
–xarcsin(x)
x + . (.)
Let
H(x) =log – ( – p)x–
√
–xarcsin(x)
x + . (.)
Then simple computations lead to
H+ = , (.)
H() = + log +logp–p , (.)
H(x) =
x√ –xh(x), (.)
whereh(x) is defined by (.). We divide the proof into four cases.
Case .p= / –√/. Then (.)-(.) and (.) together with Lemma .() lead to
X(a,b) <H
–
√
a+
+
√
b,
–
√
b+
+
√
a
.
Case . <p< / –√/. Letq= ( – p) andx→+, then / <q< and power
series expansion leads to
H(x) = –
q–
x+ox . (.)
Equations (.), (.) and (.) lead to the conclusion that there exists <δ< such that
X(a,b) >Hpa+ ( –p)b,pb+ ( –p)a (.)
for alla>b> with (a–b)/(a+b)∈(,δ).
Case .p= / –√ – /e/. Then (.) and Lemma .() lead to the conclusion that there existsσ∈(, ) such thatH(x) is strictly decreasing on (,σ] and strictly increasing
on [σ, ).
Note that (.) becomes
Therefore,
X(a,b) >H
– – e a+ + – e b, – – e b+ + – e a
follows from (.)-(.) and (.) together with the piecewise monotonicity ofH(x). Case . / –√ – /e/ <p< /. Then (.) leads to
H() > . (.)
Equations (.) and (.) together with inequality (.) imply that there exists <
δ< such that
X(a,b) <Hpa+ ( –p)b,pb+ ( –p)a
fora>b> with (a–b)(a+b)∈( –δ, ).
Theorem . Letα,β∈(, /).Then the double inequality
Gαa+ ( –α)b,αb+ ( –α)a
<X(a,b) <Gβa+ ( –β)b,βb+ ( –β)a
holds for all a,b> with a=b if and only ifα≤/ –
√
– /e/ = . . . .andβ
≥
/ –√/ = . . . . .
Proof SinceG(a,b) andX(a,b) are symmetric and homogenous of degree one, we assume thata>b> . Letx= (a–b)/(a+b)∈(, ) andp∈(, /). Then (.) and (.) lead to
logG[pa+ ( –p)b,pb+ ( –p)a] X(a,b)
= log
– ( – p)x–
√
–xarcsin(x)
x + . (.)
Let
K(x) = log
– ( – p)x–
√
–xarcsin(x)
x + . (.)
Then simple computations lead to
K+ = , (.)
K() = +log + log
p–p , (.)
K(x) =
x√ –xJ(x), (.)
Case .p=p= / –
√
/. Then
X(a,b) <Gpa+ ( –p)b,pb+ ( –p)a
follows from (.)-(.) and (.) together with Lemma .().
Case . <p< / –√/. Letq= ( – p)andx→+, then / <q< and power
series expansion leads to
K(x) = –
q–
x+ox . (.)
From (.), (.) and (.) we clearly see that there exists <δ< such that
X(a,b) >Gpa+ ( –p)b,pb+ ( –p)a
fora>b> with (a–b)/(a+b)∈(,δ). Case .p=p= / –
√
– /e/. Then (.) and Lemma .() lead to the
conclu-sion that there existsτ∈(, ) such thatK(x) is strictly decreasing on (,τ] and strictly
increasing on [τ, ).
Note that (.) becomes
K() = . (.)
Therefore,
X(a,b) >Gpa+ ( –p)b,pb+ ( –p)a
follows from (.)-(.) and (.) together with the piecewise monotonicity ofK(x). Case . / –√ – /e/ <p< /. Then (.) leads to
K() > . (.)
Equations (.) and (.) together with inequality (.) imply that there exists <
δ< such that
X(a,b) <Gpa+ ( –p)b,pb+ ( –p)a (.)
fora>b> with (a–b)/(a+b)∈( –δ, ).
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
1School of Distance Education, Huzhou Broadcast and TV University, Huzhou, 313000, China.2School of Mathematics
Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 61374086, 11171307 and 11401191, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
Received: 23 February 2015 Accepted: 23 June 2015 References
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3. Yang, Z-H, Wu, L-M, Chu, Y-M: Sharp power mean bounds for Sándor mean. Abstr. Appl. Anal.2014, Article ID 172867 (2014)
