R E S E A R C H
Open Access
Optimal inequalities for bounding Toader
mean by arithmetic and quadratic means
Tie-Hong Zhao
1, Yu-Ming Chu
1*and Wen Zhang
2*Correspondence: [email protected] 1School 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 this paper, we present the best possible parameters
α
(r) andβ
(r) such that the double inequality
α
(r)Ar(a,b) +(
1 –α
(r))
Qr(a,b)1/r<TDA(a,b),Q(a,b)<
β
(r)Ar(a,b) +(
1 –β
(r))
Qr(a,b)1/r holds for allr≤1 anda,b> 0 witha=b, and we provide new bounds for the complete elliptic integralE(r) =0π/2(1 –r2sin2θ
)1/2dθ
(r∈(0,√2/2)) of the secondkind, whereTD(a,b) =π20π/2√a2cos2
θ
+b2sin2θ
dθ
,A(a,b) = (a+b)/2 andQ(a,b) =√(a2+b2)/2 are the Toader, arithmetic, and quadratic means ofaandb,
respectively.
MSC: 26E60
Keywords: arithmetic mean; Toader mean; quadratic mean; complete elliptic integral
1 Introduction
Forp∈[, ],q∈Randa,b> witha= b, thepth generalized Seiffert meanSp(a,b),qth Gini meanGq(a,b),qth power meanMq(a,b),qth Lehmer meanLq(a,b), harmonic mean
H(a,b), geometric meanG(a,b), arithmetic meanA(a,b), quadratic meanQ(a,b), Toader meanTD(a,b) [], centroidal meanC(a,b), contraharmonic meanC(a,b) are, respectively, defined by
Sp(a,b) =
⎧ ⎪ ⎨ ⎪ ⎩
p(a–b)
arctan[p(a–b)/(a+b)], <p≤, (a+b)/, p= ,
Gq(a,b) =
⎧ ⎨ ⎩
[(aq–+bq–)/(a+b)]/(q–), q= ,
(aabb)/(a+b), q= ,
Mq(a,b) =
⎧ ⎨ ⎩
[(aq+bq)/]/q, q= , √
ab, q= ,
Lq(a,b) =
aq++bq+
aq+bq , H(a,b) = ab
a+b, G(a,b) =
√
ab, (.)
A(a,b) =a+b
, Q(a,b) =
a+b
,
TD(a,b) = π
π/
acosθ+bsinθdθ,
C(a,b) =(a
+ab+b)
(a+b) , C(a,b) =
a+b a+b .
It is well known thatSp(a,b),Gq(a,b),Mq(a,b), andLq(a,b) are continuous and strictly increasing with respect top∈[, ] andq∈Rfor fixeda,b> witha=b, and the inequal-ities
H(a,b) =M–(a,b) =L–(a,b) <G(a,b) =M(a,b) =L–/(a,b)
<A(a,b) =M(a,b) =L(a,b) <TD(a,b) <C(a,b)
<Q(a,b) =M(a,b) <C(a,b) =L(a,b)
hold for alla,b> witha=b.
The Toader meanTD(a,b) has been well known in the mathematical literature for many years, it satisfies
TD(a,b) =RE
a,b,
where
RE(a,b) = π
∞
[a(t+b) +b(t+a)]t
(t+a)/(t+b)/ dt
stands for the symmetric complete elliptic integral of the second kind (see [–]), therefore it cannot be expressed in terms of the elementary transcendental functions.
Letr∈(, ),K(r) =π/(–rsinθ)–/dθandE(r) =π/
(–rsinθ)/dθbe,
respec-tively, the complete elliptic integrals of the first and second kind. ThenK(+) =E(+) =
π/,K(r), andE(r) satisfy the derivatives formulas (see [], Appendix E, p.-)
dK(r)
dr =
E(r) – ( –r)K(r)
r( –r) ,
dE(r)
dr =
E(r) –K(r)
r ,
d[K(r) –E(r)]
dr = rE(r) –r,
the valuesK(√/) andE(√/) can be expressed as (see [], Theorem .)
K
√
=
(/)
√π = . . . . , E
√
=
(/) +(/)
where(x) =∞tx–e–tdt(Rex> ) is the Euler gamma function, and the Toader mean
TD(a,b) can be rewritten as
TD(a,b) =
⎧ ⎨ ⎩
aE( – (b/a))/π, a≥b,
bE( – (a/b))/π, a<b. (.)
Recently, the Toader meanTD(a,b) has been the subject of intensive research. Vuorinen [] conjectured that the inequality
TD(a,b) >M/(a,b)
holds for all a,b> witha=b. This conjecture was proved by Qiu and Shen [], and Barnard, Pearce and Richards [], respectively.
Alzer and Qiu [] presented a best possible upper power mean bound for the Toader mean as follows:
TD(a,b) <Mlog/(logπ–log)(a,b)
for alla,b> witha=b.
Neuman [], and Kazi and Neuman [] proved that the inequalities
(a+b)√ab–ab
AGM(a,b) <TD(a,b) <
(a+b)√ab+ (a–b)
AGM(a,b) ,
TD(a,b) <
( +√)a+ ( –√)b+
( +√)b+ ( –√)a
hold for alla,b> witha=b, whereAGM(a,b) is the arithmetic-geometric mean ofa
andb.
In [–], the authors presented the best possible parameters λ,μ ∈ [, ] and
λ,μ,λ,μ ∈ R such that the double inequalities Sλ(a,b) < TD(a,b) < Sμ(a,b),
Gλ(a,b) <TD(a,b) <Gμ(a,b) andLλ(a,b) <TD(a,b) <Lμ(a,b) hold for alla,b> with
a=b.
Letλ,μ,α,β∈(/, ). Then Chu, Wang and Ma [], and Hua and Qi [] proved that the double inequalities
Cλa+ ( –λ)b,λb+ ( –λ)a<TD(a,b) <Cμa+ ( –μ)b,μb+ ( –μ)a,
Cαa+ ( –α)b,αb+ ( –α)a<TD(a,b) <Cβa+ ( –β)b,βb+ ( –β)a
hold for alla,b> witha=bif and only ifλ≤/,μ≥/ +√π( –π)/(π),α≤/ + √
/ andβ≥/ +√/π– /.
In [–], the authors proved that the double inequalities
αQ(a,b) + ( –α)A(a,b) <TD(a,b) <βQ(a,b) + ( –β)A(a,b),
Qα(a,b)A(–α)(a,b) <TD(a,b) <Qβ(a,b)A(–β)(a,b),
α A(a,b)+
–α C(a,b)<
TD(a,b)< β A(a,b)+
–β C(a,b),
αC(a,b) + ( –α)H(a,b) <TD(a,b) <βC(a,b) + ( –β)H(a,b),
α
C(a,b) –H(a,b)+A(a,b) <TD(a,b) <β
C(a,b) –H(a,b)+A(a,b),
αC(a,b) + ( –α)A(a,b) <TD(a,b) <βC(a,b) + ( –β)A(a,b),
α A(a,b)+
–α C(a,b)<
TD(a,b)< β A(a,b)+
–β C(a,b),
αQ(a,b) + ( –α)H(a,b) <TD(a,b) <βQ(a,b) + ( –β)H(a,b),
α H(a,b)+
–α Q(a,b)<
TD(a,b)< β H(a,b)+
–β Q(a,b)
hold for alla,b> witha=bif and only ifα≤/,β≥( –π)/[(
√
– )π],α≤/,
β≥ – logπ/log,α≤/,β≥/π– ,α≤π/ – ,β≥/,α≤/,β≥/π,
α≤/,β≥/π– /,α≤/,β≥/π– ,α≤π– ,β≥/,α≤/,β≥
√/π,α≤, andβ≥/.
The main purpose of this paper is to present the best possible parametersα(r) andβ(r) such that the double inequality
α(r)Ar(a,b) + –α(r)Qr(a,b)/r<TDA(a,b),Q(a,b)
<β(r)Ar(a,b) + –β(r)Qr(a,b)/r
holds for allr≤ anda,b> witha=b.
2 Lemmas
In order to prove our main result we need two lemmas, which we present in this section.
Lemma . Let p∈(, ),t∈(,√/),λ= ( +√)[ – E(√/)/π] = . . . .and
f(t) =πp
√
–t+π
( –p) –E(t). (.)
Then f(t) < for all t∈(,√/)if and only if p≥/and f(t) > for all t∈(,√/)if and only if p≤λ.
Proof It follows from (.) that
f+= , (.)
f
√
=π
– √
(λ–p), (.)
f(t) = f(t)
t√ –t, (.)
where
f(t) =
√
–tK(t) –E(t)–πp
t
, f
f √ = √ K √ –E √
–πp
, (.)
f(t) =t[E√(t) –K(t)]
–t –πpt, (.)
f+= , (.)
f
√
= E
√ –K √ – √
πp
, (.)
f(t) = ( – t
)E(t) – ( –t)K(t)
( –t)/ –πp, (.)
f+=π
–p
, (.) f √ =√ E √
– K
√
–πp, (.)
f(t) = –( +t
)[K(t) –E(t)] +tK(t)
t( –t)/ < (.)
for allt∈(,√/).
It follows from (.) thatf(t) is strictly decreasing on (,√/). We divide the proof into three cases.
Case p≥/. Then (.) leads to
f+≤. (.)
From (.) and the monotonicity off(t) we clearly see thatf(t) is strictly decreasing on (,√/). Therefore,f(t) < for allt∈(,√/) follows easily from (.), (.), (.), (.), and the monotonicity off(t).
Case <p≤λ. Then from (.) and (.) together with E(√/) – K(√/) = –. . . . we clearly see that
f+> , f
√
< . (.)
It follows from (.) and the monotonicity off(t) that there existst∈(,
√
/) such thatf(t) is strictly increasing on (,t] and strictly decreasing on [t,
√ /). Letλ∗=
√ π [E(
√ ) –K(
√
)] = . . . . andλ∗∗= √
π [K( √
) –E(
√
)] = . . . . . We
divide the proof into three subcases.
Subcase . <p≤λ∗. Then (.) leads to
f √ ≥. (.)
It follows from (.) and (.) together with the piecewise monotonicity off(t) that
f(t) > (.)
Therefore,f(t) > for allt∈(,√/) follows easily from (.), (.), (.), and (.).
Subcase .λ∗<p≤λ∗∗. Then (.) and (.) lead to
f √ ≥, (.) f √
< . (.)
It follows from (.) and (.) together with the piecewise monotonicity off(t) that there existst∈(,
√
/) such thatf(t) is strictly increasing on (,t] and strictly
de-creasing on [t,
√ /).
Equation (.) and inequality (.) together with the piecewise monotonicity off(t)
lead to the conclusion that
f(t) > (.)
for allt∈(,√/).
Therefore, f(t) > for all t∈(,√/) follows easily from (.) and (.) together with (.).
Subcase .λ∗∗<p≤λ. Then (.), (.), and (.) lead to
f √ ≥, (.) f √
< , (.)
f
√
< E
√ –K √ – √ π λ ∗∗
< E
√ –K √ – √ π λ ∗= . (.)
It follows from (.) and (.) together with the piecewise monotonicity off(t) that there existst∈(,
√
/) such thatf(t) is strictly increasing on (,t] and strictly
de-creasing on [t,
√ /).
From (.), (.), and (.) together with the piecewise monotonicity off(t) we clearly
see that there existst∈(,
√
/) such thatf(t) is strictly increasing on (,t] and strictly
decreasing on [t,
√ /).
Therefore,f(t) > for allt∈(,√/) follows easily from (.) and (.) together with the piecewise monotonicity off(t).
Case λ<p< /. Then (.), (.), (.), (.), and (.) lead to
f
√
< , (.)
f √ < √ K √ –E √
–π λ
∗∗ = , (.) f √
< E
√ –K √ – √
π λ∗
f+> , (.) f √
<√
E √
– K
√
–π λ∗
= –√ K √ –E √
< . (.)
It follows from (.) and (.) together with the monotonicity off(t) that there
ex-istst∈(,
√
/) such thatf(t) is strictly increasing on (,t] and strictly decreasing on
[t,
√ /).
Equation (.) and inequality (.) together with the piecewise monotonicity off(t) lead to the conclusion that there existst∈(,
√
/) such thatf(t) is strictly increasing
on (,t] and strictly decreasing on [t,
√ /).
From (.), (.), (.), and the piecewise monotonicity off(t) we clearly see that there
existst∈(,
√
/) such thatf(t) is strictly increasing on (,t] and strictly decreasing on
[t,
√ /).
Therefore, there existst∈(,
√
/) such thatf(t) > fort∈(,t) andf(t) < for t∈(t,
√
/) follows from (.) and (.) together with the piecewise monotonicity of
f(t).
Lemma . Let r∈R,a,b> with <b/a<√,c= E(
√
/)/π= . . . . ,c=
√ /, λ(r)and U(r;a,b)be defined by
λ(r) = –c r
–cr
(r= ), λ= logc logc
, (.)
and
U(r;a,b) =λ(r)ar+ –λ(r)br/r (r= ), U(;a,b) =aλb–λ, (.)
respectively.Then the function r→U(r;a,b)is strictly decreasing on(–∞,∞).
Proof Letx=b/a∈(,√),r= , and
V(r,x) = –λ(r)logx–logλ(r). (.)
Then from (.)-(.) one has
logU(r;a,b) =loga+
rlog
λ(r) + –λ(r)xr,
∂logU(r;a,b) ∂r =
λ(r)( –xr) + ( –λ(r))xrlogx
r(λ(r) + ( –λ(r))xr) –
log(λ(r) + ( –λ(r))xr)
r ,
∂logU(r;a,b) ∂r
x=
= , (.)
λ(r) =(c r
– )crlogc– (cr– )crlogc
(cr
– )
,
λ(r) + –λ(r)xr|x=√= –c r
–cr
+
– –c r
–cr
λ(r)( –xr) + ( –λ(r))xrlogx
r(λ(r) + ( –λ(r))xr)
x=√
=
rlog c c
,
∂logU(r;a,b) ∂r
x=√
= , (.)
∂logU(r;a,b) ∂x∂r =
λ(r)xr–
(λ(r) + ( –λ(r))xr)V(r,x), (.)
V(r, ) = log
c
(
c)
r– –
logc
(
c)
r– < , (.)
V(r,√) =cr
logc cr– –
logc cr–
> , (.)
where inequalities (.) and (.) hold due toc>cand the functiont→logt/(tr– )
is strictly decreasing on (,∞).
Note thatλ(r)∈(, ) and the functionx→V(r,x) is strictly increasing on (,√). Then (.)-(.) lead to the conclusion that there existsx∈(,
√
) such that the function
x→∂logU(r;a,b)/∂ris strictly decreasing on (,x) and strictly increasing on (x,
√ ). It follows from (.) and (.) together with the piecewise monotonicity of the func-tionx→∂logU(r;a,b)/∂ron the interval (,√) that
∂logU(r;a,b)
∂r < (.)
for alla,b> with <b/a<√.
Therefore, Lemma . follows from (.).
3 Main result
Theorem . Let c= E(
√
/)/π= . . . . ,c=
√
/andλ(r)be defined by(.).
Then the double inequality
α(r)Ar(a,b) + –α(r)Qr(a,b)/r<TDA(a,b),Q(a,b)
<β(r)Ar(a,b) + –β(r)Qr(a,b)/r
holds for all r≤and a,b> with a=b if and only ifα(r)≥/andβ(r)≤λ(r),where r= is the limit value of r→.
Proof We first prove that Theorem . holds forr= .
Since A(a,b) <TD[A(a,b),Q(a,b)] <Q(a,b) for all a,b > with a=b, and A(a,b),
TD(a,b) andQ(a,b) are symmetric and homogeneous of degree , without loss of gen-erality, we assume thatα(),β()∈(, ) anda>b. Lett= (a–b)/(a+b)∈(,√/)
andp∈(, ). Then (.) and (.) lead to
A(a,b) =Q(a,b)√ –t, TDA(a,b),Q(a,b)=
πQ(a,b)E(t),
(.)
pA(a,b) + ( –p)Q(a,b) –TDA(a,b),Q(a,b)=
πQ(a,b)f(t),
Therefore, Theorem . forr= follows easily from Lemma . and (.).
Next, letr< anda,b> witha=b, then it follows from Theorem . forr= that
A(a,b) +Q(a,b) <TD
A(a,b),Q(a,b)<λ()A(a,b) + –λ()Q(a,b). (.)
Note that
<Q(a,b)
A(a,b)< √
, (.)
TD[A(a,b),Q(a,b)] [Ar(a,b)+Qr(a,b)]/r =
+/r π
E(t)
[ + ( –t)r/]/r, (.)
TD[A(a,b),Q(a,b)]
[λ(r)Ar(a,b) + ( –λ(r))Qr(a,b)]/r= π
E(t)
[λ(r)( –t)r/+ –λ(r)]/r, (.)
lim
t→+
+/r π
E(t)
[ + ( –t)r/]/r = lim t→√/
π
E(t)
[λ(r)( –t)r/+ –λ(r)]/r= . (.)
Therefore, Theorem . forr< follows from (.)-(.) and Lemma . together with the monotonicity of the functionr→[(ar+br)/]/r.
Letr= . Then Theorems . leads to Corollary . immediately.
Corollary . Letλ= ( +√)[ – E(√/)/π].Then the double inequality
π
√
–t+π
<E(t) < π λ
√
–t+π
( –λ)
holds for all t∈(,√/).
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 Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China.2Icahn School of Medicine at Mount Sinai, Friedman Brain Institute, New York, 10033, United States.
Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 61673169, 11371125, 11401191, and 61374086.
Received: 28 November 2016 Accepted: 17 January 2017 References
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