R E S E A R C H
Open Access
On approximating the modified Bessel
function of the first kind and Toader-Qi mean
Zhen-Hang Yang and Yu-Ming Chu
**Correspondence:
chuyuming2005@126.com School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China
Abstract
In the article, we present several sharp bounds for the modified Bessel function of the first kindI0(t) =
∞
n=0
t2n
22n(n!)2 and the Toader-Qi meanTQ(a,b) = 2
π
π/2
0 a
cos2θbsin2θd
θ
for allt> 0 anda,b> 0 witha=b.
MSC: 33C10; 26E60
Keywords: modified Bessel function; Toader-Qi mean; logarithmic mean; identric mean
1 Introduction
Leta,b> ,p: (,∞)→R+be a strictly monotone real function,θ∈(, π) and
rn(θ) =
(ancosθ+bnsinθ)/n, n= ,
acosθbsinθ, n= . (.)
Then the meanMp,n(a,b) was first introduced by Toader in [] as follows:
Mp,n(a,b) =p–
π
π
prn(θ)
dθ =p–
π
π/
prn(θ)
dθ , (.)
wherep–is the inverse function ofp.
From (.) and (.) we clearly see that
M/x,(a,b) =
π
π/√ dθ
acosθ+bsinθ
=AGM(a,b)
is the classical arithmetic-geometric mean, which is related to the complete elliptic integral of the first kindK(r) =π/( –rsinθ)–/dθ. The Toader mean
Mx,(a,b) =
π
π/
acosθ+bsinθdθ=T(a,b)
is related to the complete elliptic integral of the second kindE(r) =π/( –rsinθ)/dθ.
We have
Mxq,(a,b) =
π
π/
aqcosθbqsinθdθ
/q
(q= ).
In particular,
Mx,(a,b) =
π
π/
acosθbsinθdθ=TQ(a,b) (.)
is the Toader-Qi mean.
Recently, the arithmetic-geometric meanAGM(a,b) and the Toader meanT(a,b) have attracted the attention of many researchers. In particular, many remarkable inequalities forAGM(a,b) andT(a,b) can be found in the literature [–].
Forq= , the meanMxq,(a,b) seems to be mysterious, Toader [] said that he did not know how to determine any sense for this mean.
Letz∈C,ν∈R\{–, –, –, . . .} and(z) =limn→∞n!nz/[∞
k=(z+k)] be the classical
gamma function. Then the modified Bessel function of the first kindIν(z) [] is given by
Iν(z) =
∞
n=
zn+ν
n!n+ν(ν+n+ ). (.)
Very recently, Qiet al.[] proved the identity
Mxq,(a,b) =
π
π/
aqcosθbqsinθdθ
/q
=√abI/q
q
log
a
b (.)
and inequalities
L(a,b) <TQ(a,b) <A(a,b) +G(a,b)
<
A(a,b) +G(a,b)
<I(a,b) (.)
for allq= anda,b> witha=b, whereL(a,b) = (b–a)/(logb–loga),A(a,b) = (a+b)/,
G(a,b) =√ab, andI(a,b) = (bb/aa)/(b–a)/eare, respectively, the logarithmic, arithmetic, geometric, and identric means ofaandb.
Letb>a> ,p∈R,t= (logb–loga)/ > , and thepth power meanAp(a,b) be defined
by
Ap(a,b) =
ap+bp
/p
(p= ), A(a,b) =
√
ab=G(a,b).
Then the logarithmic meanL(a,b), the identric meanI(a,b), and the pth power mean
Ap(a,b) can be expressed as
L(a,b) =√absinht
t , I(a,b) =
√
abet/tanht–,
Ap(a,b) =
√
abcosh/p(pt) (p= )
(.)
and (.)-(.) lead to
TQ(a,b) √
ab =
Mx√,(a,b) ab =
π
π/
etcos(θ)dθ=I(t)
=
π
π/
cosh(tcosθ)dθ=
π
π/
The main purpose of this paper is to present several sharp bounds for the modified Bessel function of the first kindI(t) and the Toader-Qi meanTQ(a,b).
2 Lemmas
In order to establish our main results we need several lemmas, which we present in this section.
Lemma .(See []) Letnkbe the number of combinations of n objects taken k at a time,
that is,
n k =
n!
k!(n–k)!.
Then
∞
k=
n k
=
n n .
Lemma .(See []) Let{an}∞n= and{bn}∞n= be two real sequences with bn> and limn→∞an/bn=s.Then the power series∞n=antnis convergent for all t∈Rand
lim t→∞
∞
n=antn
∞
n=bntn
=s
if the power series∞n=bntnis convergent for all t∈R.
Lemma . The Wallis ratio
Wn=
(n+)
()(n+ ) (.)
is strictly decreasing with respect to all integers n≥and strictly log-convex with respect to all real numbers n≥.
Proof It follows from (.) that
Wn+ Wn
= –
(n+ )< (.)
for all integersn≥.
Therefore,Wnis strictly decreasing with respect to all integersn≥ follows from (.).
Letf(x) =(x+ /)/(x+ ) andψ(x) =(x)/(x) be the psi function. Then it follows from the monotonicity ofψ(x) that
logf(x)=ψ
x+
–ψ
(x+ ) > (.)
for allx≥.
Therefore,Wnis strictly log-convex with respect to all real numbersn≥ follows from
Lemma .(See []) The double inequality
(x+a)–a <
(x+a)
(x+ ) <
x–a
holds for all x> and a∈(, ).
Lemma . Let sn= (n)!(n+)!/[n(n!)].Then the sequence{sn}∞n=is strictly decreasing and
lim n→∞sn=
π. (.)
Proof The monotonicity of the sequence{sn}∞n=follows from
sn+ sn
=(n+ )(n+ ) (n+ ) < .
To prove (.), we rewritesnas
sn= (n+ )
(n– )!! nn!
=n+
( )
(n+)
(n+ )
=(n+
)
π
(n+)
(n+ )
. (.)
It follows from Lemma . and (.) that
π =
π
n+
n+ <sn<
π
n+
n . (.)
Therefore, equation (.) follows from (.).
Lemma .(See []) Let A(t) =∞k=aktkand B(t) =
∞
k=bktkbe two real power series converging on(–r,r) (r> )with bk> for all k.If the non-constant sequence{ak/bk}is in-creasing(decreasing)for all k,then the function A(t)/B(t)is strictly increasing(decreasing)
on(,r).
Lemma .(See []) Let A(t) =∞k=aktkand B(t) =
∞
k=bktkbe two real power series converging on Rwith bk > for all k.If there exists m∈Nsuch that the non-constant sequence{ak/bk}is increasing(decreasing)for≤k≤m and decreasing(increasing)for k≥m,then there exists t∈(,∞)such that the function A(t)/B(t)is strictly increasing
(decreasing)on(,t)and strictly decreasing(increasing)on(t,∞).
Lemma . The identity
I(t) =
∞
n=
(n)! n(n!)t
n
Proof From (.) and Lemma . together with the Cauchy product we have
I(t) =
∞
n=
n
k=
k(k!)
(n–k)[(n–k)!]
tn
=
∞
n=
n(n!)
n
k=
(n!)
(k!)[(n–k)!]
tn=
∞
n=
(n)! n(n!)t
n.
Lemma .(See []) Let–∞<a<b<∞and f,g: [a,b]→R.Then
b
a
f(x)g(x)dx≥ b–a
b
a
f(x)dx b
a g(x)dx
if both f and g are increasing or decreasing on(a,b).
Lemma .(See []) Let–∞<a<b<∞and f,g: (a,b)→R.Then
b
a
f(x)g(x)dx–
b–a b
a f(x)dx
b
a
g(x)dx
≥
(b–a) b
a
x–a+b
f(x)dx
b
a
x–a+b
g(x)dx (.)
if both f and g are convex on the interval(a,b),and inequality(.)becomes an equality if and only if f or g is a linear function on(a,b).
3 Main results
Theorem . The double inequalities
et
+ t<I(t) < et
√
+ t (.)
and
b
+log(b/a)<TQ(a,b) <
b
+log(b/a) (.)
hold for all t> and b>a> .
Proof From (.) we have
I(t) =
π
π/
cosh(tsinθ)dθ=
π
cosh(tx) √
–x dx (.)
and
e–tI(t) =
π
π/
et[cos(θ)–]dθ=
π
π/
dθ
etsinθ
<
π
π/
dθ
+ tsinθ =
√
We clearly see that bothcosh(tx) and /√ –xare increasing with respect toxon (, ).
Then Lemma . and (.) lead to
I(t)≥
π
dx
√ –x
cosh(tx)dx=sinht
t
= e
t
t
–
et > et
t
– + t =
et
+ t. (.)
Therefore, inequality (.) follows from (.) and (.). Lett=log(b/a)/. Then it follows from (.) and (.) that
√
b/a
+log(b/a)<
TQ(a,b) √
ab <
√
b/a
+log(b/a). (.)
Therefore, inequality (.) follows from (.).
Remark . From Theorem . we clearly see that
lim t→∞e
–tI
(t) = lim
x→+TQ(x, ) = .
Theorem . The double inequalities
α
sinh(t)
t <I(t) <β
sinh(t)
t (.)
and
α
L(a,b)A(a,b) <TQ(a,b) <β
L(a,b)A(a,b) (.)
hold for all t> and a,b> with a=b if and only ifα≤/√π,β≥
√
/,α≤
√ /π
andβ≥.
Proof Let
R(t) = I
(t)
sinh(t)/(t), (.)
an=
(n)!
n(n!), bn=
n
(n+ )!. (.)
Then simple computation leads to
an bn
=(n)!(n+ )!
n(n!) . (.)
It follows from Lemma . and (.) that the sequence{an/bn}∞n=is strictly decreasing
and
lim n→∞
an bn
=
From Lemma . we have
R(t) =
∞
n=antn
∞
n=bntn
. (.)
Lemma . and (.) together with the monotonicity of the sequence{an/bn}∞n=lead to
the conclusion thatR(t) is strictly decreasing on the interval (,∞). Therefore, we have
lim
t→∞R(t) <R(t) <t→lim+R(t) =
a b
= . (.)
From Lemma ., (.), and (.) we know that
lim t→∞R(t) =
π. (.)
Therefore, inequality (.) holds for allt> if and only ifα≤/
√
π andβ≥
√ / follows easily from (.), (.), and (.) together with the monotonicity ofR(t).
Letb>a> andt=log(b/a)/. Then inequality (.) holds fora,b> witha=bif and only ifα≤
√
/π, andβ≥ follows from (.) and (.) together with inequality (.)
for allt> if and only ifα≤/√πandβ≥
√
/.
Remark . Equations (.) and (.) imply that
lim t→∞e
–t√tI (t) =
√
π
or we have the asymptotic formula
I(t)∼ et
√
πt (t→ ∞).
Theorem . Letλ,λ> ,t= . . . .be the unique solution of the equation
d dt
tI(t) –sinht
(cosht– )sinht
= (.)
on(,∞)and
λ=
tI(t) –sinht
(cosht– )sinht
= . . . . . (.)
Then the double inequality
(λcosht+ –λ) sinht
t <I(t) <
(λcosht+ –λ) sinht
t (.)
or
λA(a,b) + ( –λ)G(a,b)
L(a,b) <TQ(a,b) <λA(a,b) + ( –λ)G(a,b)
L(a,b)
Proof Let
R(t) =
I(t) –sinht t
(cosht–)sinht t
, (.)
cn=
(n)! n(n!)–
(n+ )!, dn= n–
(n+ )!, sn=
(n)!(n+ )!
n(n!) , (.)
and
sn=n+ n+ n+ sn–
n+ n+ . (.)
Then it follows from Lemma ., Lemma ., Lemma ., and (.)-(.) that
R(t) =
∞
n=cntn
∞
n=dntn
, (.)
lim
t→∞R(t) =n→∞lim cn dn
= lim n→∞
ns n–
n– =
π, (.)
c d
= <
c d
= >
c d
=
, (.)
cn+ dn+
– cn
dn
= –
ns n
(n+ )(n+– )(n– ), (.)
and we have the inequality
sn>
π
n+ n+ n+ –n+ n+
>
n+ n+ n+ –n+ n+ =
n–n+ n+ > (.)
for alln≥.
From (.)-(.) we know that the sequence{cn/dn}∞n= is strictly increasing for ≤ n≤ and strictly decreasing forn≥. Then Lemma . and (.) lead to the conclusion that there existst∈(,∞) such thatR(t) is strictly increasing on (,t) and decreasing
on (t,∞). Therefore, we have
min
R
+,lim t→∞R(t)
<R(t)≤R(t) (.)
for allt> , andtis the unique solution of equation (.) on (,∞).
Note that
R
+=c
d
=
. (.)
From (.), (.), (.), (.), and (.) we get
π <R(t)≤R(t) =λ. (.)
Therefore, inequality (.) holds for allt> if and only ifλ≤/π,λ≥λfollows from
(.) and (.) together with the piecewise monotonicity ofR(t) on (,∞). Numerical
Theorem . Let p,q∈R.Then the double inequality
cosh–pt
sinht t
p
<I(t) <q sinht
t + ( –q)cosht (.)
or
Lp(a,b)A–p(a,b) <TQ(a,b) <qL(a,b) + ( –q)A(a,b)
holds for all t> or a,b> with a=b if and only if p≥/and q≤/.
Proof If the first inequality of (.) holds for allt> , then
lim t→+
I(t) –cosh–pt(sinhtt )p
t =
p–
≥,
which implies thatp≥/.
It is not difficult to verify that the functioncosh–pt(sinht/t)pis strictly decreasing with
respect to p∈Rfor any fixedt> , hence we only need to prove the first inequality of (.) for allt> andp= /, that is,
I(t) >
sinht
t
cosht. (.)
Making use of the power series and Cauchy product formulas together with Lemma . we have
I(t) –
sinht
t
cosht
=
∞
n= n
k=
(k)! k(k!)
((n–k))! (n–k)((n–k)!) –
n+– n+
(n+ )!
tn. (.)
LetWnandsnbe, respectively, defined by Lemma . and Lemma ., and
un= n
k=
(k)! k(k!)
((n–k))! (n–k)((n–k)!) –
n+– n+
(n+ )! . (.)
Then simple computations lead to
u=u= , u=
, u=
. (.)
It follows from Lemma . and Lemmas .-. together with (.) that
un= n
k=
(k!)((n–k)!)
(k)! k(k!)
((n–k))! (n–k)((n–k)!) –
n+– n+
(n+ )!
=
n
k=
(k!)((n–k)!)WkWn–k –
n+– n+
> n k=
(k!)((n–k)!)W n/ –
n+– n+ (n+ )!
= n k= (k!)((n–k)!)
(n/ + /)
(/)(n/ + )
–
n+– n+
(n+ )!
> n k=
π(n/ + /)(k!)((n–k)!) –
n+– n+
(n+ )!
=
π(n+ )(n!) n
k=
(n!)
(k!)((n–k)!) –
n+– n+
(n+ )!
=
π(n+ )(n!)
(n)! (n!) –
n+– n+ (n+ )!
=
n+(n+– )
π(n+ )!
n+
n+–
n+ sn–π
>
n+(n+– )
π(n+ )!
+
π –π
> (.)
for alln≥.
Therefore, inequality (.) follows from (.)-(.).
If the second inequality of (.) holds for allt> , then we have
lim t→+
I(t) –qsinhtt – ( –q)cosht
t =
q–
≤,
which implies thatq≤/.
Sincecosht>sinht/t, we only need to prove that the second inequality of (.) holds for allt> andq= /, that is,
cosht–I(t) cosht–sinht/t>
. (.)
Let
αn=
nn! – (n– )!!
nn!(n)! , βn=
n
(n+ )!, γn=
(n+ )(n+ ) (n+ ) Wn,
andWnbe defined by (.).
Then simple computations lead to
cosht–I(t) cosht–sinht/t=
∞
n=αntn
∞
n=βntn
, (.)
αn+
βn+
–αn
βn
=n+
n+ ( –Wn+) – n+
n ( –Wn) =
γn–
n(n+ ), (.)
γn+
γn
= + n+
(n+ ) > , γ=
> . (.)
From (.) and (.) we clearly see that the sequence{αn/βn}∞n=is strictly increasing,
sinht/t] is strictly increasing on the interval (,∞). Therefore, inequality (.) follows from the monotonicity of (cosht–I(t))/[cosht–sinht/t] and the fact that
lim t→+
cosht–I(t) cosht–sinht/t=
α
β
=
.
Theorem . Let p,q> ,tbe the unique solution of the equation
d[p(I(t)–)
cosh(pt)–]
dt = (.)
and
μ= p(I
(t) – ) cosh(pt) –
. (.)
Then the following statements are true:
(i) The double inequality
– p +
pcosh(pt) <I(t) < –
q +
qcosh(qt) (.)
or
–
p G(a,b) +
pA
p
p(a,b)G–p(a,b)
<TQ(a,b) <
–
q G(a,b) +
qA
q
q(a,b)G–q(a,b)
holds for allt> ora,b> witha= bif and only ifp≤√/andq≥. (ii) The inequality
I(t)≥ –
μ p +
μ
p cosh(pt) (.)
or
TQ(a,b)≥
–μ
p G(a,b) +
μ pA
p
p(a,b)G–p(a,b)
holds for allt> ora,b> witha=bifp∈(√/, ).
Proof (i) Let
R(t) =
p(I(t) – )
cosh(pt) – , (.)
un=
n(n!), vn= pn–
(n)!.
Then simple computations lead to
R(t) =
∞
n=untn
∞
n=vntn
un+ vn+
–un
vn
= – (n)! (p)n(n!)
p– n+
n+ . (.)
From (.) we clearly see that the sequence{un/vn}∞n=is strictly decreasing ifp≥ and
strictly increasing ifp≤√/. Then Lemma . and (.) lead to the conclusion that the functionR(t) is strictly decreasing ifp≥ and strictly increasing ifp≤
√
/. Hence, we have
R(t) < lim t→+R(t) =
u v
=
(.)
for allt> ifp≥ and
R(t) > lim t→+R(t) =
u v
=
(.)
for allt> ifp≤√/.
Therefore, inequality (.) holds for allt> ifp≤√/ andq≥ follows easily from (.) and (.) together with (.).
If the first inequality (.) holds for allt> , then we have
lim t→+
I(t) – ( –p +pcosh(pt))
t =
–p
≥,
which implies thatp≤√/. If there existsq∈(
√
/, ) such that the second inequality of (.) holds for allt> , then we have
lim t→∞
I(t) – ( –q
+
qcosh(qt))
eqt ≤. (.)
But the first inequality of (.) leads to
I(t) – ( –q +
q
cosh(qt))
eqt
>e
(–q)t
+ t –
– q e
–qt– +e
–qt
q → ∞ (t→ ∞),
which contradicts inequality (.).
(ii) Ifp∈(√/, ), then from (.) we know that there exists n∈Nsuch that the
sequence {un/vn}∞n= is strictly decreasing forn≤nand strictly increasing for n≥n.
Then (.) and Lemma . lead to the conclusion that there existst∈(,∞) such that
the function R(t) is strictly decreasing on (,t] and strictly increasing on [t,∞). We
clearly see thattsatisfies equation (.). It follows from (.) and (.) together with
the piecewise monotonicity ofR(t) that
R(t)≥R(t) =μ. (.)
It is not difficult to verify that the function
– p +
pcosh(pt)
is strictly increasing with respect topon the interval (,∞) and
cosh
t
– >cosh
/t
fort> .
Lettingp=√/, /,√/, /, / andq= in Theorem .(i), then we get Corol-lary . immediately.
Corollary . The inequalities
cosh/(t) < cosh
t
– < cosh
t
–
<cosh
√
t
< cosh
t
+
<
cosh
√
t
+
<I(t) <
+cosht
or
G/(a,b)A/(a,b) < A//(a,b)G/(a,b) –G(a,b)
< A
/
/(a,b)G/(a,b) –
G(a,b)
<A √
/ √
/(a,b)G
–√/(a,b)
< A
/
/(a,b)G/(a,b) +
G(a,b)
< A
√ / √
/(a,b)G
–√/(a,b) +
G(a,b)
<TQ(a,b) <A(a,b) +G(a,b)
hold for all t> or all a,b> with a=b.
Theorem . Let p> .Then the following statements are true:
(i) The inequality
I(t) >
cosh(pt)
p (.)
or
TQ(a,b) >G–p(a,b)A p
p (a,b) (.)
(iii) The inequalities
cosh/t<cosh √
t
<
cosh
√
t
/
<I(t) <cosh
t
<e
t/
(.)
or
G/(a,b)A/(a,b) <A √
/ √
/(a,b)G
–√/(a,b)
<A √
/ √
/(a,b)G
–√/(a,b)
<TQ(a,b) <A(a,b) +G(a,b)
<G(a,b)e(A(a,b)–G(a,b))/(L(a,b))
hold for allt> or alla,b> witha=b.
Proof (i) If inequality (.) holds for allt> , then we have
lim t→+
I(t) – [cosh(pt)]
p
t =
p–
≥,
which implies thatp≥√/.
It follows from Lemma of [] that the function [cosh(pt)]/(p)is strictly decreasing with respect top∈(,∞) for any fixedt> , hence we only need to prove that inequality (.) holds for allt> andp=√/. From the sixth inequality of Corollary . we clearly see that it suffices to prove that
cosh
√
t
+
>
cosh
√
t
/
for allt> , which is equivalent to
log
cosh(
√ x) +
>
log(coshx) (.)
for allx> , wherex=√t/. Let
f(x) =log
cosh(
√ x) +
–
log(coshx), (.)
f(x) =
√
sinh(√x)coshx– cosh(√x)sinhx– sinhx,
ξn= (
√
– )(√ + )n–+ (√ + )(√ – )n–– ,
ηn= (
√
+ )n–.
Then simple computations lead to
f(x) = f(x)
coshx[cosh(√x) + ], (.)
f(x) = ∞
n=
ξn
(n– )!x
n–, (.)
ξ=ξ= , (.)
ηnξn= (
√
– )(ηn–η)(ηn–η). (.)
From (.)-(.) andηn>η>η> forn≥ we know that
f(x) > (.)
for allx> .
Therefore, inequality (.) follows easily from (.)-(.) and (.).
(ii) The sufficiency follows easily from the monotonicity of the function p →
[cosh(pt)]/(p) and the last inequality in Corollary . together with the identity ( + cosht)/ =cosh(t/).
Next, we prove the necessity. If there exists p ∈ (/,
√
/) such that I(t) <
[cosh(pt)]/(p
)for allt> , then we have
lim t→∞
I(t) – [cosh(pt)]/(p
)
et/(p) ≤. (.)
But the first inequality of (.) leads to
I(t) – [cosh(pt)]/(p
)
et/(p) > + t
et et/(p)–
+e–pt
/(p)
→ ∞ (t→ ∞),
which contradicts (.).
(iii) Letp= ,√/,√/, /, +. Then parts (i) and (ii) together with the monotonicity
of the functionp→[cosh(pt)]/(p)lead to (.).
Theorem . Letθ∈[,π/].Then the inequality
I(t) >
cosh(tcosθ) +cosh(tsinθ)
(.)
or
TQ(a,b) >A
cosθ
cosθ(a,b)G–
cosθ(a,b) +Asinθ
sinθ(a,b)G–
sinθ(a,b)
holds for all t> or all a,b> with a=b if and only ifθ∈[π/, π/].In particular,the inequalities
I(t) >
cosh
–√
t +cosh
+√
t
>
cosh
√
t +cosh
t >cosh
√
or
TQ(a,b) >
A
√
–√/
√
–√/(a,b)G
–√–√/(a,b) +A
√
+√/
√
+√/(a,b)G
–√+√/(a,b)
> A
√ / √
/(a,b)G
–√/+A/
/(a,b)G/(a,b)
>A
√ / √
/(a,b)G
–√/(a,b)
hold for all t> or all a,b> with a=b.
Proof If inequality (.) holds for allt> , then we have
lim t→+
I(t) –cosh(tcosθ)+cosh(tsinθ)
t = –
cos(θ)≥,
which implies thatθ∈[π/, π/].
Next, we prove the sufficiency of inequality (.). Simple computations lead to
∂[cosh(tcosθ) +cosh(tsinθ)]
∂θ =
tsin(θ)
sinh(tsinθ)
tsinθ –
sinh(tcosθ)
tcosθ
, (.)
sinhx
x
=
x
coshx–sinhx
x > (.)
forx> .
Equation (.) and inequality (.) imply that the function θ →[cosh(tcosθ) +
cosh(tsinθ)] is decreasing on [,π/] and increasing on [π/,π/] for any fixedt> . Hence, it suffices to prove that inequality (.) holds for allt> andθ=θ=π/.
Let
ρn=
(–
√ )
n+ (+√ )
n
(n)! , σn= n(n!),
R(t) =
cosh(tcosθ) +cosh(tsinθ)
I(t)
. (.)
Then simple computations lead to
R(t) =
∞
n=ρntn
∞
n=σntn
, (.)
ρ
σ
=ρ
σ
=ρ
σ
=ρ
σ
= , (.)
ρn+ σn+ ρn σn
– = – √
[(n+√ – )(√ – )n–+ (n–√ – )(√ + )n–]
(n+ )[(√ – )n+ (√ + )n] < (.)
forn≥.
It follows from Lemma . and (.)-(.) thatR(t) is strictly decreasing on (,∞).
Therefore,
I(t) >
cosh(tcosθ) +cosh(tsinθ)
(.)
Letθ =π/,π/,π/. Then inequality (.) follows easily from (.) and the mono-tonicity of the functionθ→[cosh(tcosθ) +cosh(tsinθ)].
Theorem . The inequality
I(t) > sinht
t +
( –π)(tsinht– cosht+ )
πt (.)
holds for all t> .
Proof It is easy to verify that
d dx
√
–x =
+ x
( –x)/ > ,
∂cosh(tx)
∂x =t
cosh(tx) >
for allt> andx∈(, ), which implies that the two functions /√ –xandcosh(tx) are
convex with respect toxon the interval (, ). Then from Lemma . and (.) we have
π
I(t) –
π
sinht t
=
cosh(tx) √
–x dx–
dx
√ –x
cosh(tx)dx
>
x– √
–xdx
x–
cosh(tx)dx
=( –π)(tsinht– cosht+ )
t . (.)
Therefore, inequality (.) follows from (.).
Remark . The inequalityI(t) >sinh(t)/tin (.) is equivalent to the first inequality TQ(a,b) >L(a,b) in (.). Therefore, Theorem . is an improvement of the first inequality in (.).
Letp∈RandM(a,b) be a bivariate mean of two positiveaandb. Then thepth power-type meanMp(a,b) is defined by
Mp(a,b) =M/p
ap,bp (p= ), M(a,b) =
√
ab.
We clearly see that
Mλp(a,b) =M/pλ
aλ,bλ
for allλ,p∈Randa,b> ifMis a bivariate mean.
Theorem . The inequality
TQ(a,b) <Ip(a,b)
Proof The second inequality (.) can be rewritten as
TQ(a,b) <A/(a,b). (.)
In [, ], the authors proved that the inequality
I(a,b) >A/(a,b) (.)
holds for all distinct positive real numbersaandbwith the best possible constant /. Inequalities (.) and (.) lead to
TQ(a,b) <A/(a,b) =A//
a/,b/<I/a/,b/=I/(a,b) (.)
for alla,b> witha=b.
Ifp≥/, thenTQ(a,b) <I/(a,b)≤Ip(a,b) follows from (.) and the functionp→ Ip(a,b) is strictly increasing [].
IfTQ(a,b) <Ip(a,b) for alla,b> witha=b. Then
I(t) –et/tanh(pt)–/p< (.)
for allt> .
Inequality (.) leads to
lim t→+
I(t) –et/tanh(pt)–/p
t =
–p ≤,
which implies thatp≥/.
Remark . For alla,b> witha= b, the Toader meanT(a,b) satisfies the double in-equality [, ]
A/(a,b) <T(a,b) <Alog/(logπ–log)(a,b) (.)
with the best possible constants / andlog/(logπ–log), and the one-sided inequal-ity []
T(a,b) <I/(a,b). (.)
It follows from (.) and (.) that
A//(a,b) =A/
a/,b/<Ta/,b/
=T//(a,b) <I/
a/,b/=I//(a,b),
which can be rewritten as
Inequalities (.) and (.) lead to the inequalities
TQ(a,b) <A/(a,b) <T/(a,b) <I/(a,b) (.)
for alla,b> witha=b.
Remark . For alla,b> witha=b, Theorem . shows that
L/(a,b)A/(a,b) <TQ(a,b) <L(a,b) +A(a,b)
. (.)
It follows fromL(a,b) <A(a,b)/ + G(a,b)/, given by Carlson in [], andA(a,b) >
L(a,b) that
L(a,b) <L/(a,b)A/(a,b), A(a,b) +G(a,b)
>
L(a,b) +A(a,b)
.
Therefore, inequality (.) is an improvement of the first and second inequalities of (.).
Remark . In [, , ], the authors proved that the inequalities
L(a,b) <AGM(a,b) <L/(a,b)A/(a,b) <L/(a,b) (.)
hold for alla,b> witha=b.
Inequalities (.)-(.) lead to the chain of inequalities
L(a,b) <AGM(a,b) <L/(a,b)A/(a,b)
<TQ(a,b) <A/(a,b) <T/(a,b) <I/(a,b) (.)
for alla,b> witha=b.
Motivated by the first inequality in (.) and the third inequality in (.), we propose Conjecture ..
Conjecture . The inequality
TQ(a,b) >L/(a,b)
holds for all a,b> with a=b.
For alla,b> witha=b, inspired by the double inequality
A(a,b)G(a,b) <TQ(a,b) <A(a,b) +G(a,b)
given in Corollary . and the inequalities
A(a,b)G(a,b) <I(a,b)L(a,b) <I(a,b) +L(a,b)
<
A(a,b) +G(a,b)
Conjecture . The inequality
TQ(a,b) <I(a,b)L(a,b)
holds for all a,b> with a=b.
Remark . LetWnbe the Wallis ratio defined by (.), andcn,dn, andsnbe defined by
(.). Then it follows from Lemma . and the proof of Theorem . that the sequence {sn}∞n= is strictly decreasing andlimn→∞sn= /π, and the sequence{cn/dn}∞n= is strictly
increasing forn= , and strictly decreasing forn≥. Hence, we have
π <sn= (n+ )W
n≤s=
(.)
and
π =min
c d
, lim n→∞
cn dn
< cn
dn
=
ns n–
n– ≤ c d
=
(.)
for alln∈N.
Inequalities (.) and (.) lead to the Wallis ratio inequalities
π(n+) <Wn≤
√
n+
and
–n(π– ) +
π(n+ ) <Wn≤
+ ×–n
(n+ )
for alln∈N.
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.
Acknowledgements
The authors would like to thank the anonymous referee for his/her valuable comments and suggestions. The research was supported by the Natural Science Foundation of China under Grants 11371125, 11401191, and 61374086.
Received: 14 October 2015 Accepted: 21 January 2016 References
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