R E S E A R C H
Open Access
Monotonicity properties of a function
involving the psi function with applications
Tie-Hong Zhao
1, Zhen-Hang Yang
2and Yu-Ming Chu
3**Correspondence:
[email protected] 3School 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 parametera∈(1/15,∞) such that the functions
ψ
(x+ 1) –Lx(x,a) andψ
(x+ 1) –Lxx(x,a) are strictly increasing or decreasing with respect tox∈(0,∞), whereL(x,a) =90a12+2log(x2+x+3a+13 ) +45a2
90a2+2log(x2+x+15a–145a ) and
ψ
(x) is the classical psi function. As applications, we getseveral new sharp bounds for the psi function and its derivatives.
MSC: 33B15
Keywords: gamma function; psi function; monotonicity
1 Introduction
For real and positive values ofx, Euler’s gamma functionand its logarithmic derivative
ψ, the so-called psi function, are defined by
(x) = +∞
tx–e–tdt, ψ(x) =
(x)
(x),
respectively. For extensions of these functions to complex variables and for basic proper-ties see []. Recently, the gamma functionand psi functionψ have been the subject of intensive research. In particular, many remarkable inequalities and monotonicity proper-ties for these functions can be found in the literature [–].
Recently, Yang [] introduced the function
L(x,a) = a+ log
x+x+a+
+ a
a+ log
x+x+a– a
(.)
and proved that the double inequality
L(x,a) <ψ(x+ ) <L(x– ,b) +
x
holds for allx> if and only ifa≤a= . . . . andb≥( +
√
)/ = . . . . if
a∈(/,∞) andb∈(/,∞), whereais the unique solution of the equationL(,a) =
ψ().
Partial derivative computations give
Lx(x,a) =
a+
x+
x+x+a+
+ a
a+
x+
x+x+a– a
, (.)
Lxx(x,a) = –
a+
x+x–a+
(x+x+a+ )
– a
a+
x+x+ a+
(x+x+a– a )
, (.)
Lxxx(x,a) =
a+
(x+ )(x+x– a) (x+x+a+
)
+ a
a+
(x+ )(x+x+ a)
(x+x+a– a )
. (.)
It is not difficult to verify that
lim
x→∞
ψ(x+ ) –L(x,a)
x– = –
(a–+
√
)(a–
–√ )
,a (.)
by use of the L’Hôspital’s rule and the formula
ψ(x)∼
x+
x +
x –
x+
x –
x +· · · (x→ ∞) (.)
given in [].
The main purpose of this paper is to present the best possible parametera∈(/,∞) such that the functionsψ(x+ ) –Lx(x,a) andψ(x+ ) –Lxx(x,a) are strictly
increas-ing or decreasincreas-ing with respect tox∈(,∞), and establish several new sharp bounds for the psi function and its derivatives. All numerical computations are carried out using the MATHEMATICA software.
2 Lemmas
In order to prove our main results we need several lemmas, which we present in this sec-tion.
Lemma .(see []) LetL(x,a)be defined on(,∞)×(/,∞)by(.).Then the follow-ing statements are true:
(i) the functionsa→∂L(x,a)/∂xis strictly decreasing,a→∂L(x,a)/∂xis strictly
increasing anda→∂L(x,a)/∂xis strictly decreasing on(/,∞);
(ii) the functiona→Lxx(x,a) –Lxx(y,a)is strictly decreasing on(/,∞)ifx>y> ;
(iii) the inequalitiesψ(x+ ) –Lx[x, ( +
√
)/] > ,
ψ(x+ ) –Lxx[x, ( +
√
)/] < ,and
ψ(x+ ) –Lxxx[x, ( +
√
)/] > hold for allx> .
Lemma .(see []) The identity
ψn(x+ ) –ψn(x) =(–)
nn!
xn+
holds for all x> and n∈N.
Lemma .(see []) Letλ∈Rand f be a real-valued function defined on the interval I= (λ,∞)withlimx→∞f(x) = .Then f(x) < if f(x+ ) –f(x) > for all x∈I,and f(x) >
3 Main results
Theorem . LetL(x,a)be defined on(,∞)×(/,∞)by(.)and Fa(x) =ψ(x+ ) –
L(x,a).Then the following statements are true:
(i) Fa(x)is strictly increasing with respect toxon(,∞)if and only if a≥a= ( +
√
)/ = . . . .;
(ii) Fa(x)is strictly decreasing with respect toxon(,∞)if and only if a≤a= ( – π+
√
π– π+ )/[(π– )] = . . . ..
Proof (i) If Fa(x) is strictly increasing with respect to x on (,∞), then
limx→∞[xFa(x)]≥. Making use of L’Hôspital’s rule and (.) we get
lim
x→∞
ψ(x+ ) –L(x,a)
x– = –
xlim→∞
xFa(x)
= –(a–
+√ )(a–
–√ )
,a ≤. (.)
Therefore,a≥a= ( +
√
)/ follows easily from (.) anda∈(/,∞). Ifa≥a= ( +
√
)/, then Lemma .(i) and (iii) lead to
Fa(x) =ψ(x+ ) –Lx(x,a)≥ψ(x+ ) –Lx(x,a) >
for allx∈(,∞). Therefore,Fa(x) is strictly increasing with respect toxon (,∞).
(ii) IfFa(x) is strictly decreasing with respect toxon (,∞), then
Fa() =ψ() –Lx(,a)≤. (.)
It follows from (.) andψ() =π/ that
ψ() –Lx(,a) =
(π– )a– ( – π)a–π+
(a+ )(a– ) . (.)
Therefore,a≤a= ( – π+
√
π– π+ )/[(π– )] follows from (.)
and (.) together witha∈(/,∞).
Next, we prove thatFa(x) is strictly decreasing with respect toxon (,∞) ifa≤a. From
Lemma .(i) we clearly see that it is enough to prove thatFa(x) <ψ(x+ ) –Lx(x,a) <
for allx∈(,∞).
Letx> anda> /. Then it follows from (.), (.), and Lemma . that
lim
x→∞F
a(x) =xlim→∞
ψ(x+ ) –Lx(x,a)
= , (.)
Fa(x+ ) –Fa(x)
=ψ(x+ ) –ψ(x+ ) –Lx(x+ ,a) +Lx(x,a)
= – (x+ ) +
(a+ )[(x+ )+ (x+ ) +a+ ]
– a
[(x+ ) + ]
+ x+ (a+ )(x+x+a+
)
+ a
(x+ )
(a+ )(x+x+a– a )
–
(x+ )
=q(x,a)
p(x,a), (.)
where
q(x,a) =(a+
√– )(
+√ –a)
a (x+ )
–(a+ )
(a– )
a , (.)
p(x,a) = (x+ )
x+ x+a+
x+x+a+
×
x+x+ –
a
x+ x+ –
a
> (.)
and
∂q(x,a)
∂a = –
(a+ )
,a (x+ )
–(a+ )(a–
)(,a + )
,a < . (.)
We divide the proof into two cases.
Case.x∈(/,∞). Then from (.) and (.) together witha< / we get
q(x,a) >q
x,
>q
,
= ,,
,,,> . (.)
Equation (.) and inequalities (.) and (.) lead to
Fa(x+ ) –Fa(x) > . (.) Therefore,Fa(x) < follows from Lemma . and (.) together with (.).
Case.x∈(, /]. Then Lemma .(i) anda> / lead to
Lxx(x,a) >Lxx
x,
. (.)
It follows from (.) and Lemma .(iii) together with (.) that
Fa(x) =ψ(x+ ) –Lxx(x,a) <Lxx(x,a) –Lxx(x,a) =
P(x)
Q(x), (.)
where
Q(x) =x+ x+ x+ x+
×x+ x+ x+ x+ (x+ )> (.)
and
P(x) = ,,,x+ ,,,x+ ,,,,x
+ ,,,,x+ ,,,,x+ ,,,,x
+ ,,,,x+ ,,,x– ,,,. (.)
From (.) we clearly see that
P(x) < (.)
forx∈(, /] sinceP(x) is strictly increasing on (, /] and
P
= –,,,,,,, ,,,, < .
Equation (.) and inequalities (.) and (.) lead to the conclusion thatFa(x) is strictly decreasing on (, /]. Therefore,Fa(x) <Fa() = .
Theorem . LetL(x,a)be defined on(,∞)×(/,∞)by(.),Fa(x) =ψ(x+ ) –L(x,a) and a= . . . .is the unique solution of the equationLxx(,a) =ψ().Then the fol-lowing statements are true:
(i) Fa(x)is strictly decreasing with respect toxon(,∞)if and only if
a≥a= ( +
√
)/ = . . . .;
(ii) Fa(x)is strictly increasing with respect toxon(,∞)if and only ifa≤a.
Proof (i) If Fa(x) is strictly decreasing with respect to x on (,∞), then
limx→∞[xFa(x)]≤. Making use of L’Hôspital’s rule and (.) we get
lim
x→∞
ψ(x+ ) –L(x,a)
x– =
xlim→∞
xFa(x)
= –(a–
+√ )(a–
–√ )
,a ≤. (.)
Therefore,a≥a= ( +
√
)/ follows easily from (.) anda∈(/,∞). Ifa≥a= ( +
√
)/, then
Fa(x) =ψ(x+ ) –Lxx(x,a)≤ψ(x+ ) –Lxx(x,a) <
follows easily from Lemma .(i) and (iii).
(ii) IfFa(x) is strictly increasing with respect toxon (,∞), then
Fa() =ψ() –Lxx(,a)≥. (.)
It follows from Lemma .(i) that the function a→Fa() is strictly decreasing on (/,∞). Note that
Fa() = , a∈(/, /), (.)
F/ () = –
∞
n=
n = . . . . > , (.)
F/ () =, , –
∞
n=
Therefore,a≤afollows from (.)-(.) and the monotonicity of the functiona→
Fa().
Ifa≤a, then we only need to prove thatFa(x) > for allx∈(,∞) by Lemma .(i). We divide the proof into two cases.
Case.x∈(/,∞). Then from (.), (.), Lemma .(ii), Lemma . anda< /
we get
lim
x→∞F
a(x) =xlim→∞
ψ(x+ ) –Lxx(x,a)
= , (.)
Fa(x+ ) –Fa(x)
=ψ(x+ ) –ψ(x+ ) –Lxx(x+ ,a) –Lxx(x,a)
< (x+ ) –
Lxx
x+ ,
–Lxx
x,
= –r(x)
s(x) , (.)
where
s(x) =x+ x+ x+ x+
×x+ x+ x+ x+ (x+ )> (.)
and
r(x) = ,,,,x+ ,,,,,x
+ ,,,,,x+ ,,,,,x
+ ,,,,,x+ ,,,,,x
+ ,,,,,x+ ,,,,,x
+ ,,,,,x+ ,,,,x
– ,,,,.
We clearly see that
r(x) > (.)
forx∈(/,∞) sincer(x) is strictly increasing on (/,∞) and
r
=,,,,,,,, ,,,, > .
Therefore,Fa(x) > forx∈(/,∞) follows from (.)-(.) and Lemma ..
Case.x∈(, /]. Then fromFa() =ψ() –Lxx(,a) = we know that it is enough
to prove thatFa(x) > .
It follows from (.) and Lemma .(i) and (iii) together witha> / that
Fa(x) =ψ(x+ ) –Lxxx(x,a)
>Lxxx(x,a) –Lxxx
x,
= –R(x)
where
S(x) = (x+ )x+ x+ x+ x+
×x+ x+ x+ x+ > (.)
and
R(x) = ,,,,,,x
+ ,,,,,,x
+ ,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,,x
+ ,,,,,,x
– ,,,,,,x
– ,,,,,,.
It follows from
R(x) > , R() < , and
R
= –,,,,,,,,,,,,,, ,,,,,,,, <
that
R(x) </ –x / R() +
x
Therefore,Fa(x) > follows from (.)-(.). Leta= ( +
√
)/,a= ( – π+
√
π– π+ )/[(π– )], and
a= . . . . be the unique solution of the equationLxx(,a) =ψ(). Then (.) and
(.) lead to
Lx(x,a) =
x+
x+x+ /
x+ x+ x/ + x/ + /, (.)
Lx(x,a) =
x+
x+x+ π
(π–)
x+ x+π–
(π–)x+
π–
(π–)x+(π–)
, (.)
Lxx(x,a)
= –
(,x+ ,x+ ,x+ ,x+ ,x+ ,x+ )
(x+ x+ x+ x+ ) , (.)
Lxx
x,
= –
x+ ,x+ ,x+ ,x+ x+ x+
(x+ x+ )(x+ x+ ) . (.)
From Lemma .(i), Theorems . and ., (.)-(.), anda> / we have the
fol-lowing.
Corollary .
(i) The double inequalities
Lx(x,a) <ψ(x+ ) <Lx(x,a)
and
Lxx(x,a) <ψ(x+ ) <Lxx(x,a)
hold for allx> with the best possible constantsa,a,anda.
(ii) The double inequalities
x+
x+x+
x+ x+x
+ x
+
<ψ(x+ )
<
x+
x+x+ π
(π–)
x+ x+π–
(π–)x+
π–
(π–)x+(π–)
–
x+ ,x+ ,x+ ,x+ x+ x+
(x+ x+ )(x+ x+ )
<ψ(x+ ) < –
(,x+ ,x+ ,x+ ,x+ ,x+ ,x+ )
(x+ x+ x+ x+ )
Leta∈(,–π+
√
π–π+
(π–) ] andγ = . . . . be the Euler-Mascheroni con-stant. Then from Lemma .(ii) and the fact thatFa() = –γ–L(,a) andlimx→∞Fa(x) =
we get Corollary . immediately.
Corollary . The double inequality
L(x,a) <ψ(x+ ) <L(x,a) –γ –L(,a)
holds for all x> and a∈( ,
–π+√π–π+
(π–) ]with the best possible constant–γ – L(,a).
In particular,taking a= /, /,√/, /and using(.)one has
log
x+x+
+ log
x+x+
<ψ(x+ ) < log
x+x+
+ log
x+x+ + log + log –γ,
log
x+
+
log
x+x+
<ψ(x+ ) < log
x+
+ log
x+x+
+ log +
log
–γ,
log
x+x+
–
<ψ(x+ ) < log
x+x+ – + log
–γ
and
ψ(x+ ) > log
x+x+ log
x+x+
for x> .
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
1Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China.2Power Supply Service Center,
Zhejiang Electric Power Corporation Research Institute, Hangzhou, 310007, China.3School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China.
Acknowledgements
The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. The research was supported by the Natural Science Foundation of China under Grants 11301127, 61374086 and 11171307, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
Received: 8 April 2015 Accepted: 1 June 2015 References
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