On rational bounds for the gamma function

R E S E A R C H

Open Access

On rational bounds

for the gamma function

Zhen-Hang Yang

_{, Wei-Mao Qian}

_{, Yu-Ming Chu}

_{and Wen Zhang}

*_{Correspondence:}

chuyuming2005@126.com

1_{Department of Mathematics,}

Huzhou University, Huzhou, 313000, China

Full list of author information is available at the end of the article

Abstract

In the article, we prove that the double inequality

x2+p0

x+p0

<

(x

2_{+ 9/5}

x+ 9/5

holds for allx∈(0, 1), we present the best possible constants

λ

μ

λ(x

x+ 9/5 ≤

(x

μ(x

x+p0

for allx∈(0, 1), and we find the value ofx∗in the interval (0, 1) such that

(x

_(x

x∈(x∗, 1), where

(x) is the classical gamma function,

γ

p0=

γ

γ

MSC: 41A60; 33B15; 26D07

Keywords: gamma function; psi function; rational bound; completely monotonic function

1 Introduction

Forx> , the classical Euler gamma function(x) and its logarithmic derivative, the so-called psi functionψ(x) [] are defined by

(x) =

∞



tx–_e–t_{dt, ψ(x) =} (x) (x), respectively.

A real-valued functionf is said to be completely monotonic [] on an intervalI iff

has derivatives of all orders onIand (–)n_f(n)_{(x)≥ for alln≥ andx∈I. The}

well-known Bernstein theorem [] states that a functionf on [,∞) is completely monotonic if and only if there exists a bounded and non-decreasing functionω(t) such thatf(x) =

∞

 e–xtdω(t) converges for allx∈[,∞).

Recently, the gamma function have attracted the attention of many researchers. In par-ticular, many remarkable inequalities and properties for(x) can be found in the literature [–].

Due to(x+ ) =x(x) and(n+ ) =n!, we will only need to focus our attention on

(x+ ) withx∈(, ). Gautschi [] proved that the double inequality

n–s<(n+ )

(n+s)<e

(–s)ψ(n+) _(.)

holds for alls∈(, ) andn∈N.

Inequality (.) was generalized and improved by Kershaw [] as follows:

x+s 

–s

<(x+ )

(x+s)<e

(–s)ψ[x+(+s)/]

for allx>  ands∈(, ).

Elezović, Giordano and Pečarić [] established the double inequality

 +

 +x

–x

xx<(x+ ) < –xxx (.)

for the gamma function being valid for allx∈(, ), and asked for ‘other bounds for the gamma function in terms of elementary functions’.

Ivády [] provided the bounds for gamma function in terms of very simple rational functions as follows:

x+ 

x+  <(x+ ) <

x+ 

x+  (.)

for allx∈(, ). Inequality (.) can be regarded as a simple estimation of the value of the gamma function.

In [], Zhao, Guo and Qi proved that the function

x→Q(x) = log(x+ )

log(x_{+ ) –log(x+ )}

is strictly increasing on (, ). The monotonicity ofQ(x) on the interval (, ) and the facts thatQ(+_{) =γ andQ(}–_{) = ( –γ) lead to the conclusion that}

x_{+ }

x+ 

(–γ)

<(x+ ) <

x_{+ }

x+ 

γ

(.)

for allx∈(, ), whereγ =limn→∞( n

k=/k–logn) = . . . . is the Euler-Mascheroni

constant. Let

L(x) =

x+ 

x+ , L(x) =

x+ 

x+ 

(–γ)

, (.)

U(x) =

x_{+ }

x+ , U(x) =

x_{+ }

x+ 

γ

Then we clearly see that

L(x) <L(x) (.)

for allx∈(, ), and numerical computations show that

U(/) = . . . . >U(/) = . . . . ,

U(/) = . . . . >U(/) = . . . . .

(.)

Motivated by (.)-(.), it is natural to ask what the better parameterspandqon the interval (, ) are such that the double inequality

x_+p

x+p <(x+ ) < x_+q

x+q

holds for allx∈(, ). The main purpose of the article is to deal with this questions. Some complicated computations are carried out using the Mathematica computer algebra sys-tem.

2 Lemmas

In order to establish our main results we need several lemmas, which we present in this section.

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

(decreasing)on(a,b),then so are the functions f(x) –f(a)

g(x) –g(a),

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 ]) Let n∈Nand m∈N∪ {}with n>m,ai≥for all

≤i≤n,anam> and

Pn(t) = – m

i=

aiti+ n

i=m+

aiti.

Then there exists t∈(,∞)such that Pn(t) = ,Pn(t) < for t∈(,t)and Pn(t) > for

t∈(t,∞).

Lemma .(See [, Corollary .]) The inequality

ψ(x+ ) <  log

x+x+ 

+ log

x+x+ 

Lemma .(See [, Corollary .(ii)]) The double inequality

(x+_)(x+x+_)

x_{+ x}₊ x+

 x+

 

<ψ(x+ )

< (x+



)(x+x+

π

(π_–))

x_{+ x}₊π– (π–)x+

π– (π–)x+

 (π–)

holds for all x> .

Lemma . The inequalities



x+_ – 

(x+_) ≤ψ

_{(x+ )≤} 

x+_, (.)

– 

(x+_) ≤ψ

_{(x+ )≤–} 

(x+_) +



(x+_), (.)

 (x+_)–

 (x+_) ≤ψ

_{(x+ )≤} 

(x+_) (.)

hold for all x> –/.

Proof Letx> –/, andR(x) andR(x) be defined by

R(x) =ψ(x+ ) –log

x+ 

, (.)

R(x) = –ψ(x+ ) +log

x+ 

+ 

(x+_), (.)

respectively. Then making use of the well-known formulas

ψ(x) =

∞



e–t

t – e–xt

 –e–t

dt, logx=

∞



e–t_–e–xt

t dt

we get

R(x) =

∞



e–t t –

e–(x+)t

 –e–t

dt–

∞



e–t–e–(x+/)t t dt = ∞   t – e–t/

 –e–t

e–(x+/)tdt

= ∞   t –  sinh(t/)

e–(x+/)tdt

=

∞



sinh(t/) –t/

tsinh(t/) e

–(x+/)t_dt,

(.)

R(x) = –

∞   t –  sinh(t/)

e–(x+/)tdt+  

∞



te–(x+/)tdt

=

∞



(t– )sinh(t/) + t

tsinh(t/)

e–(x+/)tdt, (.)

Note that

tsinh(t/)(t

_{– )sinh(t/) + t}

tsinh(t/) =

∞

n=

(n– )(n+ ) (n– )!

t



n–

> , (.)

sinh(t/) – t

>  (.)

fort> .

It follows from (.)-(.) and the Bernstein theorem for complete monotonicity prop-erty that the two functions R(x) and R(x) are completely monotonic on the interval

(–/,∞).

Therefore, Lemma . follows easily from (.), (.) and the complete monotonicity of

R(x) andR(x) on the interval (–/,∞) together with the facts that

log

x+ 

(n)

=(–)

n–_{(n– )!}

(x+_)n ,

 (x+_)

(n)

=(–)

n_{(n+ )!}

(x+_)n+ .

Lemma . The double inequality

x+ 

x+ 

/(q+)

<x

_+q

x+q <

x+ 

x+ 

/q

(.)

holds for all x∈(, )and q> .

Proof Letx∈(, ),q> , andH(x) andH(x) be defined by

H(x) =log

x+q–log(x+q), H(x) =log

x+ –log(x+ ), (.) respectively. Then simple computations lead to

lim

x→+

H(x)

H(x)

=

q, xlim→–

H(x)

H(x)

= 

q+ , (.)

H

+=H

+= , (.)

H

–=H

–= , (.)

H_(x)

H_(x)=

(x+ )(x_{+ )(x}_{+ qx–q)}

(x+q)(x_+q)(x_{+ x– )},

H_(x)

H_(x)

= (q– )(x,q)

(x+q)_(x_+q)_(x_{+ x– )}, (.)

where

(x,q) =x+ x+ ( –x)x+ x+ q+ xx+ x+ ( –x)q

–xx– x– x– x+  >(x, )

= ( –x)x+ x+ x+x– + x+ x+ x(x– )

From (.) and (.) we clearly see that H_(x)/H_(x) is strictly increasing on (,√ – )∪(√ – , ). We assert that the functionH(x)/H(x) is strictly increasing

on (, ). Indeed, ifx∈(,√ – ), thenH_(x)= , and Lemma . and (.) together with the monotonicity ofH_(x)/H_(x) on (,√ – ) lead to the conclusion thatH(x)/H(x) is

strictly increasing on (,√ – ); ifx∈(√ – , ), thenH_(x) = , and Lemma . and (.) together with the monotonicity ofH_(x)/H_(x) on (√ – , ) lead to the conclusion thatH(x)/H(x) is strictly increasing on (

√  – , ).

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

monotonic-ity of the functionH(x)/H(x) on (, ).

Letp> ,x∈(, ), andf(p;x),f(p;x),f(p;x) andf(p;x) be defined by

f(p,x) =log(x+ ) –log

x_+p

x+p

, (.)

f(p,x) =

∂f(p,x)

∂x =ψ(x+ ) –

x x_+p+



x+p, (.)

f(p,x) =

∂f(p,x)

∂x =ψ

_{(x+ ) +} x

(x_+p) –



x_+p–



(x+p), (.)

f(p,x) =

∂f(p,x)

∂x =ψ

_{(x+ ) –} x

(x_+p) +

x

(x_+p) +



(x+p). (.)

Lemma . Let f(p,x)be defined by(.).Then

f(p, /) <  (.)

for p∈[/, /].

Proof From (.) and the second inequality in Lemma . we have

f(p, /) = ψ(x+ ) +

x

(x_+p)–



x_+p–

 (x+p)

x=/

< (x+



)(x+x+

π

(π_–))

x_{+ x}₊π_–

(π_–)x+

π_–

(π_–)x+_(π_–)

x=/

+ x



(x_+p) –



x_+p–

 (x+p)

x=/

= (π

_{– )}

(π_{– ,)}–

(p_{+ p}_{+ p– )}

(p+ )_{(p+ )} . (.)

Elaborated computations lead to

(π– ) (π_{– ,)}–

(p+ p+ p– ) (p+ )_{(p+ )}

=(p

_{+ ,p}_{+ p}_{– p– )}

(p+ )_{(p+ )} >  (.)

From (.) and (.) we get

f(p, /) <

(π– ) (π_{– ,)}–

(p_{+ p}_{+ p– )}

(p+ )_{(p+ )}

p=/

= (π

_{– )}

(π_{– ,)}–

,,

,, = –. . . . < .

Lemma . Let f(p,x)be defined by(.).Then

f(/, /) > . (.)

Proof From (.) and the first inequality in Lemma . we have

f(/, /) = ψ(x+ ) +

x

(x_+p)–



x_+p–

 (x+p)

p=/,x=/

> (x+

 )(x

_+x+ )

x_{+ x}₊ x+

 x+

 

+ x



(x_+p) –



x_+p–

 (x+p)

p=/,x=/

= ,,,,,

,,,,,,> .

Lemma . Let f(p,x)be defined by(.).Then

f(/,x) < 

for x∈(/, /).

Proof It follows from Lemma . and (.) that

f(/,x) <

 log

x+x+ 

+ log

x+x+  

– x

x_{+ /}+



x+ /. (.) Elaborated computations lead to

 log

x+x+ 

+ log

x+x+ 

– x

x_{+ /}+



x+ /

= h(x)

(x+ )_(x_{+ )}_(x_{+ x+ )(x}_{+ x+ )}, (.)

where

h(x) = ,x+ ,,x+ ,,x+ ,,x+ ,,x

+ ,,x+ ,x– ,,x– ,x+ ,,

h(/) = –,,,,

,,, < , (.)

h(x) = ,,x+ ,,x+ ,,x+ ,,x+ ,,x

+ ,,x+ ,x– ,,x– ,, (.)

h(/) = –,,

From Lemma ., (.) and (.) we know thath(x) is strictly decreasing on (/, /), then (.) leads to the conclusion thath(x) <  forx∈(/, /).

Therefore,

f(/,x) <

 log

x+x+ 

+ log

x+x+  

– x

x_{+ /}+



x+ /

x=/

=  log

, , +

 log

, ,+

,

, = –. . . . < 

for x ∈ (/, /) follows from (.) and (.) together with h(x) <  for x ∈

(/, /).

Lemma . Let p∈[/, ]and f(p,x)be defined by(.).Then there existsη(p)∈(, )

such that f(p,x) < for x∈(,η(p))and f(p,x) > for x∈(η(p), ).

Proof Let

g(p,x) =x+pf(p,x) =

x+pψ(x+ ) +(x

_+p)

(x+p) – x

_{+ px. (.)}

Then simple computations lead to

gp, +=pψ() + , gp, –= (p+ )ψ() + p– , (.)

∂g(p,x)

∂x =

x+pψ(x+ ) + xx+pψ(x+ )

–(x

_+p)

(x+p) +

x(x_+p)

(x+p) – x

_{+ p. (.)}

It follows from the first inequalities in (.) and (.) together with the identity

ψ(n)(x+ ) =ψ(n)(x) + (–)nn!/xn+that

ψ(x+ ) =ψ(x+ ) –  (x+ ) ≥–

 (x+ /) –



(x+ ), (.)

ψ(x+ ) =ψ(x+ ) +  (x+ ) ≥

 (x+ /)–

 (x+ /) +



(x+ ). (.)

From (.)-(.) we have

∂g(p,x)

∂x ≥

x+p 

(x+ /)–

 (x+ /) +

 (x+ )

+ xx+p –  (x+ /) –

 (x+ )

–(x

_+p)

(x+p) +

x(x+p) (x+p) – x

_{+ p}

= g(p,x)

where

g(p,x) = 

k=

bkxk–



k=

bkxk (.)

with

b=

p+ ,p– p,

b= 

,p+ p+ ,p+ ,p– ,p,

b=

,p+ ,p+ ,p+ ,p+ ,p– p,

b= 

,p+ ,p+ ,p+ ,p+ ,p+ ,p,

b=

,p+ ,p+ ,p+ ,p+ ,p+ ,p+ ,p,

b= 

p+ ,p+ ,p+ ,p+ ,p+ ,p+ ,p,

b= p+ p+ ,p+ ,p+ ,p+ ,p+ ,p– ,

b= –p+ ,p+ ,p+ ,p+ ,p+ ,p+ ,,

b= ,p+ ,p+ ,p+ ,p+ ,p+ ,,

b= p+ ,p+ ,p+ ,p+ ,p+ ,,

b= p+ ,p+ ,p+ ,p+ ,p+ ,,

b= ,p+ ,p+ ,p+ ,p+ ,,

b= p+ ,p+ ,p+ ,p+ ,,

b= p+ ,p+ ,p+ ,,

b= p+ ,p+ ,,

b= p+ ,

b= ,

g(p, ) = –, – ,p– ,p– ,p+ ,p

+ ,p+ ,p+ ,p, (.)

g(/, ) =

,,

 > . (.)

From Lemma ., (.) and (.) we clearly see that

g(p, ) >  (.)

forp∈[/, ].

Making use of Lemma . again, and (.) and (.) together with the facts thatbk> 

forp∈[/, ] andk= , , , . . . ,  we know thatg(p,x) >  forp∈[/, ] andx∈(, ).

From (.) and the identityψ(n)_{(x) =ψ}(n)_{(x+ ) + (–)}n+_n!/xn+_{we get}

– 

(x+ /) ≤ψ

_{(x+ ) =ψ(x+ ) –} 

(x+ )

≤– 

(x+ /)+

 (x+ /)–



(x+ ). (.)

Takingx=  in the first inequality of (.) andx=  in the second inequality of (.), one has

ψ()≥–

, ψ

_()≤–

 . (.)

It follows from (.) and (.) that

gp, +≤–  ×

 



+  = –

< , (.)

gp, –≥– (p+ )

_{+ p–  (.)}

forp∈[/, ]. Note that

– (p+ )

_{+ p– }

=

(p+ )( –p). (.)

Inequality (.) and equation (.) imply that

gp, –_≥–



 + 



+ ×  –  =



 >  (.)

forp∈[/, ].

Therefore, Lemma . follows easily from (.), (.), (.) and the monotonicity

of the functionx→g(p,x) on the interval (, ).

Lemma . Let p ∈ [/, /] and f(p,x) be defined by (.). Then there exist

η(p),η(p)∈(, )withη(p) <η(p)such that f(p,x) > for x∈(,η(p))∪(η(p), )and

f(p,x) < for x∈(η(p),η(p)).

Proof It follows from (.) that

f

p, += π



p

p–

(π_{+ ) + }

π

p+

(π_{+ ) – }

π

> , (.)

f

p, –=(π

_{– )p}_{+ (π}_{– )p+π}

(p+ ) >  (.)

forp∈[/, /].

on (η(p), ). Then Lemma . leads to the conclusion that

f

p,η(p)≤f(p, /) < . (.)

Therefore, there existη(p)∈(,η(p)) andη(p)∈(η(p), ) such thatf(p,x) >  forx∈

(,η(p))∪(η(p), ) andf(p,x) <  forx∈(η(p),η(p)) follow from (.)-(.) and the

piecewise monotonicity of the functionx→f(p,x) on the interval (, ).

3 Main results

Theorem . Let p> and p=γ/( –γ) = . . . . .Then the inequality

(x+ ) >x

_+p

x+p (.)

holds for all x∈(, )if and only if p≤p,and the inequality

(x+ )≤μ(x

_+p )

x+p

(.)

holds for all x∈(, )if and only ifμ≥μ,where

μ=

(x+p)(x+ )

(x +p)

= . . . . (.)

and x= . . . .is the unique solution of the equation

ψ(x+ ) – x

x_+p 

+ 

x+p

=  (.)

on the interval(, ).

Proof If inequality (.) holds for allx∈(, ), thenp≤pfollows easily from

lim

x→–

log(x+ ) –log(x_x₊+_pp)

 –x = –ψ() +

  +p=γ–

p

 +p≥.

Next, we prove that inequality (.) holds for allx∈(, ) andp=pand (.) holds for

allx∈(, ) if and only ifμ≥μ.

Letf(p,x),f(p,x),f(p,x) be defined by (.)-(.) and

g(x) =(x

_+p )

x f(p,x) =

(x_+p )

x ψ

_{(x+ ) –}p

x –

(x_+p )

x_(x+p )

+ . (.)

Then elaborated computations lead to

fp, +

=fp, –

= , (.)

f

p, +

= –γ–γ



γ > , f

p, –

= , (.)

g+= –∞, g–=(π

_{– )p}

+ (π– )p+π

g(x) =(x

_+p

)ψ(x+ )

x –

(p

–x)ψ(x+ )

x

–p[x

_{– x}_{– p}

x– p(p+ )x– (p+ )p]

(x+p)x

. (.)

It follows from the second inequality in (.) and the first inequality in (.) together with (.) that

g(x)≥– (x

_+p )

x_{(x+ /)} –

(p –x)

(x+ /)x

–p[x

_{– x}_{– p}

x– p(p+ )x– (p+ )p]

(x+p)x

= 

x_{(x+ )}_(x+p )

g(x),

where

g(x) = x+ (p+ )x+ p(p– )x+ p

p_– p+ 

x

–p

p_– p– 

x– p(p+ )

p_– p– 

x

– p_p_– p– 

x– p_(p+ )

p_– p– 

x

–p_p_– p– 

.

It is easy to verify that all the coefficients of the polynomial g(x) are positive, which

implies that g(x) is strictly increasing on (, ), then from (.) and (.) we know that there existsη∈(, ) such that the functionf(p,x) is strictly decreasing on (,η) and

strictly increasing on (η, ).

It follows from (.) and (.) together with the piecewise monotonicity of the function

f(p,x) on the interval (, ) that there existsx∈(, ) such thatf(p,x) is strictly

increas-ing on (,x) and strictly decreasing on (x, ) andxis the unique solution of equation

(.) on the interval (, ).

Therefore, the desired results follow easily from (.), (.), (.) and the piecewise monotonicity of the functionf(p,x) on the interval (, ) together with the fact that the

functionp→(x_{+p)/(x+p) is strictly increasing.}

Numerical computations show thatx= . . . . andμ= (x+p)(x+ )/(x+p) =

. . . . .

Theorem . The inequality

(x+ ) >x

₊ 

γ

x+ 

γ

holds for all x∈(,x∗),and its reverse inequality

(x+ ) <x

₊ 

γ

holds for all x∈(x∗, ),where x∗= . . . .is the unique solution of the equation

(x+ ) –x

₊

γ

x+_γ = 

on the interval(, ).

Proof Letf(p,x),f(p,x) andf(p,x) be, respectively, defined by (.), (.) and (.).

Then simple computations lead to

f



γ, 

+

=f



γ, 

–

= , (.)

f



γ, 

+

= , f



γ, 

–

= –γ–γ



 +γ > . (.)

From Lemma . and /γ = . . . . ∈ [/, /] we know that there exist

η(/γ),η(/γ)∈(, ) withη(/γ) <η(/γ) such thatf(/γ,x) is strictly increasing on

(,η(/γ))∪(η(/γ), ) and strictly decreasing on (η(/γ),η(/γ)). We claim that

f

/γ,η(/γ)

< . (.)

Indeed, iff(/γ,η(/γ))≥, then the piecewise monotonicity of the functionf(/γ,x)

on the interval (, ) and (.) lead to the conclusion thatf(/γ,x) is strictly increasing on (, ), which contradicts (.).

It follows from (.) and (.) together with the piecewise monotonicity of the function

f(/γ,x) on the interval (, ) that there existη∗(/γ)∈(η(/γ),η(/γ)) andη∗(/γ)∈

(η(/γ), ) such thatf(/γ,x) is strictly increasing on (,η∗(/γ))∪(η∗(/γ), ) and strictly

decreasing on (η∗_(/γ),η∗_(/γ)).

Therefore, Theorem . follows easily from (.) and (.) together with the piecewise monotonicity off(/γ,x) on (, ). Numerical computations show thatx∗= . . . . .

Theorem . The double inequality

λ(x₊ )

x+ 

≤(x+ ) <x

₊ 

x+ 

(.)

holds for all x∈(, )with the best possible constant

λ=(τ+ )(τ+ ) τ+ 

= . . . . , (.)

whereτ= . . . .is the unique solution of the equation

ψ(x+ ) – x

x₊ 

+ 

x+_ =  (.)

Proof Letf(p,x),f(p,x) andf(p,x) be, respectively, defined by (.), (.) and (.).

Then simple computations lead to

f

 , 

+

=f

 , 

–

= , (.)

f

 , 

+

=

–γ < , f

 , 

–

= 

–γ > . (.)

It follows from Lemma . that there exist η(/),η(/)∈ (, ) with η(/) <

η(/) such that f(/,x) >  for x∈(,η(/))∪(η(/), ) and f(/,x) <  for

x∈(η(/),η(/)), andf(/,x) is strictly increasing on (,η(/))∪(η(/), ) and

strictly decreasing on (η(/),η(/)). Then Lemmas .-. lead to the conclusion that

η(/)∈(/, /) and

f

/,η(/)

< . (.)

From (.), (.), (.) and the piecewise monotonicity off(/,x) on (, ) we clearly

see that there existsτsuch thatτis the unique solution of equation (.) on the interval

(, ), andf(/,x) is strictly decreasing on (,τ) and strictly increasing on (τ, ).

Equation (.) and the piecewise monotonicity of the functionf(/,x) on the interval (, ) lead to the conclusion that

f(/,τ)≤f(/,x) <  (.)

for allx∈(, ).

Therefore, inequality (.) holds for allx∈(, ) follows from (.) and (.). We clearly see that the parameter λgiven by (.) is the best possible constant such that the first inequality in (.) holds for all x∈(, ). Numerical computations show that

τ= . . . . andλ= . . . . .

Remark . From Theorems . and . we clearly see that the double inequality

x_+p 

x+p

<(x+ ) <x

_+p 

x+p

(.)

holds for allx∈(, ) withp=γ/( –γ) = . . . . andp= /, the constantpappears to

be the best possible, but this is not true forp, and a slightly smaller value forpis possible.

Unfortunately, we cannot find the best possible constantpin the article; we leave this as

an open problem for the reader.

Remark . From Lemma .,γ+γ_{< , /γ > ,γ/( –γ) >  and (x}_{+ )/(x+ ) <  for}

x∈(, ) one has

x₊ γ

–γ

x+_–γ_γ >

x_{+ }

x+ 

(–γ)

,

x_{+ }

x+ 

γ

>x

₊

γ

x+_γ >

x_{+ }

x+ 

γ/(+γ)

>

x_{+ }

x+ 

(–γ)

.

Therefore, the lower bound for(x+ ) given in (.) is better than that given in (.), the first inequality in Theorem . is an improvement of the first inequality in (.) for

x∈(,x∗) and the second inequality in Theorem . is an improvement of the second inequality in (.) forx∈(x∗, ), wherex∗= . . . . is given by Theorem ..

Remark . It is not difficult to verify that

min

x∈(,)

x_{+ }

x+ 

γ

=(√ – )γ = . . . . ,

min

x∈(,)

–xxx= e–/e= . . . . ,

x₊ 

x+_ < . forx∈(., .) and

x₊ 

x+_ < .

forx∈(θ,θ), whereθ= (. –

√

.)/ = . . . . andθ= (. +

√

.)/ = . . . . . Therefore, the upper bound (x_{+ /)/(x+ /) for(x+ ) given in (.) is}

better than that given in (.) forx∈(., .), and it is also better than that given in (.) forx∈(θ,θ).

Remark . Let

L(x) =

  +

 +x

–x

xx, L(x) =

x₊ γ

–γ

x+_–γ_γ . Then numerical computations show that

L(/) = . . . . <L(/) = . . . . ,

L(/) = . . . . <L(/) = . . . . ,

L(/) = . . . . <L(/) = . . . . ,

L(/) = . . . . <L(/) = . . . . ,

L(/) = . . . . <L(/) = . . . . ,

L(/) = . . . . <L(/) = . . . . ,

Therefore, there existsδ∈(, /) such that the lower bound for(x+ ) given in (.) is better than that given in (.) forx∈(δ, / +δ)∪(/ –δ, / +δ)∪(/ –δ, / +δ)∪ (/ –δ, / +δ)∪(/ –δ, / +δ)∪(/ –δ, / +δ)∪(/ –δ, / +δ).

4 Results and discussion

In this paper, we provide the accurate bounds for the classical gamma function in terms of very simple rational functions, which can be used to estimate the value of the gamma function in the area of engineering and technology.

5 Conclusion

In the article, we present several very simple and practical rational bounds for the gamma function, which can be regarded as a simple estimation of the value of the gamma function. The given results are improvements of some well-known results.

Acknowledgements

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191) and the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101).

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_{Department of Mathematics, Huzhou University, Huzhou, 313000, China.}2_{Customer Service Center, State Grid Zhejiang}

Electric Power Research Institute, Hangzhou, 310009, China.3_{School of Distance Education, Huzhou Broadcast and TV}

University, Huzhou, 313000, China.4_{Friedman Brain Institute, Icahn School of Medicine at Mount Sinai, New York, 10029,}

USA.

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Received: 1 May 2017 Accepted: 25 August 2017 References

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