Monotonic Properties of Differences for Remainders of Psi Function

REMAINDERS OF PSI FUNCTION

FENG QI, DA-WEI NIU, AND BAI-NI GUO

Abstract. Let Λp,q(x) = λ(px)−qλ(x) and Φp,q(x) = φ(px)−qφ(x) in

x ∈ (0,∞) for p > 0 and q ∈ R, where λ(x) = R0∞

tdt

(t2_+x2_)(e2πt−1) and

φ(x) =R∞

0

tdt

(t2+4x2)(eπt+1) are related toψ(x) andψ x+ 1 2

. In this article, some sufficient conditions onp >0 andq∈Rsuch that Λp,q(x) and Φp,q(x) are monotonic inx∈(0,∞) are obtained. Moreover, as by-product, an inequality involving the exponential function is established.

1. Introduction

Recall [7, 11] that a functionf is said to be completely monotonic on an interval

Iiff has derivatives of all orders and 0≤(−1)kf(k)(x)<∞for allk≥0 onI. For

our own convenience, the class of completely monotonic functions onIis denoted by

C[I]. The well known Bernstein’s Theorem [11] states thatf ∈ C[(0,∞)] if and only

iff(x) = R∞

0 e

−xt_{dµ(t), whereµ(t) is a nonnegative measure on [0,∞) such that}

the integral converges for allx >0. Note that a completely monotonic function in

(0,∞) which is non-identically zero cannot vanish at any point in (0,∞), see [7, 8]

and the references therein.

The noted Binet’s formula (see [9] and [10, p. 103]) states that forx >0,

ln Γ(x) =

x−1

2

lnx−x+ ln√2π+θ(x), (1)

where

θ(x) = Z ∞

0

_t

et₋₁ −1 + t

2 _e−xt

t2 dt (2)

is called the remainder of Binet’s formula (1).

2000Mathematics Subject Classification. Primary 33B15; Secondary 26D07, 26D20. Key words and phrases. Monotonicity, difference, psi function, remainder, inequality. The first author was supported in part by the Science Foundation of Project for Fostering Innovation Talents at Universities of Henan Province, China.

This paper was typeset usingAMS-LA_TEX.

Letp >0 andq∈Rbe real numbers and

hp,q(x) =θ(px)−qθ(x) (3)

in (0,∞).

It is clear thath1,q(x)∈ C[(0,∞)] forq≤1 and−h1,q(x)∈ C[(0,∞)] forq≥1.

Among other things, the following was proved in [1, 2].

Theorem A ([1, 2]). hp,p(x)∈ C[(0,∞)]if 0< p <1 and −hp,p(x)∈ C[(0,∞)] if p >1.

As a further generalization of Theorem A above, the following conclusion was

obtained recently.

Theorem B ([4]). hp,q(x) ∈ C[(0,∞)] if either q ≤ 1

p ≤ 1 or q ≤ 1 ≤

1

p and

−hp,q(x)∈ C[(0,∞)]if eitherq≥1

p ≥1 orq≥1≥

1

p.

In [3, p. 892] and [6, p. 17], it is given that forx >0,

ψ(x) = lnx− 1

2x−2

Z ∞

0

tdt

(t2_+x2_)(e2πt₋₁₎ (4)

and

ψx+1 2

= lnx+ 2 Z ∞

0

tdt

(t2_{+ 4x}2_)(eπt_{+ 1)}. (5)

Let

Λp,q(x) =λ(px)−qλ(x) (6)

and

Φp,q(x) =φ(px)−qφ(x) (7)

inx∈(0,∞) forp >0 andq∈R, where

λ(x) = Z ∞

0

tdt

(t2_+x2_)(e2πt₋₁₎ (8)

and

φ(x) = Z ∞

0

tdt

(t2_{+ 4x}2_)(eπt_{+ 1)}. (9)

Theorem 1. The functionΛp,q(x)is positive and decreasing inx∈(0,∞)if either p≥1andq≤0or0< p <1andpq≤1; it is negative and increasing inx∈(0,∞)

if p≥1 andpq≥1.

The functionΦp,q(x)is positive and decreasing inx∈(0,∞)if eitherp≥1and

q≤0 or 0< p <1 and q≤1; it is negative and increasing inx∈(0,∞)if p >1

andq≥1.

Theorem 2. The function Λp,q(x) is positive and decreasing in x ∈ (0,∞) for

either q ≤ 0 or 0 < q = 1

p2 ≤1, it is negative and increasing in x∈ (0,∞) for

1

p2 =q≥1.

The functionΦp,q(x)is positive and decreasing in x∈(0,∞)for either p2q <1

and q(p2_{−1)[(1 + 3q)p}2_{−4] ≤ 0 or p}2_{q = 1 and 0 < q ≤ 1, it is negative and}

increasing in x∈(0,∞) for either4≤p2_{(1 + 3q)≤1 + 3qor} 1

p2 =q≥1.

As by-product, we obtain the following inequality.

Theorem 3. Let τ∈Rbe a nonzero constant. Then inequality

ea+b> b

τ_eb_−aτ_ea

bτ_−aτ (10)

for alla >0 andb > 0 with a 6=b holds if and only ifτ ≥1 and reverses if and

only ifτ <0.

In particular, inequality

ea+b>be b_−aea

b−a (11)

is valid for all a > 0 and b > 0 with a 6= b, which is equivalent to the following

integral inequality

ea+b > 1 a−b

Z a

b

(1 +u)eudu. (12)

2. _{Proofs of main results}

Proof of Theorem 1. Direct calculation arrives at

Λp,q(x) =

Z ∞

0

t t2_+x2

₁

e2πpt₋₁− q e2πt₋₁

dt

= Z ∞

0 1

t2_+x2

t e2πt₋₁

_e2πt₋₁

e2πpt₋₁−q

dt

and

Φp,q(x) =

Z ∞

0

t t2_{+ 4x}2

₁

eπpt_{+ 1} − q eπt_{+ 1}

dt

= Z ∞

0 1

t2_{+ 4x}2

t eπt_{+ 1}

_eπt_{+ 1}

eπpt_{+ 1} −q

dt.

(14)

Let

ωr,s(t) =e

rt₋₁

est₋₁ and χr,s(t) = ert_{+ 1}

est_{+ 1} (15)

int∈(0,∞) for positive real numbersr >0 ands >0. The L’Hˆospital rule yields

lim

t→0+ωr,s(t) =

r

s, t→lim0+χr,s(t) = 1, (16)

and

lim

t→∞ωr,s(t) = limt→∞χr,s(t) = 

 

 

0, r < s,

∞, r > s.

(17)

Direct differentiation and standard argument gives that

dωr,s(t)

dt =

(r−s)e(r+s)t−(rert−sest) (est₋₁₎2

and

dχr,s(t)

dt =

(r−s)e(r+s)t_−(sest_−rert₎

(est_{+ 1)}2 .

The requirement dωr,s(t)

dt Q0 is equivalent with

(r−s)e(r+s)t_Qrert−sest,

(u−v)eu+v_Queu−vev,

ueu(ev−1)_Qvev(eu−1),

ueu eu₋₁ Q

vev ev₋₁,

whereu=rt >0 andv=st >0. Since

d dx

_xex

ex₋₁

=e

x_(ex_−x−1)

(ex₋₁₎2 >0

forx >0, the function _exex₋x₁ is increasing inx∈(0,∞). This implies that

dωr,s(t)

dt Q

0 in (0,∞) if and only if r_Qs, and then ωr,s(t) is 

 

 

decreasing

increasing

and only ifr_Qs. Therefore, whenp >1,

−q < e

2πt₋₁

e2πpt₋₁−q <

1

p−q;

when 0< p <1,

1

p−q <

e2πt₋₁

e2πpt₋₁ −q <∞.

Thus, ifp >1 andq≤0 or 0< p <1 andpq ≤1, the function Λp,q(x) is positive

and decreasing; ifp >1 andpq≥1, it is negative and increasing in (0,∞).

The requirement dχr,s(t)

dt Q0 is equivalent with

(r−s)e(r+s)t_Qsest−rert,

(u−v)eu+v_Qvev−ueu,

ueu(ev+ 1)_Qvev(eu+ 1),

ueu eu_{+ 1} Q

vev ev_{+ 1},

whereu=rt >0 andv=st >0. Since

d dx

_xex

ex_{+ 1}

=e

x_(ex_{+x+ 1)}

(ex_{+ 1)}2 >0

forx >0, the function _exex₊₁x is increasing inx∈(0,∞). This implies that

dχr,s(t)

dt Q

0 in (0,∞) if and only if r_Qs, and then χr,s(t) is 

 

 

decreasing

increasing

int ∈(0,∞) if

and only ifr_Qs. Therefore, whenp >1,

−q < e πt_{+ 1}

eπpt_{+ 1} −q <1−q;

when 0< p <1,

1−q < e πt_{+ 1}

eπpt_{+ 1}−q <∞.

Thus, if p >1 andq≤0 or 0< p <1 and q≤1, the function Φp,q(x) is positive

and decreasing; if p > 1 and q ≥1, it is negative and increasing in (0,∞). The

Proof of Theorem 2. Straightforward computation yields

Λp,q(x) =

Z ∞

0

t e2πt₋₁

₁

t2_+p2_x2−

q t2_+x2

dt = Z ∞ 0 t e2πt₋₁

(1−q)t2+ (1−p2q)x2

(t2_+p2_x2_)(t2_+x2₎ dt

,

Z ∞

0

t

e2πt₋₁ρp,q;t(x) dt,

(18)

Φp,q(x) =

Z ∞

0

t eπt_{+ 1}

1

t2_{+ 4p}2_x2 −

q t2_{+ 4x}2

dt = Z ∞ 0 t eπt_{+ 1}

(1−q)t2_{+ 4(1−p}2_q)x2 (t2_+p2_x2_)(t2_{+ 4x}2₎ dt

,

Z ∞

0

t

eπt_{+ 1}%p,q;t(x) dt.

(19)

By standard argument, we have

dρp,q;t(x)

dx =

2xt4_[p2_(p2_q−1)u2_{+ 2p}2_{(q−1)u+ (q−p}2_)] (t2_+p2_x2₎2_(t2_+x2₎2 ,

d%p,q;t(x)

dx =

2xt4[16p2(p2q−1)u2+ 8p2(q−1)u+ (4q−p2−3p2q)] (t2_+p2_x2₎2_(t2_{+ 4x}2₎2 ,

whereu= x

t

2

>0. Hence, if either



 

 

p2_q−1

≶0

q≤0

or           

p2_{q−1 = 0}

q−1_Q0

q−p2_Q0,

then the derivative dρp,q;t(x)

dx Q0; if either



 

 

p2_q−1

≶0

q(p2−1)[(1 + 3q)p2−4]≤0 or           

p2_{q−1 = 0}

q−1_Q0

4q−p2_−3p2_q

Q0,

then the derivative d%p,q;t(x)

dx Q0.

Consequently, if eitherq ≤0 or 0 < q= _p12 ≤1 and p4≥1 then

dρp,q;t(x)

dx ≤0

and the function Λp,q(x) is decreasing in x ∈ (0,∞); if q = _p12 ≥ 1 and p

4 _{≤ 1}

then dρp,q;t(x)

dx ≥0 and the function Λp,q(x) is increasing in x∈ (0,∞); if either

p2q <1 andq(p2−1)[(1 + 3q)p2−4]≤0 orp2q= 1 and 0< q ≤1 then d%p,q;t(x)

dx ≤

(p2_{−1)[(1 + 3q)p}2_{−4]≤0 orp}2_{q= 1 andq≥1 then}d%p,q;t(x)

dx ≥0 and the function

Φp,q(x) is increasing in x∈(0,∞). The proof of Theorem 2 is complete.

Proof of Theorem 3. Without loss of generality, assumeb > a >0 in (10). Then it

can be rearranged as

bτ_eb eb₋₁ >

aτ_ea ea₋₁.

Direct calculation gives

d dx

_xτ_ex

ex₋₁

= x

τ−1_ex _τ− x ex₋₁

ex₋₁ .

It is easy to see that the function _exx₋₁ is decreasing in (0,∞), with

lim

x→0+

x

ex₋₁ = 1 and _xlim_→∞ x ex₋₁ = 0.

Hence, the function _exxτ₋ex₁ is increasing (or decreasing, respectively) inx∈(0,∞) if

and only ifτ≥1 (orτ <0, respectively). Inequality (10) follows.

References

[1] Ch.-P. Chen and F. Qi, Completely monotonic functions related to the gamma functions, RGMIA Res. Rep. Coll.8 (2005), no. 2, Art. 3. Available online athttp://rgmia.vu.edu. au/v8n2.html.

[2] Ch.-P. Chen and F. Qi, Logarithmically completely monotonic functions related to gamma function and remainder of Binet’s formula with an application, submitted.

[3] I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, Sixth Edition, Academic Press, New York, 2000.

[4] S. Guo,Complete monotonicity of difference of remainder of Binet’s formula, private com-munication.

[5] J.-Ch. Kuang,Ch´angy`ong B`udˇeng Sh`ı(Applied Inequalities), 3rd ed., Shandong Science and Technology Press, Ji’nan City, Shandong Province, China, 2004. (Chinese)

[6] W. Magnus, F. Oberhettinger and R. P. Soni,Formulas and Theorems for the Special Func-tions of Mathematical Physics, Springer, Berlin-New York, 1966.

[7] F. Qi Certain logarithmicallyN-alternating monotonic functions involving gamma andq -gamma functions, RGMIA Res. Rep. Coll.8(2005), no. 3, Art. 5. Available online athttp: //rgmia.vu.edu.au/v8n3.html.

[9] Zh.-X. Wang and D.-R. Guo,Special Functions, Translated from the Chinese by D.-R. Guo and X.-J. Xia, World Scientific Publishing, Singapore, 1989.

[10] Zh.-X. Wang and D.-R. Guo, T`esh¯u H´ansh`u G`ail`un, The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000. (Chinese)

[11] D. V. Widder,The Laplace Transform, Princeton Univ. Press, Princeton, New Jersey, 1941.

(F. Qi)Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

E-mail address: [email protected], [email protected], [email protected],

[email protected]

URL:http://rgmia.vu.edu.au/qi.html

(D.-W. Niu)School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

E-mail address:[email protected]

(B.-N. Guo)School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China