International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 123-131
http://www.etamaths.com

ABSOLUTE MONOTONICITY OF FUNCTIONS RELATED TO

ESTIMATES OF FIRST EIGENVALUE OF LAPLACE

OPERATOR ON RIEMANNIAN MANIFOLDS

FENG QI1,2,∗ AND MIAO-MIAO ZHENG1

Abstract. The authors find the absolute monotonicity and complete mono-

tonicity of some functions involving trigonometric functions and related to
estimates the lower bounds of the first eigenvalue of Laplace operator on Rie-

mannian manifolds.

1. Background and mail results

In [38, 39], J. Q. Zhong and H. C. Yang obtained that the first eigenvalue λ1 of
Laplace operator on a compact Riemannian monifold M with non-negative Ricci
curvature satisfies

(1.1) λ1 ≥
π2

d2
,

where d denotes the diameter of M. The inequality (1.1) improves corresponding
results in [11, 12]. For proving the inequality (1.1), the authors introduced in [38,
Lemma 4] and [39, Lemma 4] the function

(1.2) ψ(θ) =




4
π

(θ + sin θ cos θ) − 2 sin θ
cos2 θ

, θ ∈
(
−
π

2
,
π

2

)
±1, θ = ±

π

2

and obtained that the function y(θ) = ψ(θ) satisfies ψ′(θ) ≥ 0, the differential
equation

(1.3) y(θ) − sin θ + y′ sin θ cos θ −
1

2
y′′(θ) cos2 θ = 0,

and the inequality

(1.4) 0 ≤ ψ′(θ) cos θ ≤ 2
(

4

π
− 1
)

on
[
−π

2
, π
2

]
. These results were ever employed in [37, p. 348, Lemma 4]. In [8,

p. 3], it was pointed out that ψ′(θ) ≥ 0 and |ψ(θ)| ≤ 1 on
[
−π

2
, π
2

]
. For more

information, please refer to [18, Lemma 4], [23, Lemma 1 and Proposition 7], [26,
Lemma 4], and [27, Proposition 3].

2010 Mathematics Subject Classification. Primary 26A48, 33B10; Secondary 11B68, 34A05,

44A10, 58C40.
Key words and phrases. Absolutely monotonic function; completely monotonic function; trigono-
metric function; Laplace operator; eigenvalue.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

123



124 F. QI AND M.-M. ZHENG

Let M be a m-dimensional compact Riemannian manifold with boundary ∂M,
the inner radius of M be ρ, the Ricci curvature of M be not less than −R, and the
mean curvature of ∂M be not less than −H0, where R and H0 are positive scalars.
Theorem 3 in [35, p. 331] reads that the first eigenvalue µ1 of M under Dirichlet
boundary condition satisfies

(1.5) µ1 ≥
π2

4ρ2
−

1

2
R−

2

3
(m− 1)H0

π

ρ
.

For proving the inequality (1.5), the author considered the functions

(1.6) p(θ) =




2

cos2 θ

∫ π/2
θ

t cos2 t d t, θ ∈
(
−
π

2
,
π

2

)
0, θ = ±

π

2

and

(1.7) φ(θ) =




1

cos2 θ

∫ π/2
θ

cos2 t d t, θ ∈
(
−
π

2
,
π

2

)
0, θ =

π

2

and obtained in [35, pp. 338–340] that p′(θ) ≤ 0 and φ′(θ) ≤ 0 on
[
0, π

2

]
, that

(1.8)

∫ π/2
0

p(θ) d θ =
π

2
,

∫ π/2
0

φ(θ) d θ =
1

2
,

and that the function Z(θ) = 1 + αp(θ) + βφ(θ) satisfies Z
(
π
2

)
= 1 and

(1.9) Z(θ) = 1 + α cos2 θ −Z′(θ) cos θ sin θ +
1

2
Z′′(θ) cos2 θ, θ ∈

[
0,
π

2

]
.

In [18, Propositions 11 and 12], [23, Propositions 2, 3, and 5], and [27, Propositions 1
and 2], it was obtained that the function Y (θ) = p(θ) satisfies the differential
equation

(1.10) Y ′′(θ) cos2 θ − 2Y ′(θ) sin θ cos θ − 2Y (θ) + 2 cos2 θ = 0

and the inequalities

(1.11) p′(θ) sin θ ≤ 0, |p′(θ) cos θ| ≤
8

3
, p(θ) ≤

π2

8
−

1

2

for θ ∈
[
− π

2
, π
2

]
. In [9], it was established that the function

p(θ)
cos θ

is increasing on[
0, π

2

]
, that the function p′(θ) is decreasing, and that

(1.12)
π2

8
−

1

2
≤
p(θ)

cos θ
≤
π

3
, p(θ) ≤

1

5
+ cos2 θ

on
[
−π

2
, π
2

]
. See also [34, p. 699].

In [13, Theorem 1.1], it was obtained that the first positive eigenvalue λ of
Laplace operator on a closed n-dimensional Riemannian manifold with Ricci cur-
vature Ric(M) ≥ (n− 1)K > 0 has the lower bound

(1.13) λ ≥
1

2
(n− 1)K +

π2

4r2
,

where r is the largest interior radius of the nodal domains of eigenfunctions of the
eigenvalue λ. For verifying the above conclusion, the author considered in [13,



ABSOLUTE AND COMPLETE MONOTONICITY OF SOME FUNCTIONS 125

Lemma 3.1] the function ξ(t) = −2p(t) and obtained some conclusions on ξ(t),
which may be reformulated as follows.

(1) For t ∈
(
−π

2
, π
2

)
, the function ξ(t) meets

1

2
ξ′′(t) cos2 t− ξ′(t) cos t sin t− ξ(t) = 2 cos2 t,(1.14)

ξ′(t) cos t− 2ξ(t) sin t = 4t cos t;(1.15)

(2) For t ∈
[
−π

2
, π
2

]
,

(1.16) 1 −
π2

4
= ξ(0) ≤ ξ(t) ≤ ξ

(
±
π

2

)
= 0 and

∫ π/2
0

ξ(t) d t = −
π

2
;

(3) The derivative ξ′(t) is increasing on
[
−π

2
, π
2

]
,

(1.17) ξ′
(
±
π

2

)
= ±

2π

3
, and ξ′(t)


< 0, t ∈

(
−
π

2
, 0
)
,

> 0, t ∈
(

0,
π

2

)
;

(4) For t ∈
[
−π

2
, π
2

]
,

(1.18) 2

(
3 −

π2

4

)
≤
ξ′(t)

t
≤

4

3
,

and for t ∈
(
0, π

2

)
,

(1.19)

[
ξ′(t)

t

]′
> 0;

(5) For t ∈
[
−π

2
, π
2

]
,

(1.20) ξ′′(t) > 0, ξ′′
(
±
π

2

)
= 2, and ξ′′(0) = 2

(
3 −

π2

4

)
;

(6) For t ∈
[
−π

2
, π
2

]
,

(1.21) ξ′′′
(
π

2

)
=

8π

15
, ξ′′′(t)


< 0, t ∈

(
−
π

2
, 0
)
,

> 0, t ∈
(

0,
π

2

)
.

By calculus, it is easy to see that

ψ(θ) =




2

π
[2θ + sin(2θ) −π sin θ] sec2 θ, θ ∈

(
−
π

2
,
π

2

)
,

±1, θ = ±
π

2
,

(1.22)

p(θ) =



(
π2

8
−

1

2
θ2
)

sec2 θ −θ tan θ −
1

2
, θ ∈

(
−
π

2
,
π

2

)
,

0, θ = ±
π

2
,

(1.23)

and

(1.24) φ(θ) =


−

1

4
[2θ + sin(2θ) −π] sec2 θ, θ ∈

(
−
π

2
,
π

2

)
,

0, θ =
π

2
.

See also [28, pp. 6–7]. For more information, please read [4, 5, 10, 17, 24, 25, 30, 36]
and closely related references therein.



126 F. QI AND M.-M. ZHENG

A function f is said to be completely monotonic on an interval I if it has deriva-
tives of all orders on I and satisfies

(1.25) 0 ≤ (−1)k−1f(k−1)(x) < ∞
for x ∈ I and k ∈ N, where f(0)(x) means f(x) and N stands for the set of all
positive integers. See [14, Chapter XIII], [31, Chapter 1], or [33, Chapter IV].
The class of completely monotonic functions may be characterized by the famous
Hausdorff-Bernstein-Widder theorem [33, p. 161, Theorem 12b]: A necessary and
sufficient condition that f(x) should be completely monotonic for 0 < x < ∞ is
that

(1.26) f(x) =

∫ ∞
0

e−xt d α(t),

where α(t) is non-decreasing and the above integral converges for 0 < x < ∞.
Recall from [14, Chapter XIII] or [33, Chapter IV] that a function f is said to

be absolutely monotonic on an interval I if it has derivatives of all orders and

(1.27) f(k−1)(t) ≥ 0
for t ∈ I and k ∈ N. Theorem 12c in [33, p. 162] states that a necessary and
sufficient condition that f(x) should be absolutely monotonic in −∞ < x < 0 is
that

(1.28) f(x) =

∫ ∞
0

ext d α(t),

where α(t) is non-decreasing and the integral converges for −∞ < x < 0.
For more information on completely and absolutely monotonic functions, please

refer to [6, 7, 19, 20, 21, 22, 29] and closely related references therein.
In this paper, we will prove the following absolute and complete monotonicity of

functions related to estimates of first eigenvalue of Laplace operator on Riemannian
manifolds.

Theorem 1.1. The functions ψ(θ) and 8
π
−2−ψ′(θ) cos θ are absolutely monotonic

on
(
0, π

2

)
.

Theorem 1.2. The function −p′(θ) is absolutely monotonic on
(
0, π

2

)
.

Theorem 1.3. The function φ(θ) is completely monotonic on
(
−π

2
, π
2

)
.

2. Proofs of Theroems 1.1 to 1.3

Proof of Theorem 1.1. The function ψ(θ) may be rewritten as

ψ(θ) =
4

π
tan θ +

4

π
θ sec2 θ − 2 tan θ sec θ

=
4

π
tan θ +

4

π
θ(tan θ)′ − 2(sec θ)′

=
4

π
(θ tan θ)′ − 2(sec θ)′.

It is well known [1, p. 75, 4.3.67 and 4.3.69] that the tangent tan x and the secant
sec x can be expanded into power series

(2.1) tan z =

∞∑
n=1

(−1)n−122n(22n − 1)B2n
z2n−1

(2n)!



ABSOLUTE AND COMPLETE MONOTONICITY OF SOME FUNCTIONS 127

and

(2.2) sec z =

∞∑
n=0

(−1)nE2n
z2n

(2n)!

for |z| < π
2

, where Bn for n ≥ 0 are Bernoulli numbers which may be defined by
the power series expansion

(2.3)
z

ez − 1
=

∞∑
n=0

Bn
zn

n!
= 1 −

z

2
+

∞∑
k=1

B2k
z2k

(2k)!
, |z| < 2π

and En for n ≥ 0 stand for Euler numbers which are integers and may be defined
by

(2.4)
2ez

e2z + 1
=

∞∑
n=0

En
n!
zn =

∞∑
n=0

E2n
z2n

(2n)!
, |z| < π,

see [1, p. 804, 23.1.1 and 23.1.2] or [32, p. 3, (1.1) and p. 15]. Consequently,

ψ(θ) =
4

π

∞∑
n=1

(−1)n−122n(22n − 1)B2n
(2n− 1)!

θ2n−1 − 2
∞∑
n=1

(−1)n

(2n− 1)!
E2nθ

2n−1

= 2

∞∑
n=1

1

(2n− 1)!
(−1)n−1

[
2

π
22n(22n − 1)B2n + E2n

]
θ2n−1.

In [1, p. 805, 23.1.15], it was listed that

(2.5)
4n+1(2n)!

π2n+1
> (−1)nE2n >

1

1 + 3−1−2n
4n+1(2n)!

π2n+1
, n ∈{0}∪N.

In [2], it was obtained that the double inequality

(2.6)
2(2n)!

(2π)2n
1

1 − 2α−2n
≤ (−1)n−1B2n ≤

2(2n)!

(2π)2n
1

1 − 2β−2n

holds for n ∈ N if and only if α ≤ 0 and β ≥ 2 + ln(1−6/π
2)

ln 2
= 0.649 . . . . As a result,

(−1)n−1
[

2

π
22n(22n − 1)B2n + E2n

]
>

2

π
22n(22n − 1)

2(2n)!

(2π)2n
1

1 − 2−2n
−

4n+1(2n)!

π2n+1

= 0.

This implies that the function ψ(θ) is absolutely monotonic on
[
0, π

2

]
.

Direct calculation and utilization of (2.1) and (2.2) yield

8

π
− 2 −ψ′(θ) cos θ = 4 sec2 θ −

8

π
(θ tan θ sec θ + sec θ) − 4 +

8

π

= 4(tan θ)′ −
8

π
[θ(sec θ)′ + sec θ] − 4 +

8

π

= 4

∞∑
n=1

(−1)n−1(2n− 1)22n(22n − 1)B2n
(2n)!

θ2n−2

−
8

π

[
∞∑
n=1

(−1)n

(2n− 1)!
E2nθ

2n +

∞∑
n=0

(−1)n

(2n)!
E2nθ

2n

]
− 4 +

8

π



128 F. QI AND M.-M. ZHENG

= 4

∞∑
n=1

2n + 1

(2n)!

[
22(n+1)(22(n+1) − 1)(−1)nB2(n+1)

2(n + 1)(2n + 1)
−

2

π
(−1)nE2n

]
θ2n.

Employing the inequalities (2.5) and (2.6) reveals

22(n+1)(22(n+1) − 1)(−1)nB2(n+1)
2(n + 1)(2n + 1)

−
2

π
(−1)nE2n

>
22(n+1)(22(n+1) − 1)

2(n + 1)(2n + 1)

2(2n + 2)!

(2π)2n+2
1

1 − 2−2n−2
−

2

π

4n+1(2n)!

π2n+1

= 0.

This means that the function 8
π
−2−ψ′(θ) cos θ is absolutely monotonic on

[
0, π

2

]
.

The proof of Theorem 1.1 is complete. �

Proof of Theorem 1.2. Straightforward computation and utilization of (2.1) yield

−p′(θ) =
1

2
[θ2(tan θ)′]′ + (θ tan θ)′ −

π2

8
(tan θ)′′

=
1

2

∞∑
n=1

(−1)n−122n(22n − 1)B2n
θ2n−1

(2n− 2)!

+

∞∑
n=1

(−1)n−122n(22n − 1)B2n
θ2n−1

(2n− 1)!

−
π2

8

∞∑
n=2

(−1)n−1(2n− 1)(2n− 2)22n(22n − 1)B2n
θ2n−3

(2n)!

=

∞∑
n=1

22n−1
[
(2n + 1)(22n − 1)(−1)n−1B2n

−
π2

2(n + 1)
(22n+2 − 1)(−1)nB2n+2

]
θ2n−1

(2n− 1)!
.

Accordingly, to prove the absolute monotonicity of the function −p′(θ), it suffices
to show the inequality

(2.7)
|B2n+2|
|B2n|

=
(−1)nB2n+2
(−1)n−1B2n

≤
22n − 1

22n+2 − 1
2(n + 1)(2n + 1)

π2
, n ∈ N.

In [32, p. 5, (1.14)], it was listed that

(2.8) B2n =
(−1)n+12(2n)!

(2π)2n

∞∑
m=1

1

m2n
, n ∈ N.

Then

(2.9)
(−1)nB2n+2
(−1)n−1B2n

=
2(n + 1)(2n + 1)

π2
1

4

∑∞
m=1

1
m2n+2∑∞

m=1
1

m2n

, n ∈ N.

Hence, to prove the inequality (2.7), it is sufficient to verify

1

4

∑∞
m=1

1
m2n+2∑∞

m=1
1

m2n

≤
22n − 1

22n+2 − 1
, n ∈ N,



ABSOLUTE AND COMPLETE MONOTONICITY OF SOME FUNCTIONS 129

which may be rearranged as(
1 −

1

22n+2

) ∞∑
m=1

1

m2n+2
≤
(

1 −
1

22n

) ∞∑
m=1

1

m2n
, n ∈ N.

This inequality is a special case of Lemma 2.1 in [3, 40], which may be slightly
modified as follows: the sequence(

1 −
1

2n

) ∞∑
m=1

1

mn
=

∞∑
m=1

1

mn
−
∞∑
m=1

1

(2m)n
=

∞∑
m=1

1

(2m− 1)n
, n ≥ 2

is decreasing in n. The proof of Theorem 1.2 is complete. �

Remark 2.1. For more information on the inequality (2.7), please refer to [15, 16]
and closely related references therein.

Proof of Theorem 1.3. By definition, it is easy to see that a function f(x) is com-
pletely monotonic in (a,b) if and only if f(−x) is absolutely monotonic in (−b,−a).
See [33, p. 145, Definition 2c]. Hence, it is sufficient to prove that the function
φ(−θ) is absolutely monotonic on

(
−π

2
, π
2

)
.

It is easy to see that

φ(−θ) =
1

4
[2θ + sin(2θ) + π] sec2 θ

=
1

4
[2θ(tan θ)′ + 2 tan θ + π(tan θ)′]

=
1

4
[2(θ tan θ)′ + π(tan θ)′].

Utilization of (2.1) leads to

φ(−θ) =
1

4

[
2

∞∑
n=1

22n(22n − 1)(−1)n−1B2n
θ2n−1

(2n− 1)!

+ π

∞∑
n=1

(2n− 1)22n(22n − 1)(−1)n−1B2n
θ2n−2

(2n)!

]
.

Since (−1)n−1B2n > 0 for all n ∈ N, all the coefficients of θk for k ≥ 0 in the power
series expansion of φ(−θ) are positive. Therefore, the function φ(−θ) is absolutely
monotonic on

(
−π

2
, π
2

)
. The proof of Theorem 1.3 is complete. �

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1Department of Mathematics, College of Science, Tianjin Polytechnic University,
Tianjin City, 300387, China

2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao
City, Inner Mongolia Autonomous Region, 028043, China

∗Corresponding author