R E S E A R C H
Open Access
Wirtinger inequality using Bessel
functions
Tatjana Z. Mirkovi´c
1**Correspondence:
1College of Applied Professional
Studies, Vranje, Serbia
Abstract
This paper presents of some new Wirtinger-type integral inequalities by using Bessel functions. We establish one weighted Wirtinger inequality.
MSC: 26D20; 33C10
Keywords: Wirtinger’s inequality; Bessel functions
1 Introduction
The Wirtinger inequality plays a very important role in the theory of approximation, the theory of Sobolev’s spaces, the theory of function of several variables and functional anal-ysis. In 1916. Wirtinger established an integral inequality.
Theorem 1.1(Wirtinger inequality) Let f:R→Rbe a continuous periodic function with period2πand let f∈L2.Then,if02πf(x)dx= 0the following inequality holds:
2π
0
f2(x)dx≤
2π
0
f2(x)dx,
with equality if and only if f(x) =acosx+bsinx,where a and b are constants.
Theorem 1.2 Let f(x)be a smooth function with period2π.Then,for all real t,
2π
0
f(x) –f(x+t)2dx≤4sin2 t
2 2π
0
f2(x)dx. (1)
Equality is attained if and only if f(x) =acosx+bsinx+c,where a,b,c are real constants
(for t= 0equality holds always).
In [1], Beesack obtained the following generalization of the Wirtinger inequality: Ifk> 1,
f(x)∈C1([0,π]),f(0) = 0, then
π
0
f(x)2kdx≥ 2k– 1
(ksin2πk)2k
π
0
f2k(x)dx, k≥1. (2)
In [2], Hall proved the following theorem:
Theorem 1.3 Suppose that k∈N,f(x)∈C2[0,π]and f(0) =f(π) = 0.Let H(u)be an even
function,increasing and strictly convex on R+,and such that H(0) =H(0) = 0;moreover,
uH(u)→0as u→0.Then we have
π
0
Hf(x)/f(x)f2k(x)≥(2k– 1)λ
π
0
f2k(x)dx, λ=λ(k,H), (3)
whereλ=λ(k,H)is determined by the equation
∞
0
G(u)
G(u) + (2k– 1)λ
du
u =kπ, G(u) :=uH
(u) –H(u). (4)
For each non-negative constantp, the associated Bessel equation is
x2d
2y
dx2 +x
dy dx +
x2–p2y= 0. (5)
Since Bessel’s differential equation is a second-order equation, there must be two linearly independent solutions, which are called Bessel functions. These functions play important roles in many areas of applied mathematics (see [3,4]). Typically the general solution is given as
y=a1Jp(x) +a2Yp(x),
wherea1anda2are arbitrary constants.
Special functionsJp(x) are Bessel functions of the first kind, which are finite atx= 0 for all real values ofp, andYp(x) are Bessel functions of the second kind, which are singular
atx= 0.
The Bessel function of the first kind of orderpcan be determined using an infinite power series expansion as follows:Jp(x) =+k=0∞k!(–1)(k+pk+1)(2x)2k+p. Since(k+ 1) =k!, it follows that
Jp(x) =
+∞
k=0
(–1)k
k!(k+p)!
x
2 2k+p
. (6)
For integer orderp, functionsJp andJ–p are not linearly independent,J–p= (–1)pJp. In
contrast, for non-integer orders,JpandJ–pare linearly independent.
The most important Bessel functions areJ0(x) andJ1(x). Forp= –12andp=12, this
func-tions expansion as follows:
J–1/2(x) =
2
πxcosx, (7)
J1/2(x) =
2
πxsinx. (8)
2 Main results
Theorem 2.1 Let f∈L2k on[0,π],with f(0) =f(π) = 0.Then the following inequality holds:
π
0
f2k(x)dx≤ 1
2k– 1
π 2
2k π
2
0
J0
π 2kcost
cost dt
2k π
0
Proof SinceJn(z) = (2z)n+∞
Using the integration by parts formula on the integralI2r+1=
π
The above equality becomes
π
Proof Starting with the right side of (10), we obtain
=
Proof From the equationJ2
= 2t
+∞
n=0
(–1)n
n!(n+ 1)!
t
2
2n+1 π
2
0
cos2n+1x dx
2π
0
f2(x)dx
= 2t
+∞
n=0
(–1)n n!(n+ 1)!
t2n+1
2·22n
22nn!n!
(2n+ 1)! 2π
0
f2(x)dx
=t
+∞
n=0
(–1)nn!t2n+1
(n+ 1)n!(2n+ 1)! 2π
0
f2(x)dx
=t
+∞
n=0
(–1)nt2n+1
(n+ 1)(2n+ 1)! 2π
0
f2(x)dx
=t
t
1!–
t3
2·3!+
t5
3·5!–
t7
4·7!+· · ·
2π
0
f2(x)dx
=t
2t2
2!t –
4t4
2·4!t+
6t6
3·6!t–
8t8
4·8!t+
10t10
5·10!t–· · ·
2π
0
f2(x)dx
= 2t
t2
2!t– t4
4!t+ t6
6!t– t8
8!t+ t10
10!t–· · ·
2π
0
f2(x)dx
= 2t
+∞
n=1
(–1)n+1 t
2n t(2n)!
2π
0
f2(x)dx
=t
1 –
+∞
n=0
(–1)n t
2n
(2n)!
2π
0
f2(x)dx
= 2(1 –cost) 2π
0
f2(x)dx= 4sin2t
2 2π
0
f2(x)dx.
Equation (1) implies the desired inequality (11).
Acknowledgements
The author would like to thank the anonymous referees for their constructive comments. The author states that no funding source or sponsor has participated in the realization of this work.
Competing interests
The author declares that he has no competing interests.
Authors’ contributions
The work as a whole is a contribution of the author. All authors read and approved the final manuscript.
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Received: 21 November 2017 Accepted: 7 May 2018
References
1. Beesack, P.: Hardy’s inequality and its extension. Pac. J. Math.11, 39–61 (1961)
2. Hall, R.: Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function. J. Number Theory97, 397–409 (2002)
