Journal of Inequalities in Pure and Applied Mathematics

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Journal of Inequalities in Pure and Applied Mathematics

http://jipam.vu.edu.au/

Volume 5, Issue 4, Article 94, 2004

INEQUALITIES INVOLVING BESSEL FUNCTIONS OF THE FIRST KIND

EDWARD NEUMAN DEPARTMENT OFMATHEMATICS

SOUTHERNILLINOISUNIVERSITY

CARBONDALE, IL 62901-4408, USA.

[email protected]

URL:http://www.math.siu.edu/neuman/personal.html

Received 25 September, 2004; accepted 13 October, 2004 Communicated by A. Lupa¸s

ABSTRACT. An inequality involving a functionf_α(x) = Γ(α+ 1)(2/x)^αJ_α(x)(α >−¹₂)is obtained. The lower and upper bounds for this function are also derived.

Key words and phrases: Bessel functions of the first kind, Inequalities, Gegenbauer polynomials.

2000 Mathematics Subject Classification. 33C10, 26D20.

1. INTRODUCTION ANDDEFINITIONS

In this note we deal with the function

(1.1) f_α(x) = Γ(α+ 1)

2 x

α

J_α(x),

x ∈ R, α >−¹₂ andJ_α stands for the Bessel function of the first kind of orderα. It is known (see, e.g., [1, (9.1.69)]) that

f_α(x) =₀F₁

−; α+ 1; −x² 4

=

∞

X

n=0

1 n!(α+ 1)_n

−x² 4

n

,

where(a)_k = Γ(a+k)/Γ(a)(k = 0,1, . . .). It is obvious from the above representation that f_α(−x) = f_α(x) and also that f_α(0) = 1. The function under discussion admits the integral representation

(1.2) f_α(x) =

Z 1

−1

cos(xt)dµ(t)

(see, e.g., [1, (9.1.20)]) wheredµ(t) =µ(t)dtwith (1.3) µ(t) = (1−t²)^α−¹² 2^2αB

α+1

2, α+1 2

ISSN (electronic): 1443-5756

171-04

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2 EDWARDNEUMAN

being the Dirichlet measure on the interval [−1,1] and B(·,·) stands for the beta function.

Clearly (1.4)

Z 1

−1

dµ(t) = 1.

Thusµ(t)is the probability measure on the interval[−1,1].

In [2], R. Askey has shown that the following inequality

(1.5) f_α(x) +f_α(y)≤1 +f_α(z)

holds true for all α ≥ 0 and z² = x² + y². This provides a generalization of Grünbaum’s inequality ([4]) who has established (1.5) forα= 0.

In this note we give a different upper bound for the sum f_α(x) +f_α(y) (see (2.1)). Also, lower and upper bounds for the function in question are derived.

2. MAINRESULTS

Our first result reads as follows.

Theorem 2.1. Letx, y ∈R. Ifα >−¹₂, then

(2.1) [f_α(x) +f_α(y)]² ≤[1 +f_α(x+y)][1 +f_α(x−y)].

Proof. Using (1.2), some elementary trigonometric identities, Cauchy-Schwarz inequality for integrals, and (1.4) we obtain

|f_α(x) +f_α(y)| ≤ Z 1

−1

|cos(xt) + cos(yt)|dµ(t)

= 2 Z 1

−1

cos(x+y)t

2 cos(x−y)t 2

dµ(t)

≤2 Z 1

−1

cos²(x+y)t 2 dµ(t)

¹2 Z 1

−1

cos² (x−y)t 2 dµ(t)

¹2

= 2 1

2 Z 1

−1

(1 + cos(x+y)t)dµ(t) ¹₂

× 1

2 Z 1

−1

(1 + cos(x−y)t)dµ(t) ¹2

= [1 +f_α(x+y)]¹²[1 +f_α(x−y)]¹².

Hence, the assertion follows.

Whenx=y, inequality (2.1) simplifies to2f_α²(x)≤1 +fα(2x)which bears resemblance of the double-angle formula for the cosine function2 cos²x= 1 + cos 2x.

Our next goal is to establish computable lower and upper bounds for the function f_α. We recall some well-known facts about Gegenbauer polynomials C_k^α (α > −¹₂, k ∈ N) and the Gauss-Gegenbauer quadrature formulas. They are orthogonal on the interval [−1,1] with the weight functionw(t) = (1−t²)^α−¹². The explicit formula forC_k^αis

C_k^α(t) =

[k/2]

X

m=0

(−1)^m Γ(α+k−m)

Γ(α)m!(k−2m)!(2t)^k−2m

J. Inequal. Pure and Appl. Math., 5(4) Art. 94, 2004 http://jipam.vu.edu.au/

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INEQUALITIESINVOLVINGBESSELFUNCTIONS OF THEFIRSTKIND 3

(see, e.g., [1, (22.3.4)]). In particular,

(2.2) C₂^α(t) = 2α(α+ 1)t²−α, C₃^α(t) = 2

3α(α+ 1)[2(α+ 2)t³−3t].

The classical Gauss-Gegenbauer quadrature formula with the remainder is [3]

(2.3)

Z 1

−1

(1−t²)^α−¹²g(t)dt=

k

X

i=1

w_ig(t_i) +γ_kg^(2k)(η),

whereg ∈C^2k([−1,1]),γ_kis a positive number and does not depend ong, andηis an interme- diate point in the interval(−1,1). Recall that the nodest_i(1 ≤i ≤n) are the roots ofC_k^αand the weightsw_i are given explicitly by [5, (15.3.2)]

(2.4) w_i =π2^2−2α Γ(2α+k)

k![Γ(α)]² · 1

(1−t²_i)[(C_k^α)⁰(ti)]² (1≤i≤k).

We are in a position to prove the following.

Theorem 2.2. Letα >−¹₂. If|x| ≤ ^π₂ , then

cos x

p2(α+ 1)

!

≤fα(x) (2.5)

≤ 1 3(α+ 1)

"

2α+ 1 + (α+ 2) cos

s 3 2(α+ 2)x

!#

. Equalities hold in (2.5) ifx= 0.

Proof. In order to establish the lower bound in (2.5) we use the Gauss-Gegenbauer quadrature formula (2.3) withg(t) = cos(xt)andk = 2. Since g⁽⁴⁾(t) = x⁴cos(xt) ≥ 0fort ∈ [−1,1]

and|x| ≤ ^π₂ ,

(2.6) w₁g(t₁) +w₂g(t₂)≤ Z 1

−1

(1−t²)^α−¹² cos(xt)dt.

Making use of (2.2) and (2.4) we obtain

−t₁ =t₂ = 1 p2(α+ 1)

andw₁ =w₂ = ¹₂2^2αB(α+ ¹₂, α+¹₂). This in conjunction with (2.6) gives 2^2αB

α+1

2, α+ 1 2

cos x

p2(α+ 1)

!

≤ Z 1

−1

(1−t²)^α−¹² cos(xt)dt.

Application of (1.3) together with the use of (1.2) gives the asserted result. In order to derive the upper bound in (2.5) we use again (2.3). Letting g(t) = cos(xt) and k = 3 one has g⁽⁶⁾(t) = −x⁶cos(xt)≤0for|t| ≤1and|x| ≤ ^π₂ . Hence

(2.7)

Z 1

−1

(1−t²)^α−¹² cos(xt)dt ≤w₁g(t₁) +w₂g(t₂) +w₃g(t₃).

It follows from (2.2) and (2.4) that

−t₁ =t₃ = s

3

2(α+ 2), t₂ = 0

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4 EDWARDNEUMAN

and

w₁ =w₃ = 2^2αB

α+ 1

2, α+1 2

α+ 2 6(α+ 1), w₂ = 2^2αB

α+1

2, α+ 1 2

2α+ 1 3(α+ 1).

This in conjunction with (2.7), (1.3), and (1.2) gives the desired result. The proof is complete.

Sharper lower and upper bounds forf_αcan be obtained using higher order quadrature formulas (2.3) with even and odd numbers of knots, respectively.

REFERENCES

[1] M. ABRAMOWITZANDI.A. STEGUN (Eds.), Handbook of Mathematical Functions with Formu- las, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1965.

[2] R. ASKEY, Grünbaum’s inequality for Bessel functions, J. Math. Anal. Appl., 41 (1973), 122–124.

[3] K.E. ATKINSON, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989.

[4] F. GRÜNBAUM, A property of Legendre polynomials, Proc. Nat. Acad. Sci., USA, 67 (1970), 959–

960.

[5] G. SZEGÖ, Orthogonal polynomials, in Colloquium Publications, Vol. 23, 4th ed., American Math- ematical Society, Providence, RI, 1975.