Table Of ContentVAN DER CORPUT INEQUALITIES FOR BESSEL FUNCTIONS
A´RPA´DBARICZ,ANDREALAFORGIA,ANDTIBORK.POGA´NY
Abstract. In this note we offer some log-concavity properties of certain functions related to Bessel
functionsofthefirstkindandmodifiedBesselfunctionsofthefirstandsecondkind,bysolvingpartially
a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the
firstkindandtheirderivatives. Moreover,wegiveanapplicationofthementionedresultsbyextending
two inequalities of van der Corput to Bessel and modified Bessel functions of the first kind. Similar
4
inequalities are proved alsofor modifiedBessel functions of the second kind, as well as for log-concave
1
probabilitydensityfunctions.
0
2
1. Introduction
l
u Forν unrestrictedrealnumberthemodifiedBesselfunctionofthefirstkindoforderν,denotedusually
J
by Iν, is the particular solution of the differential equation [16, p. 77]
8
(1.1) x2y′′(x)+xy′(x) (x2+ν2)y(x)=0.
−
]
A Modified Bessel equation (1.1), which is of frequent occurrence in problems of mathematical physics,
differs from the well-known Bessel’s equation only in the coefficient of y. In fact, the modified Bessel
C
function I has the infinite series expansion [16, p. 77]
. ν
h
at (1.2) Iν(x)= x2 2n+ν = x2 ν 0F1 ν+1,x2
m n≥0n!Γ(cid:0)(ν(cid:1)+n+1) Γ((cid:0)ν+(cid:1) 1) · (cid:18) 4 (cid:19)
X
[ for all ν 6= −1,−2,... and x ∈ R, where Γ denotes the Euler gamma function and x 7→0F1 ν+1,x42
2 is the hypergeometric function defined below. We note that the function ν :R [1, ), defi(cid:16)ned by (cid:17)
v I → ∞
4 x2 1 n
6 (1.3) ν(x)=0F1 ν+1, =2νΓ(ν+1)x−νIν(x)= 4 x2n,
I 4 (ν+1) n! ·
7 (cid:18) (cid:19) n≥0 (cid:0) (cid:1) n
X
7
for ν = 1, 2,..., (where for a = 0 the notation (a) = a(a+1)(a+2)...(a+n 1), (a) = 1 is the
. 6 − − 6 n − 0
1 well-knownPochhammer(or Appell) symbol)is strictly log-convex,when ν > 1, see [13]. Observealso
0 −2
that in particular we obtain
4
π
:1 I−21(x)= 2 ·x12I−21(x)=coshx,
v r
Xi xandthu(sxa)ctisuasltlryicItνlyislologg-c-oconnvvexexfoforraallllνν>≥−121.aMndorxeoverR,,resceeen[t6l,ypit.w2a7s1c].onWjeectnuorteedtthhaatttinhe[8fu],nucstiionng
ν
7→ I − ∈
r Ho¨lder’s inequality for integrals, the authors claimed that all derivatives of the function x (x) are
a strictly log-convex for ν > 1 and x R. In [10] it was proved that in fact just the derivat7→iveIsνof even
−2 ∈
order of the function will be strictly log-convex. Recall that (x) = coshx, thus in this case it
Iν I−21
is clear that for all k 0,1,... the function x (2k)(x) = coshx is strictly log-convex on R and
∈ { } 7→ I−12
(2k+1)
x (x)=sinhx is strictly log-concaveon(0, ). This suggestedthe followingconjecture (see [3,
7→I−21 ∞
p. 101], [4, p. 187]): the function x (2k)(x) is strictly log-convex for all k 0,1,... , ν > 1 and
ν
7→ I ∈ { } −
x R, while the function x (2k+1)(x) is strictly log-concavefor all k 0,1,... , ν > 1 and x>0.
ν
∈ 7→I ∈{ } −
Inthispaperouraimistogivethefull(negative)answertotheaboveconjecturesforthecasek =0.We
willshowthat(seethe ConcludingRemark2)there existsaz j (wherej is thefirstpositivezero
ν ν,1 ν,1
≥
of the Bessel function J ) such that the function x (x) is strictly log-convex on (0,z ) and strictly
ν ν ν
7→ I
log-concaveon(z , ).Moreover,wewillprovethat(seeTheorem1anditsproof)ifν 1, 1 ,then
ν ∞ ∈ − −2
(cid:0) (cid:3)
2010 Mathematics Subject Classification. 33C10,26A51,26D07, 39B72.
Key words and phrases. Bessel functions of the first kind, modified Bessel functions of the first and second kind, log-
convexity, log-concavity, vanderCorputinequality, probabilitydensityfunctions,trigonometricandhyperbolicfunctions.
1
2 A´.BARICZ,A.LAFORGIA,ANDT.K.POGA´NY
the function x ′(x) is strictly log-concaveon (0, ), while if ν > 1, then there exists an x , which
7→Iν ∞ −2 ν
satisfies x > 2(ν+3), such that x ′(x) is strictly log-concave on (0,x ) and strictly log-convex
ν 7→ Iν ν
on (x , ). The paper is organized as follows: in section 2 we study the log-concavity of the function
ν
∞ p
x ′(x)andweapplythemainresulttoextendaninequalityofvanderCorputconcerninghyperbolic
7→Iν
functions to modified Bessel functions of the first kind. Moreover,we give the counterparts of the above
mentioned results for trigonometric functions and Bessel functions of the first kind. Motivated by these
results we present also the counterpart of these results for modified Bessel functions of the second kind.
Section3containstheproofsofthe theorems,andsection4isdevotedforconcludingremarks. Insection
4, among other things, we point out that the function x (x) cannot be strictly log-convex on R for
ν
7→I
all ν > 1, and thus we disprove the conjecture from [6, p. 271] (see also [3, p. 100], [4, p. 187]).
−
2. Inequalities for Bessel and modified Bessel functions
2.1. Modified Bessel functions of the first kind. Our first main result reads as follows.
Theorem 1. The following assertions are true:
(a) If ν 1, 1 , then the function x ′(x) is strictly log-concave on (0, ).
∈ − −2 7→Iν ∞
(b) If ν >(cid:0)−1, then(cid:3)the function x7→Iν′(x) is strictly log-concave on 0, 2(ν+3) .
(c) If ν > −12, then there exists an xν, which satisfies xν > 2(ν(cid:16)+3p), such tha(cid:17)t x 7→ Iν′(x) is
strictly log-concave on (0,x ) and strictly log-convex on (x , ).
ν ν
p∞
The following inequality involving hyperbolic sine and cosine functions of van der Corput [12, p. 270]
is of special interest in this section:
(2.1) cosha coshb a b√sinhasinhb for all a,b 0.
| − |≥| − | ≥
Taking into account the relation (x)=coshx the inequality (2.1) may be written as
I−12
(a) (b) a b ′ (a) ′ (b) for all a,b 0.
|I−12 −I−21 |≥| − | I−12 I−21 ≥
q
The following result is an extension of van der Corput’s inequality (2.1).
Corollary 1. If a,b 0 and ν 1, 1 , then
≥ ∈ − −2
(2.2) (cid:0) (a) (cid:3) (b) a b ′(a) ′(b).
|Iν −Iν |≥| − | Iν Iν
Moreover, the above inequality is valid for all ν > 1 andpa,b 0, 2(ν+3) .
− ∈
(cid:16) p (cid:17)
Since for all n 1,2,... and ν = 1, 2,... one has (ν +1)(ν+2) = (ν+1) , from (1.3) we
∈ { } 6 − − n−1 n
have the following differentiation formula for the function , namely
ν
I
(2.3) 2(ν+1) ′(x)=x (x),
Iν Iν+1
where x R and ν = 1, 2,.... Now taking into account the relations (x)=coshx and
∈ 6 − − I−12
π sinhx
1(x)= x−12I1(x)= ,
I2 2 · 2 x
r
the inequality (2.1) may be written as
(a) (b) a b ab (a) (b),
|I−12 −I−21 |≥| − | I21 I12
q
and from (2.3) it follows that the inequality (2.2) is equivalent to
(2.4) 2(ν+1) (a) (b) a b ab (a) (b).
ν ν ν+1 ν+1
|I −I |≥| − | I I
We also note that in particular we have p
π sinhx coshx
3(x)=3 x−32I3(x)= 3 ,
I2 2 · 2 − x3 − x2
r (cid:18) (cid:19)
thus from (2.4) for ν = 1 we obtain the following interesting inequality
2
bsinha asinhb a b (acosha sinha)(bcoshb sinhb) for all a,b 0,√7 .
| − |≥| − | − − ∈
p (cid:16) (cid:17)
VAN DER CORPUT INEQUALITIES FOR BESSEL FUNCTIONS 3
2.2. Bessel functions of the first kind. Now, let us consider the analogueof , that is, the function
ν
: R ( ,1], defined by (x) = 2νΓ(ν +1)x−νJ (x), where J stands foIr the Bessel function of
ν ν ν ν
J → −∞ J
the first kind. Let us denote by j the nth positive zero of the Bessel function of the first kind J . The
ν,n ν
analogue of Theorem 1 is the following result.
Theorem 2. If ν > 1 then the function x ′(x) is strictly log-concave on (0,j ).
− 7→−Jν ν+1,1
The following inequality involving sine and cosine functions of van der Corput [12, p. 237] is the
analogue of (2.1):
(2.5) cosa cosb a b√sinasinb for all a,b [0,π].
| − |≥| − | ∈
Taking into account the relation (x)=cosx the inequality (2.5) may be written as
J−12
(a) (b) a b ′ (a) ′ (b) for all a,b [0,π].
|J−12 −J−12 |≥| − | J−21 J−21 ∈
q
The following result is an extension of van der Corput’s inequality (2.5).
Corollary 2. If a,b [0,j ] and ν > 1, then
ν+1,1
∈ −
(2.6) (a) (b) a b ′(a) ′(b).
|Jν −Jν |≥| − | Jν Jν
Observe that satisfies the following differentiation fpormula
ν
J
(2.7) 2(ν+1) ′(x)= x (x),
Jν − Jν+1
where x R and ν = 1, 2,.... Now taking into account the relations
∈ 6 − −
π π sinx
J−12(x)= 2 ·x12J−12(x)=cosx and J21(x)= 2 ·x−21J12(x)= x ,
r r
the inequality (2.5) may be written as
(a) (b) a b ab (a) (b),
|J−21 −J−21 |≥| − | J21 J12
q
and from (2.7) it follows that the inequality (2.6) is equivalent to
(2.8) 2(ν+1) (a) (b) a b ab (a) (b).
ν ν ν+1 ν+1
|J −J |≥| − | J J
We also note that in particular we have p
π sinx cosx
3(x)=3 x−32J3(x)=3 ,
J2 2 · 2 x3 − x2
r (cid:18) (cid:19)
thus from (2.8) for ν = 1 we obtain the following inequality
2
bsina asinb a b (acosa sina)(bcosb sinb) for all a,b [0,j ],
| − |≥| − | − − ∈ 32,1
where j3,1 =4.493409457... is the firpst positive root of the equation sinx=xcosx.
2
2.3. Modified Bessel functions of the second kind. Finally, for ν > 0 let us consider the function
: (0, ) (0, ), defined by (x) = 2ν−1Γ(ν)xνK (x), where K is the modified Bessel function
ν ν ν ν
K ∞ → ∞ K
of the second kind, called sometimes as the MacDonald function. The analogue of Theorem 1 is the
following result.
Theorem 3. If ν 3 then the function x ′(x) is strictly log-concave on (0, ).
≥ 2 7→−Kν ∞
The following result is an analogue of van der Corput’s inequality (2.1) for modified Bessel functions
of the second kind.
Corollary 3. If a,b 0 and ν 3, then
≥ ≥ 2
(2.9) (a) (b) a b ′(a) ′(b).
|Kν −Kν |≥| − | Kν Kν
We note that in particular we have p
π π
(2.10) 3(x)= x23K3(x)= (x+1)e−x,
K2 2 · 2 2
r
thus from (2.9) for ν = 3 we obtain the following inequality
2
(a+1)e−a (b+1)e−b a b√abe−a+2b for all a,b 0.
− ≥| − | ≥
(cid:12) (cid:12)
(cid:12) (cid:12)
4 A´.BARICZ,A.LAFORGIA,ANDT.K.POGA´NY
3. Proofs of the main results
Proof of Theorem 1. (a) & (b) First observe that for ν > 1 the power series
−
1 n−1
′(x)= 4 x2n−1
Iν 2(ν+1) (n 1)! ·
n≥1 (cid:0) (cid:1)n −
X
has only positive coefficients, and thus it makessense to study the log-concavityof x ′(x). Moreover
7→Iν
observe that x ′(x) is strictly log-concave on (0, ) if and only if x ′′(x)/ ′(x) is strictly
decreasing on (07→, )I.νBy using the recurrence formula [1∞6, p. 79] [x−νI (x)]′ =7→xI−ννI I(xν) we have
ν ν+1
∞
(3.1) ′(x)=2νΓ(ν+1)x−νI (x).
Iν ν+1
On the other hand application of [16, p. 79] xI (x)=xI′(x) νI (x) and (3.1) yields
ν+1 ν − ν
1
(3.2) ′′(x)=2νΓ(ν+1)x−ν I (x)+ I (x) .
Iν ν+2 x ν+1
(cid:20) (cid:21)
ByusingthenotationR (x)=I (x)/I (x),inviewof (3.1)and(3.2),weobtainthatx ′′(x)/ ′(x)
ν ν+1 ν 7→Iν Iν
is strictly decreasing on (0, ) for all ν 1, 1 if and only if the function x Q (x), where
∞ ∈ − −2 7→ ν+1
Q (x) = 1/x+R (x), is strictly decreasing on (0, ) for all ν 1, 1 . To prove this we need to
ν ν (cid:0) ∞(cid:3) ∈ − −2
show that Q is strictly decreasing on (0, ) for all ν 0,1 . However,since the function
ν ∞ ∈ 2 (cid:0) (cid:3)
x Q1(x)=R1(x)+1/x=I3((cid:0)x)/I(cid:3)1(x)+1/x=cothx
7→ 2 2 2 2
is strictly decreasing on (0, ), we can suppose that ν 0,1 . By using the fact (see [15, p. 446]) that
∞ ∈ 2
R (x) 0 as x 0, we have that Q (x) as x 0. On the other hand, by using the asymptotic
ν ν
→ → → ∞ → (cid:0) (cid:1)
expansion of I (x) for large values of x it can be seen (see [15, p. 446]) that R (x) 1 as x , and
ν ν
→ →∞
consequently Q (x) 1 as x . Thus, for small values of x the function Q is strictly decreasing.
ν ν
→ → ∞
Moreover,we can show that Q is strictly decreasing on (0,√2ν+4), and consequently on (0,2), for all
ν
ν >0. To prove this, first observe that Q (x) can be rewritten as
ν
I′(x) 1 ν
Qν(x)= ν + −
I (x) x
ν
and by using twice the inequality [14, p. 526]
xI′(x) x2
y (x)= ν >ν+ ,
ν
Iν(x) ν+ 1 + x2+ ν+ 3 2
2 2
q
where ν >0, x>0, we obtain that (cid:0) (cid:1)
x2Q′(x)=xy′(x) y (x)+ν 1=x2+ν2 y2(x) y (x)+ν 1
ν ν − ν − − ν − ν −
2ν+1 x4
< 1 x2 1
−ν+ 21 + x2+ ν+ 32 2 − − ν+ 1 + x2+ ν+ 3 2 2 −
2 2
q (cid:0) (cid:1) (cid:18) q (cid:19)
(cid:0) (cid:1)
2(ν+1)x2 x2+ ν+ 3 2 ν+ 1 2(ν+1)x2
= 1= 2 − 2 1 w (x).
ν+ 1 + x2+ ν+ 3 2 2 − qx2+(cid:0)ν+ 32(cid:1)2+(cid:0)ν+ 12(cid:1) · 2(ν+1)+x2 − ≡ ν
2 2
(cid:18) q (cid:19) q (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1)
Now taking into account that for ν 1 both functions
≥−2
t+ ν+ 3 2 ν+ 1 2(ν+1)t
t 2 − 2 and t
7→ qt+(cid:0)ν+ 3(cid:1)2+(cid:0)ν+ 1(cid:1) 7→ 2(ν+1)+t
2 2
q
are increasing on (0, ), it follows (cid:0)that th(cid:1)eir p(cid:0)roduct(cid:1)is also increasing and this in turn implies that for
∞
ν >0 and x (0,√2ν+4) we have
∈
2ν+4+ ν+ 3 2 ν+ 1 2(ν+1)(2ν+4)
w (x)< 2 − 2 1=ϕ(ν)< lim ϕ(ν)=0.
ν q2ν+4+(cid:0)ν+ 3(cid:1)2+(cid:0)ν+ 1(cid:1) · 2(ν+1)+2ν+4 − ν→∞
2 2
q
(cid:0) (cid:1) (cid:0) (cid:1)
VAN DER CORPUT INEQUALITIES FOR BESSEL FUNCTIONS 5
Here we used that the function ϕ is increasing on (0, ). Summarizing, we have proved that for ν > 0
∞
and x (0,√2ν+4) the function Q satisfies x2Q′(x)<0, which implies that Q is strictly decreasing
∈ ν ν ν
on (0,2) for ν > 0. Thus the first extreme of the function Q , if any, is a minimum. Now, since R (x)
ν ν
satisfies the differential equation
2ν+1
u′(x)=1 u2(x) u(x),
− − x
it follows that Q (x) satisfies the differential equation
ν
2ν 1 2ν 1
v′(x)=1 v2(x) − v(x)+ − .
− − x x2
Differentiating both sides of the above relation we get
2ν 1 2 2ν 1
v′′(x)= − v(x) 2v(x)+ − v′(x).
x2 − x − x
(cid:20) (cid:21) (cid:20) (cid:21)
This shows that if Q′(x )=0 for some x 2, then for that x we have
ν ⋆ ⋆ ≥ ⋆
2ν 1 2 2ν 1 1
Q′ν′(x⋆)= x−2 Qν(x⋆)− x = x−2 Rν(x⋆)− x .
⋆ (cid:20) ⋆(cid:21) ⋆ (cid:20) ⋆(cid:21)
On the other hand, by using the inequality [1, p. 241]
x
R (x)> ,
ν
x2+(ν+1)2+ν+1
which holds for ν 0 and x>0, we obtain tqhat for ν 0,1 and x 2 we have
≥ ∈ 2 ≥
x 1
(cid:0) (cid:1)
R (x)> .
ν
x2+ 9 + 3 ≥ x
4 2
This implies that Q′′(x ) < 0. But, this is inqcontradiction with the fact that the first extreme is a
ν ⋆
minimum, and consequently the derivative of Q does not vanish when ν 0,1 . This in turn implies
ν ∈ 2
that Q is strictly decreasing on (0, ) for ν 0,1 , that is, Q is strictly decreasing on (0, ) for
ν ∞ ∈ 2 ν+1 (cid:0) (cid:1) ∞
ν 1, 1 . Thisisequivalenttothe factthatifν 1, 1 ,thenthefunctionx ′(x) isstrictly
∈ − −2 (cid:0) (cid:1)∈ − −2 7→Iν
log-c(cid:0)oncave o(cid:1)n (0, ). On the other hand, since Qν is(cid:0)strictly(cid:1)decreasing on 0, 2(ν+2) for ν > 0,
∞
we get that Q is strictly decreasing on 0, 2(ν+3) for ν > 1, and th(cid:16)is ispequivalent(cid:17)to the fact
ν+1
−
that if ν > 1, then the function x ′(x(cid:16)) ispstrictly lo(cid:17)g-concaveon 0, 2(ν+3) .
− 7→Iν
(c) By using the above things it is clear that we need to show that(cid:16)forpν > 21 the(cid:17)re exists an xν−1 >
2(ν+2) such that Q is strictly decreasing on (0,x ) and strictly increasing on (x , ). For this
ν ν−1 ν−1
∞
we note that [9, p. 276]
p
1 4ν2 1
yν(x) x + − ...,
− ∼−2 8x −
whichholdsfor largevalues ofxandfixedν. Fromthis weobtainthatfor ν > 1 andlargevalues ofx we
2
have y (x) x < ν 1, or equivalently Q (x) < 1. Since Q is strictly decreasing on 0,√2ν+4 and
ν ν ν
− −
Q (x) 1 as x , it follows that there must be a point at which Q′(x) vanishes. In other words,
ν → → ∞ ν (cid:0) (cid:1)
there exists an x > 2(ν+2) such that Q is strictly decreasing on (0,x ) and strictly increasing
ν−1 ν ν−1
on (x , ). Moreover, Q′ changes sign only once, otherwise we will have two local minima between
ν−1 ∞ p ν
which there must be a local maximum, which is impossible, since in that stationary point x⋆ we would
have
2ν 1 1
Q′ν′(x⋆)= (x⋆−)2 Rν(x⋆)− x⋆ >0,
(cid:20) (cid:21)
according to the above discussion. (cid:3)
Proof of Corollary 1. Without loss of generality it is enough to show that inequality (2.2) holds for
b>a>0.Observethatthe functionx ′(x) is convexon[a,b],since aspowerserieshasonlypositive
7→Iν
coefficients. Nowtakinginto accountthatfromTheorem1the functionx ′(x) isstrictly log-concave
7→Iν
on (0, ), from the Hermite-Hadamardinequality [12, p. 14] applied to the function x ′(x) we have
∞ 7→Iν
(b) (a) 1 b a+b
Iν −Iν = ′(x)dx ′ ′(a) ′(b),
b a b a Iν ≥Iν 2 ≥ Iν Iν
− − Za (cid:18) (cid:19)
p
6 A´.BARICZ,A.LAFORGIA,ANDT.K.POGA´NY
which completes the proof.
Another proof of Corollary 1 is as follows. Let f : [a,b] R be such that f′ : [a,b] (0, ) is
→ → ∞
log-concave. Then the inequality [7, p. 242]
1 b 1 b logf′(a)+logf′(b)
(3.3) log f′(x)dx logf′(x)dx
b−aZa ! ≥ b−aZa ≥ 2
implies that
f(b) f(a)
log − log f′(a)f′(b)
b a ≥
−
is valid. Applying (3.3) for the function f = , whichphas the property that x ′(x) is strictly
log-concaveon (0, ), we arrive at (2.2). Iν 7→ Iν (cid:3)
∞
Proof of Theorem 2. By using the recurrence relation [16, p. 45] [x−νJ (x)]′ = x−νJ (x) we get
ν ν+1
−
′(x)= 2νΓ(ν+1)x−νJ (x),
Jν − ν+1
1
′′(x)=2νΓ(ν+1)x−ν J (x) J (x) ,
Jν ν+2 − x ν+1
(cid:20) (cid:21)
and thus the function x ′(x), which maps (0,j ) into (0, ), is strictly log-concaveif and only
7→−Jν ν+1,1 ∞
ifthefunctionx 1/x J (x)/J (x)is strictlydecreasingon(0,j ). But,usingthe well-known
ν+2 ν+1 ν+1,1
7→ −
Mittag-Leffler expansion
J (x) 2x
ν+1
= ,
J (x) j2 x2
ν n≥1 ν,n−
X
we obtain that
J (x) ′ j2 +x2
ν+1 ν,n
=2 >0
J (x) (j2 x2)2
(cid:20) ν (cid:21) n≥1 ν,n−
X
for all ν > 1, x = j , n 1,2,... . Note that this can be proved also by using the Neumann’s
ν,n
− 6 ∈ { }
formula for the product of two Bessel functions of the first kind with different orders, see [11, Lemma
2.5]. In particular, the above result means that the function x J (x)/J (x) is strictly increasing on
ν+1 ν
7→
(0,j ) for ν > 1. Thus, for ν > 2 the function x 1/x J (x)/J (x) is strictly decreasing on
ν,1 ν+2 ν+1
(0,j ) as the−sum of two strictl−y decreasing functi7→ons. − (cid:3)
ν+1,1
Proof of Corollary 2. Observe that as in Corollary 1 it is enough to consider the case when b > a.
Then the proof of the result follows from Theorem 2 and (3.3) by choosing for f the function . (cid:3)
ν
−J
Proof of Theorem 3. By using the recurrence relation [16, p. 79] xK′(x)+νK (x)= xK (x) we
ν ν − ν−1
obtain that
′(x)= 2ν−1Γ(ν)xνK (x),
Kν − ν−1
ν
′′(x)= 2ν−1Γ(ν)xν K (x)+K′ (x) ,
Kν − x ν−1 ν−1
h i
and thus x ′(x) is strictly log-concave on (0, ) if and only if the function
7→−Kν ∞
′′(x) ν K′ (x) 1 K (x)
x Kν = + ν−1 = ν−2
7→ ′(x) x K (x) x − K (x)
Kν ν−1 ν−1
is strictly decreasing on (0, ). By using [11, Lemma 2.4] the function x K (x)/K (x) is strictly
ν+1 ν
∞ 7→
decreasing on (0, ) for ν > 1, which implies that the function x K (x)/K (x) is strictly
∞ −2 7→ ν−2 ν−1
increasing on (0, ) for ν > 3. Consequently, the function x ′(x) is strictly log-concaveon (0, )
∞ 2 7→−Kν ∞
for ν > 3. Finally, observe that by using (2.10) we have that x ′ (x) = πxe−x is also strictly
log-conca2veon (0, ). This completes the proof. 7→ −K32 2 (cid:3)
∞
Proof of Corollary 3. Clearly, as in Corollary 1 it is enough to consider the case when b > a. Then
the proof of the result follows from Theorem 3 and (3.3) by choosing for f the function . (cid:3)
ν
−K
VAN DER CORPUT INEQUALITIES FOR BESSEL FUNCTIONS 7
4. Concluding remarks
In this section we would like to comment the main results of this paper.
1. We note that in [3, Theorem 3.6.3] (see also [4, Theorem 3.25]) it was proved that the function
x ′(x) is strictly log-concave on 0, 2(ν+2) for ν > 1. Moreover, it was shown in [3,
7→ Iν −
Theorem 3.6.4] (see also [4, Theorem(cid:16)3.2p6]) that th(cid:17)e inequality (2.2) is valid for all ν > 1 and
−
a,b 0, 2(ν+2) .Theorem1andCorollary1complementandimprovetheabovementioned
∈
results(cid:16). p (cid:17)
2. Wealsonotethat(see[3,Theorem3.6.3]or[4,Theorem3.25])forν > 1thefunctionx (x)
ν
− 7→I
is strictly log-convex on 0, 2(ν+1)(ν+2) . Since the function x log (x) is even it
ν
7→ I
follows that in fact x (cid:16)(x)pis strictly log-co(cid:17)nvex on 2(ν+1)(ν+2), 2(ν+1)(ν+2)
ν
7→ I −
for ν > 1. Moreover, it was shown in [2, Theorem 4](cid:16)thapt the function x p ν(x) is strictl(cid:17)y
− 7→ I
log-convex on [ j ,j ] for ν > 1, where j is the first positive zero of the Bessel function
ν,1 ν,1 ν,1
− −
of the first kind J . Next, we show that the function x (x) cannot be strictly log-convexon
ν ν
the whole R for ν > 1. When ν 1 the strict log-c7→onIvexity of x (x) is certainly valid,
− ≥ −2 7→ Iν
but in the case when ν 1, 1 there exists a z j such that the function x (x)
∈ − −2 ν ≥ ν,1 7→ Iν
is strictly log-convex on (0,z ) and strictly log-concave on (x , ). For this observe that the
ν ν
(cid:0) (cid:1) ∞
function x (x) is strictly log-convex (log-concave) if and only if x R (x) is strictly
ν ν
7→ I 7→
increasing (decreasing). But, it was shown in [15, p. 446] that when ν 1, 1 the function
∈ − −2
x R (x) on (0, ) is increasing first to reach a maximum and then decreasing. This means
ν
7→ ∞ (cid:0) (cid:1)
thatthere exists a z suchthat x R (x) is strictly increasingon(0,z )andstrictly decreasing
ν ν ν
7→
on (z , ), as we required. Now, since x (x) is strictly log-convex on (0,j ] we clearly
ν ν ν,1
∞ 7→ I
have that z j .
ν ν,1
≥
3. We would like to mention that the idea of the first proof of (2.2) does not work in the case of
(2.6) because in view of (x) = cosx, the function x ′ (x) = sinx is concave and
J−12 7→ −J−21
log-concaveand consequently
′ a+b 1 b ′ (x)dx= J−21(b)−J−12(a), ′ a+b ′ (a) ′ (b)
J−12 (cid:18) 2 (cid:19)≤ b−aZa J−21 b−a −J−21 (cid:18) 2 (cid:19)≥ J−12 J−21
q
and these together does not imply the van der Corput’s inequality (2.5). However, as we have
seen in the proof of Corollary2 the inequality (3.3) applied to the function implies the van
ν
−J
der Corput inequality (2.6). In other words, there no need to assume for f′ to be convex, its
log-concavityis quite enough, according to (3.3).
4. Finally, we would like to mention that functions which have log-concave derivatives occur for
exampleinprobabilitytheory. Namely,manyprobabilitydensityfunctionsarelog-concave(seefor
example [5]), and thus applying the inequality (3.3) for the cumulative distribution functions we
cangetvanderCorputinequalities. Forexample,considertheprobabilitydensityandcumulative
distribution function of the standard normal distribution
ϕ(x)= 1 e−x22 and Φ(x)= 1 x e−t22dt.
√2π √2π
Z−∞
The function x Φ′(x) = ϕ(x) is log-concave on R, and applying the inequality (3.3) for the
7→
function f =Φ we get the following van der Corput inequality for all real a and b
Φ(a) Φ(b) a b ϕ(a)ϕ(b).
| − |≥| − |
Another example is the gamma distribution whicph has support (0, ), probability density and
∞
cumulative distribution function as follows
xα−1e−x γ(α,x) 1 x
f(x)= and F(x)= = tα−1e−tdt,
Γ(α) Γ(α) Γ(α)
Z0
where γ(α, ) is the lower incomplete gamma function and α > 0 is the shape parameter. Since
·
the probability density function of the gamma distribution is log-concaveon (0, ) for all α 1
∞ ≥
(see for example [5, p. 192]), applying (3.3) for the function F we get the following van der
Corput inequality
γ(α,a) γ(α,b) a b(ab)α−21e−a+2b,
| − |≥| − |
where a,b 0 and α 1.
≥ ≥
8 A´.BARICZ,A.LAFORGIA,ANDT.K.POGA´NY
Acknowledgement. The workofA´.Bariczwassupportedby the J´anosBolyaiResearchScholarshipof
the Hungarian Academy of Sciences. This author is very grateful to Aingeru Ferna´ndez from University
of the Basque Country, Spain, for pointing out an error in the proof of Theorem 1 in an earlier version
of the paper.
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Departmentof Economics, Babe¸s-BolyaiUniversity, 400591Cluj-Napoca, Romania
Institute of Applied Mathematics, John von Neumann Faculty of Informatics, O´buda University, 1034 Bu-
dapest, Hungary
E-mail address: [email protected]
Departmentof Mathematics,Roma Tre University,LargoSanLeonardo Murialdo,1,00146Rome,Italy
E-mail address: [email protected]
Facultyof Maritime Studies, University of Rijeka,51000Rijeka, Croatia
Institute of Applied Mathematics, John von Neumann Faculty of Informatics, O´buda University, 1034 Bu-
dapest, Hungary
E-mail address: [email protected]
