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AN ISOPERIMETRIC RESULT FOR THE FUNDAMENTAL FREQUENCY VIA DOMAIN DERIVATIVE. 2 1 0 2 n Carlo Nitsch a J DipartimentodiMatematicaeApplicazioni“R.Caccioppoli” 0 ComplessoMonteS.Angelo 3 ViaCintia,80126Napoli,Italy ] C O Abstract. TheFaber-Krahndeficitδλofanopenbounded setΩisthenor- malized gap between the values that the first Dirichlet Laplacian eigenvalue . h achievesonΩandontheballhavingsamemeasureasΩ. Foranygivenfamily t of open bounded sets of RN (N ≥2)smoothly converging to aball, itiswell a knownthat bothδλandtheisoperimetricdeficitδP arevanishingquantities. m δP It is known as well that, at least for convex sets, the ratio is bounded by δλ [ belowbysomepositiveconstant(see[3,19]),andinthisnote,usingthetech- nique of the shape derivative, we provide the explicit optimal lower bound of 2 sucharatioasδP goestozero. v 8 2 1. Introduction 3 5 GivenanopenboundedsetofRN itsfirstDirichletLaplacianeigenvalueλisthe . least positive number for which the boundary problem 1 0 ∆u=λu in Ω 2 − u=0 on ∂Ω 1 (cid:26) : admits nontrivial solutions. v i TheFaber-Krahninequalityisaremarkablepropertywhich,answeringtoacon- X jecture formulated by Lord Rayleight, states that among sets of given measure the r ballhastheleastfirstDirichletLaplacianeigenvalue. NamelyifΩ♯ denotestheball a having the same measure as Ω then (1) λ(Ω) λ(Ω♯). ≥ Inequality (1) falls in the large class of so-called isoperimetric inequalities. By antonomasia the isoperimetric inequality is the one which characterizes the ball as the sethavingminimialperimeteramongthosesetsoffixedvolume,butnowadays, inabroadsense isoperimetricinequality is aninequality wherea functionalis opti- mized under some geometricalprescription. The study of isoperimetric inequalities goes back to the beginning of mathematics and has always been a flourishing field. Recently many authors turned the attention to the study of quantitative versions of the classical isoperimetric inequalities (see for instance [2, 5, 6, 7, 8, 9, 10]), and quantitativeversionsofFaber-Krahninequalityhavebeen investigatedfor instance 1991 Mathematics Subject Classification. Primary: 35P15;Secondary: 49R05, 35J25. Key words and phrases. Domainderivative;firstlaplacianeigenvalue; isoperimetricdeficit; 1 2 CARLONITSCH in [11, 16]. Here we are interested in a recent result [3] obtained in the wake of a celebrated paper by L. E. Payne and H. F. Weinberger [19]. As custom let us denote by Per(Ω) δP(Ω)= 1, Per(Ω♯) − the isoperimetric deficit of Ω and following [11] we denote by λ(Ω) δλ(Ω)= 1, λ(Ω♯) − the Faber-Krahndeficit. The classical isoperimetric inequality and the Faber-Krahn inequality respectively infer that both δP and δλ are always non negative quantities. When Ω is convex then (1) can be improved (see [3, 19]) establishing that for any η >0 there exists C >0 depending on N such that if δP(Ω) η then ≤ (2) δP(Ω) Cδλ(Ω). ≥ The name “quantitative Faber-Krahn inequality” comes from the fact that it quantifies how “small” is the Faber-Krahn deficit when the set Ω is “close” to the ball having same measure. Itiseasytoshowthatinequality(2)isoptimalinthesensethatforanyη,C,γ >0 thereexistsaboundedconvexsetΩsuchthatδP(Ω) ηandδP(Ω)<Cδλ(Ω)1+γ. ≤ Very little is known about the optimal value of the constant C even in the limit as δP 0 and here comes the idea to exploit the technique of shape derivative to → investigate the behavior of the ratio δP/δλ along an arbitrary family of sets which convergesina suitable wayto a ball. Morepreciselyweuse the followingdefinition Definition 1.1. We say that a one parameter family Ω(t) of open bounded sets of RN smoothly converges to an open bounded set Ω as t goes to zero, if there exists a positive δ, and a one parameter family of transformations Φ (0 t < δ) of RN t ≤ in itself such that (a) Φ and Φ −1 belong to C∞(RN;RN) for all 0 t<δ; t t (b) the mappings t Φ (x) and t Φ −1(x) belo≤ng to C∞([0,δ[) for all x RN; t t → → ∈ (c) Ω(0)=Ω and Ω(t)=Φ (Ω) for all 0 t<δ; t ≤ We denote by ω the volume of the unit ball of RN, by J the Bessel function N ν of first kind and order ν, and by j the first positive zero of J . Our main result ν ν follows. Theorem 1.2. For any N 2 there exists a dimensional constant ≥ jN/2−1 N(N +1) rJ2 (r)dr N/2−1 C = Z0 N 2 2 j J′ (j ) j2 N N/2−1 N/2−1 N/2−1 N/2−1− suchthatforanygivenon(cid:16)eparameterfamilyofse(cid:17)tsΩ(cid:16)(t)ofRN,sm(cid:17)oothlyconverging to a ball as t 0, then → δP(Ω(t)) liminf C . N t→0 δλ(Ω(t)) ≥ The constant C is optimal and there exists a family Ω(t) for which the equality N sign is achieved. AN ISOPERIMETRIC RESULT FOR THE FUNDAMENTAL FREQUENCY 3 2. Proof of Theorem 1.2 We considerafamily ofopenboundedsetsΩ(t)(0 t<δ)smoothlyconverging ≤ as t 0 in the sense of Definition 1.1 to a smooth open bounded connected set → Ω. For any 0 t < δ we denote by λ(t) and u(x,t) respectively the first Dirichlet ≤ Laplacian eigenvalue and the normalized solution to ∆u(x,t)=λ(t)u(x,t) in Ω(t) −  u 0 in Ω(t) (3)  u(≥x,t)=0 on ∂Ω(t)  kukL2(Ω(t)) =1. The proof of Theorem 1.2 will be carried on by choosing an arbitrary family of smoothsets convergingto a ball as t goes to zero andby performing Taylorexpan- sionof the ratio δP(Ω(t)) aroundt=0. For the seek of simplicity we split the proof δλ(Ω(t)) of Theorem 1.2 in severalsteps. In the first step we provide the general expression offirstandsecondorderderivativesofthefirstDirichletLaplacianeigenvaluealong thefamilyΩ(t). InthesecondstepweletΩbetheaballofRN andwedifferentiate δP(Ω(t)) and δλ(Ω(t)) twice at t =0 deducing that there exists a functional on F δP(Ω(t)) ∂Φ C∞(∂Ω) such that = n t +o(1) as t goes to zero. Here δλ(Ω(t)) F · ∂t (cid:18) (cid:12)t=0(cid:19) n denotes the unit outer normal to ∂Ω. In (cid:12)the third and last step we show that (cid:12) C is exactly the minimum achieved by the(cid:12)functional when we vary Φ on the N t F whole class of admissible smooth transformations (in the sense of Definition 1.1 (a)-(b)-(c)). Step 1. We begin the proofcomputing the first and the second order derivatives of the Dirichlet Laplacian eigenvalue along the family Ω(t), using the well known Hadamard’s formula. Namely we prove the following Lemma Lemma 2.1. For 0 t < δ let u(x,t) be the family of solutions to (3) and let ≤ λ(t) be the corresponding family of eigenvalues. There exists ε>0 such that for all 0 t<ε the family λ(t) is smooth and it holds ≤ ∂Φ (4) λ′(t)= Du t(Φ−1) Dud N−1, | | ∂t t · H Z∂Ω(t) ∂Φ T ∂Φ (5) λ′′(t)= Du t(Φ−1) D2u t(Φ−1) Z∂Ω(t)| | (cid:20) ∂t t (cid:21) · · ∂t t ∂2Φ ∂Φ +Du t(Φ−1)+2Dw t(Φ−1) d N−1 · ∂t2 t · ∂t t H (cid:19) 4 CARLONITSCH where w solves ∆w(x,t)=λ′(t)u(x,t)+λ(t)w(x,t) x Ω(t) − ∈  ∂Φ (6)  w(x,t)=− ∂tt(Φ−t 1(x))·Du(x,t) x∈∂Ω(t). uw =0 Equations (4)-(Z5Ω)(-t()6) are related to other formulas which can be found in litera- ture(see[13,14,18,21]andthereferencesthereincontained). Inparticular[14,18] contain very general formulation of the notion of shape derivative with application totheDirichletLaplacianeigenvalues. Wealsoobservethatveryofteninliterature (see for instance [13, Theorem 2.5.6]) it is considered the case of a first order per- turbations of identity, namely Φ = I+tW, where W is a suitably smooth vector t field. The result is that the term Du ∂2Φt in (5), which in our case plays a crucial · ∂t2 rule, would be missing. Hence, for the seek of completeness we decided to provide here a complete proof of the statement of Lemma 2.1, and we exploit a level sets method. Proof of Lemma 2.1. Classical regularity theory [12] for elliptic equation ensures that, for all t [0,δ[, u(x,t) C∞(Ω(t)). Moreover,arguing as in [14, Chapter 5], ∈ ∈ there exists at least some positive ε < δ such that the function u(x,t) belongs to C∞([0,ε[;C∞(Ω(t))). Differentiating with respect to t the equation in (3) we get ∂u ∂u (7) ∆ =λ′(t)u+λ(t) . − ∂t ∂t Since Ω(t) is,atanytime, the zero-levelsetofu(x,t), if y ∂Ω thenΦ (y) ∂Ω(t) t ∈ ∈ and we have (8) u(Φ (y),t)=0 for all y ∂Ω and 0<t<ε. t ∈ ∂Φ (y) t Now,theboundarypointΦ (y) ∂Ω(t)moveswithvelocity . Differentiating t ∈ ∂t once (8) with respect to t we get d ∂Φ ∂u t u(Φ (y),t)=Du(Φ (y),t) (y)+ (Φ (y),t)=0, t t t dt · ∂t ∂t and hence ∂Φ ∂u (9) Du t(Φ −1)+ =0 on ∂Ω(t). t · ∂t ∂t Thereforethe projectionofthevelocity ∂Φt(y)alongthe directionofthe unitouter ∂t 1 ∂u normal n is equal to , which is pointwise defined since Ω(t) is smooth and Du ∂t | | standard barrier arguments imply that Du does not vanish on ∂Ω(t). Differentiating twice (8) with respect to t we get (10) ∂Φ T ∂Φ ∂2Φ ∂u ∂Φ ∂2u t(Φ −1) D2u t(Φ −1)+Du t(Φ −1)+2 D t(Φ −1)+ =0 ∂t t · · ∂t t · ∂t2 t ∂t · ∂t t ∂t2 (cid:20) (cid:21) (cid:18) (cid:19) which highlights the connection between the acceleration ∂2Φt of a boundary point ∂t2 x Φ (y) ∂Ω(t) and the value of ∂2u at the same point. ≡ t ∈ ∂t2 AN ISOPERIMETRIC RESULT FOR THE FUNDAMENTAL FREQUENCY 5 Now, if f(x,t) is a smooth function and J(t) = f(x,t) dx, the classical ZΩ(t) Hadamard formula gives (see, for instance, [14, 21]) ∂f 1 ∂u (11) J′(t)= (x,t)dx+ f(x,t) d N−1. ∂ Du ∂t H ZΩ(t) Z∂Ω(t) | | Therefore, since the L2 norm of u is constant with respect to t and u vanishes on ∂Ω(t) we have d ∂u (12) u2dx=2 u dx=0 dt ∂t ZΩ(t) ZΩ(t) d2 ∂2u ∂u 2 (13) u2dx=2 u dx+2 dx=0. dt2 ∂t2 ∂t ZΩ(t) ZΩ(t) ZΩ(t)(cid:18) (cid:19) Furthermore (11) applied to λ(t) provides the relation d ∂u ∂Φ λ′(t)= Du2dx= Du d N−1 = Du t(Φ −1)Dud N−1, t dt | | − | |∂t H | | ∂t · H ZΩ(t) Z∂Ω(t) Z∂Ω(t) or equivalently ∂u (14) λ′(t)= DuD dx, ∂t ZΩ(t) (cid:18) (cid:19) obtaining (4). Finally, if we differentiate λ twice, we can use (7),(12),(13) and (14) to get (15) d ∂u λ′′(t)= DuD dx dt ∂t ZΩ(t) (cid:18) (cid:19) ∂u 2 ∂2u Du ∂u ∂u = D dx+ DuD dx+ D d N−1 ∂t ∂t2 Du ∂t ∂t H ZΩ(t)(cid:12) (cid:12) ZΩ(t) Z∂Ω(t) | |(cid:18) (cid:19) (cid:12)(cid:12) (cid:12)(cid:12)∂u ∂u ∂2u ∂2u = (cid:12) ∆(cid:12) dx ∆u dx Du d N−1 − ∂t ∂t − ∂t2 − | |∂t2 H ZΩ(t)(cid:18) (cid:19) ZΩ(t) Z∂Ω(t) ∂u 2 ∂2u ∂u ∂2u =λ(t) +u dx+λ′(t) u dx Du d N−1 ZΩ(t) (cid:18)∂t(cid:19) ∂t2! ZΩ(t) ∂t −Z∂Ω(t)| |∂t2 H ∂2u = Du d N−1. − | |∂t2 H Z∂Ω(t) Once we set ∂u w(x,t)= (x,t) ∂t in (7), (9) and (12), using (10) we get (5) and the proof is complete (cid:3) We observe that the family of transformations Φ is not uniquely determined by t thefamilyΩ(t). Inparticularitisalwayspossibletochoosethevelocityvectorfield 6 CARLONITSCH ∂Φt orthogonalto ∂Ω. In such a case (4)-(5)-(6) computed at t=0 become ∂t t=0 (cid:12) ∂Φ (16)(cid:12) λ′(0)= Du2n t − | | · ∂t Z∂Ω (cid:12)t=0 (17) λ′′(0)= ω2H Du2n(cid:12)(cid:12)(cid:12) ∂2Φt 2ω∂ω d N−1, −| | · ∂t2 − ∂n H Z∂Ωh (cid:12)(cid:12)t=0 i (cid:12) ∆ω(x)=λ′(0)u(x,0)+λ((cid:12)0)ω(x) in Ω −  ∂Φ (18)  ω(x)=|Du(x,0)|n· ∂tt(x)(cid:12)(cid:12)t=0 on ∂Ω. (cid:12) (cid:12) u(x,0)ω(x)dx=0 Here H is the sumoZfΩthe principal curvatures of ∂Ω and n the unit outer normal of ∂Ω. Since ∆u(x,0) vanishes on ∂Ω we have used the identity Du DuT D2u Du= div Du3 on ∂Ω, · · − Du | | (cid:18)| |(cid:19) in conjuction with div Du =H. − |Du| Step 2. Due to the i(cid:16)nvaria(cid:17)nce of both isoperimetric and Faber-Krahn deficits with respect to homotheties we shall perform all the remaining computation under theassumptionthatthefamilyΩ(t)hasconstantvolumeintequaltoω ,therefore N from now on Ω Ω(0) is just a unit ball in Rn. ≡ ∂Φ t Without loss of generality, we also assume that the velocity field is ∂t (cid:12)t=0 ∂Φ (cid:12) orthogonal to ∂Ω and for all x ∂Ω we denote by V(x) = n(x) t a(cid:12)nd by ∈ · ∂t (cid:12) (cid:12)t=0 ∂2Φ (cid:12) A(x)=n(x) t respectivelythe initial scalarvelocityand the pr(cid:12)ojectionof · ∂t2 (cid:12) (cid:12)t=0 the initial accelerati(cid:12)on along the unit outer normal n(x) of Ω. (cid:12) Under these assu(cid:12)mptions, for t small enough, the boundary of Ω(t) can be rep- resented in polar coordinates r R+, ξ N−1 by an equation ∈ ∈S A(ξ) r(ξ,t)=1+V(ξ)t+ t2+o(t2). 2 If σ denotes the usual surface area measure on N−1 then ξ S Per(Ω(t))= r(ξ,t)N−2 r(ξ,t)2+ D r(ξ,t)2dσ , ξ ξ ZSN−1 q | | and after a taylor expansion we have Per(Ω(t))=nω +t (N 1)V(ξ)dσ N ξ ZSN−1 − t2 + (N 1)A(ξ)+(N 1)(N 2)V2(ξ)+ D V(ξ)2 dσ +o(t2). ξ ξ 2 ZSN−1 − − − | | (cid:2) (cid:3) On the other hand, since ω = Ω(t) for all t [0,δ[ N | | ∈ AN ISOPERIMETRIC RESULT FOR THE FUNDAMENTAL FREQUENCY 7 then 1 Ω(t) = r(ξ,t)N dσ ξ | | N ZSN−1 t2 =ω +t V(ξ)dσ + A(ξ)+(N 1)V2(ξ) dσ +o(t2) N ξ ξ ZSN−1 2 ZSN−1 − (cid:2) (cid:3) yields (19) V(ξ)dσ = A(ξ)+(N 1)V2(ξ) dσ =0. ξ ξ ZSN−1 ZSN−1 − (cid:2) (cid:3) As a consequence t2 Per(Ω(t))=nω + D V(ξ)2 (N 1)V2(ξ) dσ +o(t2) N ξ ξ 2 ZSN−1 | | − − (cid:2) (cid:3) and Per(Ω(t)) δP(Ω(t))= 1 Nω1/n Ω(t)(N−1)/N − N | | t2 = D V(ξ)2 (N 1)V2(ξ) dσ +o(t2). ξ ξ 2NωN ZSN−1 | | − − (cid:2) (cid:3) We consider now the series expansion t2 λ(t)=λ(0)+tλ′(0)+ λ′′(0)+o(t2). 2 The gradient Du(,0) on ∂Ω has constant modulus (see [15]) · j2 J′ (j ) N/2−1 N/2−1 N/2−1 G = , N jN/2−1 1/2 Nω rJ2 (r)dr N N/2−1 (cid:18) Z0 (cid:19) and therefore using (16) and (19) we deduce λ′(0)=0 in accordance with the fact thattheball,amongsetsoffixedmeasure,isastationarypointforthefirstDirichlet Laplacian eigenvalue. Thereafter, for all x (r,ξ) Ω, we set ≡ ∈ v(r,ξ)=G ω(x) N where ω is defined in (18). Taking into account that, for the unit ball, the sum of the principal curvatures H equals N 1, from (17) and (19) we get − ∂v(r,ξ) λ′′(0)=2G2 V(ξ) +(N 1)V2(ξ) dσ . NZSN−1(cid:20) ∂r (cid:12)r=1 − (cid:21) ξ (cid:12) Consequently (cid:12) (cid:12) λ(t) δλ(Ω(t))= 1 λ(0) − 2 G ∂v(r,ξ) =t2 N V(ξ) +(N 1)V2(ξ) dσ +o(t2). ξ (cid:18)jN/2−1(cid:19) ZSN−1(cid:20) ∂r (cid:12)r=1 − (cid:21) (cid:12) Here we have used the fact that the first Dir(cid:12)ichlet Laplacian eigenvalue on the (cid:12) unit ball of RN is λ(0)=j2 (see for instance [15]) and that λ′(0)=0. N/2−1 8 CARLONITSCH We need now an explicit representation of the function v(r,ξ) in terms of the scalarvelocityV(ξ). Tothisaim,weobservethat(18),inconjunctionwithλ′(0)=0 and Du(x,0) =G imply that v(r,ξ) satisfies N | | ∂Ω (cid:12) r(cid:12)1−N ∂ rN−1∂v r−2∆ v =j2 v (r,ξ) (0,1) N−1 −(cid:12) ∂r ∂r − ξ N/2−1 ∈ ×S (20)  (cid:0) (cid:1)  v(1,ξ)=GNV(ξ) ξ N−1 ∈S where ∆ξis the Laplace Beltrami operator on N−1. S Then we remind (see for instance [17]) that V(ξ) admits an expansion +∞ V(ξ)= a Y (ξ) ξ N−1 k k ∈S k=0 X in terms of a family of spherical harmonics Yk(ξ) k∈N which satisfy for all k 0 { } ≥ ∆ Y =k(k+N 2)Y and Y =1. ξ k k k L2 − − k k The coefficient a is the projection of V on the normalized eigenfuntion Y k k a = V(ξ)Y (ξ)dσ , k k ξ ZSN−1 so that +∞ V 2 = a2. k kL2 k k=0 X Notice that Y =(Nω )−1/2 and (19) imply 0 N a = V Y dσ =(Nω )−1/2 V dσ =0. 0 0 ξ N ξ ZSN−1 ZSN−1 Accordingly we use the separation of variables v(r,ξ)= R (r)Y (ξ) to solve k k k the Poissonproblem (20) and infer P J (j r) v(r,ξ)=r1−N/2 a ℓk N/2−1 Y (ξ) k k J (j ) k≥1 ℓk N/2−1 X where ℓ = k(k+N 2)+(N/2 1)2 =k+N/2 1. k − − − Consequently we have p ∂v N J′ (j ) (1,ξ)= 1 +j ℓk N/2−1 a Y (ξ), ∂r − 2 N/2−1J (j ) k k k≥1(cid:20)(cid:18) (cid:19) ℓk N/2−1 (cid:21) X and in view of the recurrence relations of the Bessel functions (see [1, 9.1.27]) § ℓ J′(s)= J (s) J (s) ℓ s ℓ − ℓ+1 we can write ∂v(1,ξ)= k j Jℓk+1(jN/2−1) a Y (ξ). ∂r − N/2−1 J (j ) k k k≥1(cid:20) ℓk N/2−1 (cid:21) X Finally we obtain AN ISOPERIMETRIC RESULT FOR THE FUNDAMENTAL FREQUENCY 9 N2 a2 ℓ2 δP(Ω(t)) = jN2/2−1 kX≥2 k(cid:20) k− 4 (cid:21) +o(1). δλ(Ω(t)) 2NωNG2N! a2 ℓ + N j Jℓk+1(jN/2−1) k k 2 − N/2−1 J (j ) k≥2 (cid:20) ℓk N/2−1 (cid:21) X Observe that a provides no contribution in the summation, indeed the projection 1 of V on the subspace Y corresponds to a translation of the ball Ω(0) (with no 1 deformation). Step 3. It is evident that δP(Ω(t)) j2 ℓ2 N2 (21) liminf inf  N/2−1 k− 4  t→0 δλ(Ω(t)) ≥k≥2 2NωNG2N!ℓk+ N2 −jN/2−1JJℓkℓ+k(1j(NjN/2/−2−1)1)   andtheremainderoftheproofofTheorem1.2consistsinshowingthattheinfimum on righthand side of (21) is achieved for k = 2 independently on N. In fact the constant C defined in Theorem 1.2 coincides with N j2 ℓ2 N2  N/2−1 k− 4 (cid:12) .  2NωNG2N!ℓk+ N2 −jN/2−1JJℓkℓ+k(1j(NjN/2/−2−1)1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)k=2  (cid:12) In principle, minimizing the righthand side of (21) is elemen(cid:12)tary. However it is (cid:12) worth providing the details, since the proof involves the usage of several nontrivial properties of the Bessel functions. The next Lemma concludes the proof of Theorem 1.2. In what follows we use the notion of convex sequence: Definition 2.2. We say that a sequence αk k∈N of real numbers is convex (con- { } cave) for k k if α 2α +α 0 ( 0) when k k +1. 0 k+1 k k−1 0 ≥ − ≥ ≤ ≥ Lemma 2.3. For all N 2, let k 2, ℓ =k+N/2 1, and k ≥ ≥ − j2 ℓ2 N2 = N/2−1 k− 4 . Qk 2NωNG2N!ℓ + N j Jℓk+1(jN/2−1) k 2 − N/2−1 J (j ) ℓk N/2−1 We have . k k+1 Q ≤Q Proof of Lemma 2.3. First we prove that, for any given value N 2, the sequence ≥ Jℓk+1(jN/2−1) is positive, decreasing, vanishing, and convex for k 2. De- Jℓk(jN/2−1) k∈N ≥ nnoting by z o= j , the claim follows at once from the continued fraction N N/2−1 representation(see [1, 9.1.73]) § J (z ) 1 (22) ℓk+1 N = Jℓk(zN) 2(ℓk+1) 1 zN − 2(ℓk+2) 1 zN − 2(ℓk+3) ... z − N 10 CARLONITSCH 2(ℓ +1) k In fact, after observing that > 1 (see [4, 20]) it is easy to deduce that z N thesequence Jℓk+1(jN/2−1) ispositivedecreasingandvanishing. Itremainsto Jℓk(jN/2−1) k∈N prove the connvexity and weobegin observing that if a sequence αk k∈N is concave { } then {α−k1}k∈N is convex, hence 1 , k N 2(ℓk+1) ∈ z N is convex for k 2 but also ≥ 1 1 , and k N 2(ℓ +1) 1 2(ℓ +1) 1 ∈ k k zN − 2(ℓk+2) zN − 2(ℓk+2) 1 zN zN − 2(ℓk+3) z N are convex for k 2, as well as any truncation of the continued fraction (22). By ≥ approximationwe deduce that Jℓk+1(jN/2−1) is a convex sequence for k 2. Jℓk(jN/2−1) k∈N ≥ Eventuallywededucethat,forannyN fixed, o isaconvexsequencefork 2since k itisthe ratiobetweenapositiveconvexsequQence zN2 ℓ2 N2 ≥ and 2NωNG2N k− 4 k∈N J (zn(cid:16)) (cid:17)(cid:16) (cid:17)o a positive concave sequence ℓ + N z ℓk+1 N . k 2 − N J (z ) (cid:26) ℓk N (cid:27)k∈N The convexity of for k 2 implies the increasing monotonicity of for k k Q ≥ Q k 2 if and only if . In view of the recurrence relations of the Bessel 2 3 ≥ Q ≤ Q functions [1, 9.1.27], we have § J (z ) N ℓ2 N = , J (z ) z ℓ1 N N J (z ) N +2 z ℓ3 N = N, J (z ) z − N ℓ2 N N and −1 J (z ) N +4 N +2 z ℓ4 N = N . J (z ) z − z − N ℓ3 N N (cid:18) N (cid:19) After a tedious but straightforwardcomputation we get < if and only if 2 3 Q Q (23) (z2 2)N2+5z2N 2z4 >0. N − N − N It is not difficult to prove that the last inequality holds true for large values of N, inviewofthefollowingupperandlowerboundsonthefirstzeroofBesselfunctions (see [4, 20]) N N N (24) 1 z +1+1 . N 2 − ≤ ≤r 2 r 2 !