Asymptotic relation for zeros of cross-product of Bessel functions and applications
Vladimir Bobkov

Inferring network connectivity from event timing patterns
    Jose Casadiego^1,2, Dimitra Maoutsa^2, Marc Timme^1,2,3,4
    December 30, 2023
=============================================================




	Let a_ν,k be the k-th positive zero of the cross-product of Bessel functions J_ν(R z) Y_ν(z) - J_ν(z) Y_ν(R z), where ν≥ 0 and R>1. 
	We derive an initial value problem for a first order differential equation whose solution α(x) characterizes the limit behavior of a_ν,k in the following sense:
	
    lim_k →∞a_kx,k/k = α(x),
    	   
    	x ≥ 0.

	Moreover, we show that 
	
    a_ν,k < π k/R-1 + πν/2 R.

	We use α(x) to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.
	
	Keywords: 
	cross-product of Bessel functions, asymptotic of zeros, upper bound for zeros, Bessel functions, eigenvalues, Pleijel theorem.
	
	MSC2010: 
	33C10, 	
	33C47,	
	35P20.	




§ INTRODUCTION AND MAIN RESULTS


Consider the cross-product of the ν-th order Bessel functions of the first and second kind

    f_ν,R(z) := J_ν(R z) Y_ν(z) - J_ν(z) Y_ν(R z).

Hereinafter, we always assume that ν, z ≥ 0, and R>1. 
However, since f_ν,R(z) is even with respect to ν and z (see Appendix <ref>), and f_ν,R(z) = -f_ν, 1/R(Rz), the cases ν < 0, z < 0, and R ∈ (0,1) are also covered.
It is well-known that f_ν,R is oscillating and has infinitely many zeros all of which are simple, see the discussion in <cit.>. 
We will denote by a_ν,k the k-th positive zero of f_ν,R, k = 1,2,…

The prominent role of zeros of f_ν,R for applications (see, e.g., <cit.>) reveals through the fact that a_ν,k^2's constitute the spectrum of the Laplace operator under homogeneous Dirichlet boundary conditions in a planar annulus with the inner radius 1 and outer radius R. Below, we will use this relation and the main result of the present article to obtain a Pleijel-type result on the nodal domains statistics for the Laplace eigenfunctions in annuli <cit.>.

However, unlike the widely developed theory of zeros of Bessel functions J_ν, Y_ν, and corresponding cylinder functions (see, e.g., the surveys <cit.> and references therein), significantly less inequalities and asymptotic results are known for zeros of f_ν,R. Among known ones, the inequality of McCann <cit.> reads as

    a_ν,k≥√(a_0,k^2 + ν^2/R^2),

and the following  approximation of a_ν,k for a fixed ν of McMahon <cit.> (see also <cit.>) states that 

    a_ν,k = π k/R-1 + O(1/k).


The aim of the present article is to characterize the asymptotic behavior of a_kx,k as k →∞ for x ≥ 0 and obtain an upper bound for a_ν,k. To this end, we will use the result of Willis <cit.> who derived the following formula for the derivative of a_ν,k with respect to ν:

    da_ν,k/dν   = 2a_ν,k(J_ν^2(a_ν,k)+Y_ν^2(a_ν,k))∫_0^∞ K_0(2Ra_ν,ksinh t) e^-2ν t  dt/(J_ν^2(a_ν,k)+Y_ν^2(a_ν,k)) - (J_ν^2(Ra_ν,k)+Y_ν^2(Ra_ν,k))
       -2a_ν,k (J_ν^2(Ra_ν,k)+Y_ν^2(Ra_ν,k))∫_0^∞ K_0(2a_ν,ksinh t) e^-2ν t  dt/(J_ν^2(a_ν,k)+Y_ν^2(a_ν,k)) - (J_ν^2(Ra_ν,k)+Y_ν^2(Ra_ν,k)),

where K_0 is the modified Bessel function of the second kind and zero order.
In fact, Willis assumed that neither J_ν(a_ν,k)=J_ν(Ra_ν,k)=0 nor Y_ν(a_ν,k)=Y_ν(Ra_ν,k)=0 for a considered a_ν,k. However, checking the derivation of (<ref>), it is easy to see that this assumption is redundant and (<ref>) is valid for any a_ν,k. For the convenience of the reader, we give corresponding arguments in Appendix <ref> below.

Note that the denominators in (<ref>) are positive since J_ν^2(z)+Y_ν^2(z) decreases <cit.>.
By working with the numerators in (<ref>), Willis proved that da_ν,k/dν > 0 for any ν > 0, that is, positive zeros of f_ν,R increase with respect to ν≥ 0; see also <cit.> for further results in this direction.

Let us state our main result.

	Let R> 1. Then 
	
    lim_k →∞a_kx,k/k = α(x),
    	   x ≥ 0 ,

	where α(x) is a unique solution of the initial value problem
	
    dy/dx = 
    	arccos(x/Ry) - Ac(x/y)/R √(1 - (x/Ry)^2) - Sr(1-(x/y)^2),
    	     
    	y(0) = π/R-1,

	for x ≥ 0.
	Here, the functions Ac and Sr are zero extensions of arccos and , respectively, defined as
	
    Ac(t) = 
    	{arccos t    for     |t| ≤ 1,
    
    	0   for     |t| > 1,
    	.
    	    Sr(t) = 
    	{√(t)  for     t ≥ 0,
    
    	0   for     t < 0.
    	.



We will study some basic properties of α(x) in Section <ref> below.

Note that the differential equation (<ref>) “generalizes” the formula <cit.>

    dj_ν,κ/dν = 2 j_ν,κ∫_0^∞ K_0(2j_ν,κsinh t) e^-2ν t  dt,

where j_ν,κ, κ = k - α/π, k=1,2,…, denotes the k-th positive zero of the cylinder function

    J_ν(z) cosα - Y_ν(z) sinα,
      
    0 ≤α < π.

Using (<ref>), Elbert and Laforgia <cit.> proved that 

    lim_κ→∞j_κ x, κ/κ = ι(x),    x>-1,

where ι(x) is a unique solution of the initial value problem

    dy/dx = 
    arccos(x/y)/√(1 - (x/y)^2),
         
    y(0) = π.

In fact, ι(x) admits a closed-form representation in terms of a solution of a transcendental equation <cit.>; see also <cit.>. Our proof of Theorem <ref> is inspired by the approach of Elbert and Laforgia in combination with the formula (<ref>) of Willis. 



Theorem <ref> and the formula (<ref>) can be helpful to obtain various bounds for a_ν,k. In particular, we provide the following result.

	Let R>1, ν≥ 0, and k=1,2,…
	Then the following upper bound is satisfied:
	
    a_ν,k < π k/R-1 + πν/2 R.





Finally, we use Theorem <ref> to obtain a Pleijel-type result on the nodal domains statistics for the Laplace eigenfunctions in annuli. 
Denote by {λ_k} the increasing sequence of eigenvalues of the Dirichlet eigenvalue problem

    {
    -Δ u    = λ u        in Ω, 
    
    u   =0        on  ∂Ω,
    .

where Ω⊂ℝ^2 is a bounded domain.
Let φ_k be an eigenfunction associated with λ_k, and denote by μ(φ_k) the number of nodal domains of φ_k, that is, the number of connected components of the set Ω∖{x∈Ω: φ_k(x)=0}. The famous nodal domain theorem of Courant asserts that μ(φ_k) ≤ k for any k. Pleijel <cit.> obtained the following refinement of this fact:

    Pl(Ω) := lim sup_k →∞μ(φ_k)/k≤4/j_0,1^2 = 0.69166…

Here, Pl(Ω) is called Pleijel constant of Ω.
We refer the reader to the surveys <cit.> for the overview of results in this direction.

In connection with the conjecture of Polterovich <cit.>, there is an interesting question to determine the exact value of the Pleijel constant Pl(Ω) for particular domains, see <cit.>.
In the article <cit.>, we investigated the values and expressions of Pl(Ω) for some symmetric domains like a disk, annuli (rings), and their sectors. 
In particular, under the notation 
A_R := {x ∈ℝ^2:  1 < |x| < R}, R > 1, it was proved in <cit.> that 

    Pl(A_R) = 8/R^2-1 sup_x>0{x  lim sup_k →∞k^2/a_kx,k^2},

provided any sufficiently large eigenvalue λ_k of (<ref>) on A_R has the multiplicity at most two. In the case of a higher multiplicity (which can possibly occur, see <cit.>), Pl(A_R) is estimated by the right-hand side of (<ref>) from below. 
Combining these facts with Theorem <ref>, we obtain the following result.

	Let R>1. Then 
	
    Pl(A_R) ≥8/R^2-1 sup_x>0{x/α(x)^2},
	
	where α(x) is the solution of (<ref>) defined in Theorem <ref> (see also Remark <ref>).
	The equality in (<ref>) is satisfied if any sufficiently large eigenvalue λ_k of (<ref>) on A_R has the multiplicity at most two.


The article is structured as follows. In Section <ref>, we prove Theorem <ref>. Section <ref> is devoted to the study of properties of α(x). In Section <ref>, we prove Theorem <ref>. In Section <ref>, we provide some numerical results concerning the value of Pl(A_R) for several R>1. Finally, for the convenience of the reader, we prove some basic properties of f_ν,R in Appendix <ref>.
	
	



§ PROOF OF THEOREM <REF>


	Let us denote the right-hand side of the differential equation in (<ref>) as F, that is, 
	
    F(x,y) = 
    	arccos(x/Ry) - Ac(x/y)/R √(1 - (x/Ry)^2) - Sr(1-(x/y)^2).

	Let us also introduce the set
	
    E = {(x,y) ∈ℝ^2:  x ≥ 0,  y > x/R},

	and split E as E = E_1 ∪ E_2, where
	
    E_1 = {(x,y) ∈ℝ^2:  x > 0, x/R< y ≤ x },
    	   
    	E_2 = {(x,y) ∈ℝ^2:  x ≥ 0,  y > x }.

	
	
	We start by showing that the initial value problem (<ref>) possesses a unique solution α(x) for x ≥ 0. 
	Noting that the functions Ac and Sr defined in (<ref>) are continuous on (-1,∞) and ℝ, respectively, we see that F is continuous on E. This fact yields the existence of a solution α(x) of (<ref>) on a maximal interval I = [0,b) for some b ∈ (0, ∞]. 
	Moreover, since R>1, F is positive on E for x>0, which implies that α(x) increases and hence α(x) > α(0) = π/R-1. 
	Using this fact and the inequality
	
    F(x,y) = arccos(x/Ry)/R√(1 - (x/Ry)^2) 
    	>
    	1/R  for  
    	(x,y) ∈ E_1,

	we see that α(x) cannot meet the boundary y=x/R of E. 
	Let us now suppose that α(x) →∞ as x tends to some finite x_0>0. However, since F(x,α(x)) stays finite, α(x) also stays finite on finite x-intervals, which leads to a contradiction. Thus, we conclude that b=∞.
	To prove that α(x) is the unique solution of (<ref>) for x ≥ 0, we show that F is one-sided Lipschitz with respect to y provided (x,y) ∈ E and y > ε>0 for some ε>0. More precisely, let us show that for any ε >0 and all (x,y), (x,z) ∈ E with y,z > ε there holds
	
    (F(x,y)-F(x,z))(y-z) ≤1/Rε (y-z)^2.

	Take any ε>0 and assume, without loss of generality, that y<z for a pair (x,y), (x,z) ∈ E with y,z > ε. 
	First, recalling (<ref>), we obtain for any (x,t) ∈ E_1 with t>ε that
	
    .∂ F(x,s)/∂ s|_s=t = 
    	x/R^2 t^2 - x^2(1 - x F(x,t)/t)
    	≤x/Rt(Rt+x)≤1/Rε.
	
	Thus, if (x,y), (x,z) ∈ E_1, then, applying the mean value theorem, we deduce that F satisfies (<ref>).
	If (x,y), (x,z) ∈ E_2, then F(x,y)>F(x,z), and hence (<ref>) is automatically satisfied. 
	The validity of (<ref>) in the remaining case (x,y) ∈ E_1, (x,z) ∈ E_2 follows by combining the previous two cases. 
	Indeed, by the mean value theorem there exists t_0 ∈ [y,x] such that
	
    (F(x,y)-F(x,z))(y-z)
    	=
    	(F(x,y)-F(x,x)+F(x,x)-F(x,z))(y-z)
       ≤(F(x,y)-F(x,x))(y-z)
    	= 
    	.∂ F(x,s)/∂ s|_s=t_0 (y-z)(y-x)
    	≤1/Rε (y-z)^2.

	Therefore, the standard uniqueness theorem (see, e.g.,  <cit.>) implies that α(x) is the unique solution of (<ref>).

	Now we prove the convergence result (<ref>).
	Denote
	
    α_k(x) = a_kx,k/k  for 
    	x ≥ 0,
    	 k =1,2,…

	Since .dα_k(x)/dx = da_ν,k/dν|_ν=kx, we see from Willis' formula (<ref>) that α_k(x) is the solution of the initial value problem
	
    d y/dx = F_k(x,y), 
    	   
    	y(0) = α_k(0),

	where
	
    F_k(x,y) 
    	   = 2 k y(J_kx^2(k y)+Y_kx^2(k y))  2 k y∫_0^∞ K_0(2Rk y sinh t) e^-2 kx t  dt/2 k y (J_kx^2(k y)+Y_kx^2(k y)) - 2 k y (J_kx^2(Rk y)+Y_kx^2(Rk y))
       -2k y (J_kx^2(Rk y)+Y_kx^2(Rk y))  2 k y ∫_0^∞ K_0(2k y sinh t) e^-2 kx t  dt/2 k y(J_kx^2(k y)+Y_kx^2(k y)) - 2 k y(J_kx^2(Rk y)+Y_kx^2(Rk y)).
	
	(The additional multipliers 2ky in (<ref>) are included for the simplicity of further usage.)
	Moreover, α_k(x) is the unique solution of (<ref>) due to the continuous differentiability of F_k with respect to y>0. 
	Below, we will expand F_k further via Nicholson's formula	<cit.>
	
    J_kx^2(Rky)    + Y_kx^2(Rky) =
    	8/π^2∫_0^∞ K_0(2 Rky sinh t) cosh(2kx t)   dt 
       = 4/π^2∫_0^∞ K_0(2 Rky sinh t) e^-2kx t  dt 
    	+
    	4/π^2∫_0^∞ K_0(2 Rky sinh t) e^2kx t  dt.

	Note that the lower bound (<ref>) implies that (x, α_k(x)) ∈ E for any x ≥ 0 and k. That is, we can assume that each F_k is defined on E. 
	
	We are going to show that F_k converges to F uniformly on every compact subset of E_1 and E_2 as k →∞. Then <cit.> in combination with the uniqueness of α(x) obtained above will imply that α_k(x) →α(x) for each x ≥ 0, which is equivalent to the desired result (<ref>).
	To this end, we will show the local uniform convergence and local boundedness of integrals in (<ref>) and (<ref>).
	
	We start by performing the following trivial change of variables:
	
    2ky∫_0^∞ K_0(2Rky sinh t) e^-2 kx t  dt 
    	= 
    	1/R∫_0^∞ K_0(u  sinh(u/2Rky)/u/2Rky) e^-x/Ry u  du.

	Since sinh t > t for t>0, sinh t/t→ 1 as t → 0, and K_0(t) is decreasing and integrable, we obtain that
	
    2ky∫_0^∞ K_0(2Rky sinh t) e^-2 kx t  dt 
    	< 1/R∫_0^∞ K_0(u) e^-x/R y u  du ≤ C < ∞
	
	for all (x,y) ∈ E and k, where C>0 is a uniform constant.
	To obtain a lower bound for (<ref>), we use a Cusa-Huygens-type inequality from <cit.> which reads as
	
    sinh(u/2Rky)/u/2Rky < cosh(u/2Rky) + 2/3.

	Let us introduce u_k > 0 such that 
	
    cosh(u_k/2Rky) + 2/3 = 1 + 1/k.

	Using the series expansion of arccosh  t, we get
	
    u_k = 2Rky  arccosh(1+3/k) = y   O(√(k)) →∞  as 
    	k →∞.

	Therefore, using the monotonicity of K_0(t) and cosh t for t>0, we deduce that
	
    ∫_0^∞ K_0(u  sinh(u/2Rky)/u/2Rky) e^-x/Ry u  du ≥∫_0^u_k K_0(u (1+1/k)) e^-x/Ry u  du.

	Since by <cit.>, 
	
    K_0(t) ∼√(%s/%s)π2t  e^-t  as  t →∞,

	we see that
	
    ∫_u_k^∞ K_0(u (1+1/k)) e^-x/Ry u  du → 0
    	  as  k →∞

	locally uniformly on E.
	(Note that the locality comes from the fact that u_k depends on y.)
	Thus, 
	
    ∫_0^u_k K_0(u (1+1/k)) e^-x/Ry u  du 
    	→∫_0^∞ K_0(u) e^-x/R y u  du 
    	  as  k →∞

	and hence
	
    2ky∫_0^∞ K_0(2Rky sinh t) e^-2 kx t  dt 
    	→1/R∫_0^∞ K_0(u) e^-x/R y u  du 
    	  as  k →∞

	locally uniformly on E, and the left-hand side of (<ref>) is bounded on E, see (<ref>). 
	Clearly, the same results hold true for the integral 2ky∫_0^∞ K_0(2ky sinh t) e^-2 kx t  dt.
	
	Analogously, recalling that x/Ry < 1 for (x,y) ∈ E, 
	we deduce from (<ref>) that 
	
    ∫_u_k^∞ K_0(u (1+1/k)) e^x/Ry u  du → 0
    	  as  k →∞

	and hence
	
    2ky∫_0^∞ K_0(2Rky sinh t) e^2 kx t  dt 
    	→1/R∫_0^∞ K_0(u) e^x/R y u  du 
    	  as  k →∞

	locally uniformly on E. 
	Moreover, the right-hand side of (<ref>) is locally bounded on E, and the same holds true for the left-hand side, cf. (<ref>).
	
	In its turn, the treatment of the remaining integral 2ky∫_0^∞ K_0(2ky sinh t) e^2 kx t  dt already depends on the choice of E_1 or E_2.
	Indeed, since x/y<1 on E_2, we see, as above, that 
	
    2ky∫_0^∞ K_0(2ky sinh t) e^2 kx t  dt 
    	→∫_0^∞ K_0(u) e^x/y u  du 
    	  as  k →∞

	locally uniformly on E_2, and the right-hand side (and hence the left-hand side) of (<ref>) is locally bounded away from zero and infinity on E_2.
	On the other hand, using (<ref>), we deduce as in (<ref>) that 
	
    2ky∫_0^∞ K_0(2ky sinh t) e^2 kx t  dt 
    	≥∫_0^u_k K_0(u (1+1/k)) e^x/y u  du
    	→∞

	as k →∞ locally uniformly on E_1. 
	
	Let us now rewrite F_k as F_k = G_k/H_k, where
	
    G_k(x,y) 
    	   := 2 k y ∫_0^∞ K_0(2Rk y sinh t) e^-2 kx t  dt 
       - 2 k y ∫_0^∞ K_0(2Rk y sinh t) e^2 kx t  dt · 2 k y ∫_0^∞ K_0(2k y sinh t) e^-2 kx t  dt/2 k y ∫_0^∞ K_0(2k y sinh t) e^2 kx t  dt

	and
	
    H_k(x,y)
    	   := 1 + 2 k y ∫_0^∞ K_0(2k y sinh t) e^-2 kx t  dt/2 k y ∫_0^∞ K_0(2k y sinh t) e^2 kx t  dt
       - 2 k y ∫_0^∞ K_0(2Rk y sinh t) e^2 kx t  dt/2 k y ∫_0^∞ K_0(2k y sinh t) e^2 kx t  dt
    	- 2 k y ∫_0^∞ K_0(2Rk y sinh t) e^-2 kx t  dt/2 k y ∫_0^∞ K_0(2k y sinh t) e^2 kx t  dt.

	Recall that if some functional sequences {g_k} and {h_k} are locally bounded and locally uniformly convergent to g and h, respectively, then g_k h_k → g h locally uniformly. Using this fact and the local uniform convergence and local boundedness of the integrals in (<ref>), (<ref>), and (<ref>) or (<ref>), we see that
	
    F_k(x,y) →1/R∫_0^∞ K_0(u) e^-x/R y u  du
    	  as  k →∞

	locally uniformly on E_1, and F_k →G/H locally uniformly on E_2, where
	
    G(x,y) 
    	:= 1/R∫_0^∞ K_0(u) e^-x/R y u  du
    	- 1/R∫_0^∞ K_0(u) e^x/R y u  du ·∫_0^∞ K_0(u) e^-x/y u  du/∫_0^∞ K_0(u) e^x/y u  du

	and
	
    H(x,y)
    	:= 1 
    	+ ∫_0^∞ K_0(u) e^-x/y u  du/∫_0^∞ K_0(u) e^x/y u  du 
    	- 1/R∫_0^∞ K_0(u) e^x/Ry u  du/∫_0^∞ K_0(u) e^x/y u  du
    	- 1/R∫_0^∞ K_0(u) e^-x/Ry u  du/∫_0^∞ K_0(u) e^x/y u  du.

	Simplifying the above expressions via the formula <cit.>
	
    ∫_0^∞ K_0(u) e^-a u  du = arccos a/√(1-a^2),
    	   
    	|a| < 1,
	
	we conclude that F_k converges locally uniformly on E_1 and E_2 to F.
	
	As an auxiliary fact which will be used also in Section <ref> below, let us note that the following inequality is satisfied:
	
    F_k(x,y) < arccos(x/Ry)/R√(1 - (x/Ry)^2),
    	   (x,y) ∈ E,
    	 k=1,2,…

	To obtain (<ref>), it is enough to recall that K_0(t) decreases for t>0, which implies that 
	
    ∫_0^∞ K_0(2k y sinh t) e^-2 kx t  dt 
    	>
    	∫_0^∞ K_0(2Rk y sinh t) e^-2 kx t  dt

	for any y>0 and k=1,2,… 
	Recalling also that the denominators in (<ref>) are positive, we deduce from (<ref>) that
	
    F_k(x,y) < 2ky ∫_0^∞ K_0(2 R k y sinh t) e^-2 kx t  dt.

	Therefore, using (<ref>) and (<ref>), we obtain (<ref>). 
	Finally, let us show that the obtained local uniform convergence of F_k to F implies the convergence result (<ref>). 
	Note that (0,α_k(0)) ∈ E_2 for all k, and  (<ref>) for ν=0 reads as α_k(0) →α(0).
	Therefore, applying <cit.> (with minor modifications) on E_2 and recalling that α(x) is the unique solution of (<ref>), we deduce that α_k(x) →α(x) for any x ∈ [0,x_0), where x_0 defines the right maximal interval of applicability of <cit.>. If x_0 = ∞, then we are done. If x_0 < ∞, then the only possibility is that x_0 = α(x_0), that is, (x_0, α(x_0)) ∈∂ E_2 ∩∂ E_1. 
	Since 
	arccos t = arctan(√(1-t^2)/t) < √(1-t^2)/t for t ∈ (0,1), we get
	
    F(x,y) = arccos(x/Ry)/R√(1 - (x/Ry)^2) 
    	< y/x≤ 1
    	  for  
    	(x,y) ∈ E_1.

	Thus, we see that α(x) < x and hence (x, α(x)) ∈ E_1 for all x>x_0.
	Moreover, in view of the inequalities (<ref>) and (<ref>), α_k(x) < x for any x>x_0 and sufficiently large k. 
	Fixing x_1>x_0 and applying <cit.> on E_1, we can extract a subsequence {α_k_n(x)} which converges locally uniformly on the maximal interval (x_0, x_2) to a solution α̃(x) of the differential equation in  (<ref>) with the initial value α̃(x_1) = lim_n→∞α_k_n(x_1). By continuity, α(x_0) = α̃(x_0), and the uniqueness of α(x) yields α(x) = α̃(x) for all x ∈ (x_0,x_2). Moreover, x_2=∞, as it follows from (<ref>) and (<ref>). 
	Thus, we conclude that α_k(x) →α(x) for any x ≥ 0.	




§ PROPERTIES OF A(X)


We start with auxiliary results which will be used to obtain upper bounds for α(x).
We use the notations F_k, F, and E from Section <ref>. 
Let us note that (<ref>) and the local uniform convergence of F_k to F proved in Section <ref> imply

    F(x,y) ≤arccos(x/Ry)/R√(1 - (x/Ry)^2),
       (x,y) ∈ E.


Let now ι̃(x) be a unique solution of the initial value problem

    dy/dx = 
    arccos(x/y)/√(1 - (x/y)^2),
         
    y(0) = π R/R-1,

for x > 0. 
The uniqueness of ι̃(x) follows from the fact that the right-hand side of the differential equation in (<ref>) is Lipschitz provided y > ε for some ε > 0. 
Note that, ι̃(x) can be expressed as a solution of the following system (see also <cit.>):

    ι̃(x) = x/sinα,
        sinα/cosα - (π/2-α) sinα = (R-1)x/π R,
        α∈(0, π/2),

where the second equation can be equivalently written as

    α - (π/2-α) = π R/(R-1)x.

The left-hand side of (<ref>) tends to +∞ as α→ 0+, tends to 0 as α→π/2, and decreases on (0, π/2). Thus, for any x > 0 there exists a unique α∈ (0,π/2) satisfying (<ref>), and hence ι̃(x) is also determined. Moreover, in view of the monotonicity of the right-hand side of (<ref>), we conclude that α = α(x) is increasing, which yields α'(x) ≥ 0.

It is not hard to see that ι̃(x) is increasing and concave. Indeed, substituting ι̃(x) = x/sinα(x) into (<ref>), we get

    dι̃(x)/dx = π/2-α(x)/cosα(x) > 0.

Taking the second derivative of ι̃(x) and recalling that α'(x) ≥ 0, we easily conclude that ι̃”(x) is negative, and hence ι̃(x) is concave.
By the concavity, 

    ι̃(x) ≤ι̃(0) + ι̃'(0) x 
    = 
    π R/R-1 + π x/2.


Now we are ready to collect some basic properties of α(x). 

	Let R>1 and let α(x) be the solution of (<ref>). Then α(x) is increasing and
	
    √(π^2/(R-1)^2 + x^2/R^2) < α(x) < ι̃(x)/R≤π/R-1 + π x/2 R,
    	  
    	x>0,

	where ι̃(x) is the solution of (<ref>). 


	The monotonicity of α(x) is a consequence of the positivity of F on E. Recalling that the right-hand side of (<ref>) is Lipschitz provided y > ε for some ε > 0, the first upper bound in (<ref>) follows from (<ref>) by noting that for sufficiently small x ≥ 0 the inequality in (<ref>) is strict. The last inequality in (<ref>) follows from (<ref>).
	
	To obtain a lower bound for α(x) we note that
	
    arccos(z/R) - arccos(z)
    	=
    	arccos(z^2/R+ √(1 - (z/R)^2)√(1 - z^2)), 
    	   
    	0<z<1.

	Applying now the inequality arccos t > √(1-t^2) for t ∈ [0,1), we get
	
    arccos(z/R) - arccos(z)/R √(1 - (z/R)^2) - √(1 - z^2) > z/R > z/R^2,
    	   
    	0<z<1.

	On the other hand,
	
    arccos(z/R)/R √(1 - (z/R)^2)→1/R  as 
    	z → R-,

	and the left-hand side of (<ref>) decreases on (1,R).
	Therefore, we deduce that
	
    arccos(z/R) - Ac(z)/R √(1 - (z/R)^2) - Sr(1-z^2) > z/R^2,
    	   
    	0 < z < R,

	which implies that F(x,y) > x/R^2 y for (x,y) ∈ E, x>0, and hence the lower bound for α(x) in (<ref>) follows.



	The value of α = α(x) can be also found as a unique solution of the transcendental equation
	
    R α√(1 - (x/R α)^2) - α√(1 - (x/α)^2)
    	=
    	x arccos(x/R α) - x arccos(x/α) + π

	for x ∈[0, x_0], and
	
    R α√(1 - (x/R α)^2) - x_0 √(R^2-1) 
    	=
    	x arccos(x/R α) - x_0 arccos(1/R)

	for x ∈[x_0, +∞), 
	where
	
    x_0 := π/√(R^2-1) -arccos(1/R).

	In fact, knowing that the value lim_k →∞a_kx,k/k exists by Theorem <ref>, one can obtain (<ref>) using the leading terms of trigonometric Debye asymptotics of Bessel functions J_ν and Y_ν (see <cit.>). On the other hand, since α(x_0) = x_0, we have α(x) < x for all x > x_0 (see (<ref>)), and the equation (<ref>) can be obtained by integrating (<ref>) using, e.g., the substitute y = x/sinγ; see <cit.>.





§ THE UPPER BOUND FOR A

Let us turn to the proof of the upper bound (<ref>) of Theorem <ref>. 
To this end, consider the eigenvalue problem

    {
    -(r u')'    = μ r u,    r ∈ (1,R), 
    
    u(1)    = u(R) =0.
    .

It can be checked that the k-th eigenvalue μ_k of (<ref>) is equal to a_0,k^2, and

    ψ_k(r) = J_0(a_0,k r) Y_0(a_0,k) - J_0(a_0,k) Y_0(a_0,k r)

is the unique (modulo scaling) eigenfunction associated with μ_k. 
Moreover, ψ_k has exactly k nodal domains. 
Let us denote r_0=1 and r_k=R, for convenience. Then μ_k can be characterized as

    μ_k = min_r_0<r_1<…<r_k-1<r_kmax{μ_1^(r_0,r_1), …,
    μ_1^(r_k-1,r_k)},

see <cit.> with minor modifications.
Here μ_1^(r_i,r_i+1) is the first eigenvalue of (<ref>) on (r_i,r_i+1), that is,

    μ_1^(r_i,r_i+1) = inf_u ∈ W_0^1,2(r_i,r_i+1) ∖{0}∫_r_i^r_i+1 r |u'|^2   dr/∫_r_i^r_i+1 r |u|^2   dr.


The proof of Theorem <ref> will be based on the upper estimate (<ref>) for F_k and the following fact.

	Let R>1. Then for any k=1,2,… the following inequality is satisfied:
	
    a_0,k < π k/R-1.



	We use the characterization (<ref>) to estimate μ_k = a_0,k^2 from above. 
	As an admissible function for each μ_1^(r_i,r_i+1) we use
	
    v(r) = cos(π k r/R-1) - (π k/R-1) sin(π k r/R-1)

	whenever k/R-1∉ℕ, and 
	
    v(r) = tan(π k/R-1)cos(π k r/R-1) - sin(π k r/R-1)

	otherwise.
	Denoting r_i = 1 + (R-1)i/k, we see that v ∈ W_0^1,2(r_i, r_i+1) ∖{0} and 
	
    μ_1^(r_i,r_i+1)≤∫_r_i^r_i+1 r |v'|^2   dr/∫_r_i^r_i+1 r |v|^2   dr 
    	= 
    	π^2 k^2/(R-1)^2,    i=0,…,k-1.

	Recalling that ψ_k satisfies (<ref>) and v satisfies  -v” = π^2 k^2/(R-1)^2 v, we deduce that ψ_k and v are linearly independent on any (r_i,r_i+1).
	Thus, since each μ_1^(r_i,r_i+1) possesses a unique minimizer (modulo scaling), we conclude that the strict inequality in (<ref>) holds true, and hence (<ref>) implies (<ref>).


	Let us now prove Theorem <ref>.

	Under the notation (<ref>), the inequality (<ref>) reads as α_k(0) < π/R-1. Using this fact and the inequality (<ref>), we conclude that 
	
    α_k(x) < ι̃(x)/R

	where ι̃(x) is the solution of (<ref>),
	which reads as
	
    a_ν,k < ι̃(ν/k)k/R.

	Applying the inequality (<ref>), we obtain the desired upper bound (<ref>).



	One can derive other upper bounds for a_ν,k from (<ref>) by estimating ι̃(x) from above in a different way than (<ref>).




§ PLEIJEL'S CONSTANT FOR ANNULI


For each fixed R>1, one can numerically solve the initial value problem (<ref>) (or equations (<ref>) and (<ref>)) to obtain α(x) and hence compute the value

    8/R^2-1 sup_x>0{x/α(x)^2}

from Proposition <ref>. 
Using a build-in ODE-solver of Mathematica, we obtained several approximate values of (<ref>) listed in Table <ref>. Recall that if the multiplicity of any sufficiently large eigenvalue of (<ref>) on A_R is at most two, then (<ref>) gives an exact value of the Pleijel constant Pl(A_R). 


The values from Table <ref> suggest that the following asymptotics are satisfied (see <cit.>):

    lim_R → 1[8/R^2-1 sup_x>0{x/α(x)^2}] 
       = 
    Pl(ℛ) = 2/π = 0.6366197…, 
    
    lim_R →∞[8/R^2-1 sup_x>0{x/α(x)^2}] 
       = 
    Pl(B) = 0.4613019…,

and (<ref>) lies in between these two values. 
Here ℛ is any rectangle (0,a) × (0,b) with irrational ratio a^2/b^2 (see <cit.>), and B is a planar disk (see <cit.>).

It is also informative to compare the values from Table <ref> with the behavior of the ratio μ(φ_k)/k for large k, where φ_k is the k-th eigenfunction of (<ref>) on A_R of the form

    φ_n(r,θ) = (J_ν(a_ν,n r) Y_ν(a_ν,n) - J_ν(a_ν,n) Y_ν(a_ν,n r) )cos(νθ),

for some n = 1,2,… and ν = 0,1,…, where r ∈ (1,R) and θ∈ [0,2π). 
We present the corresponding plots on Figures <ref> and <ref>.








§ APPENDIX



	Let ν,z ∈ℝ and R>0. Then f_ν,R(z) is even with respect to ν and z.


	Evenness of f_ν,R(z) with respect to ν follows by applying the equalities
	
    J_-ν(z) = (-1)^ν J_ν(z)
    	  and  
    	Y_-ν(z) = (-1)^ν Y_ν(z),
    	  provided ν∈ℤ,

	and from the relation
	
    Y_ν(z) = J_ν(z) cos (νπ) - J_-ν(z)/sin (νπ),
    	  provided ν∉ℤ.

	Evenness of f_ν,R(z) with respect to z follows from the relations <cit.>
	
    J_ν(-z)=e^iπνJ_ν(z)
    	  and  
    	Y_ν(-z)=e^-iπνY_ν(z) + 2i cos (νπ) J_ν(z),
    	   z ∈ℂ.




	Any zero a_ν,k of f_ν,R satisfies (<ref>).


	Let us take any zero z := a_ν,k.
	If neither J_ν(z)=J_ν(Rz)=0 nor Y_ν(z)=Y_ν(Rz)=0, then (<ref>) is proved in <cit.>. 
	Assume that J_ν(z)=J_ν(Rz)=0. 
	Note that zeros of J_ν and Y_ν are interlacing, which implies that Y_ν(z), Y_ν(Rz) ≠ 0. 
	Therefore, we can rewrite f_ν,R(z)=0 in the form
	
    J_ν(z)/Y_ν(z) - J_ν(Rz)/Y_ν(Rz) = 0.

	Recalling that any zero of f_ν,R is simple (see, e.g., <cit.>), the rate of change of z=z(ν) with respect to ν can be found from the total derivative
	
    .∂/∂ x(J_ν(x)/Y_ν(x) - J_ν(Rx)/Y_ν(Rx))|_x=zdz/dν + 
    	.∂/∂ν(J_ν(x)/Y_ν(x) - J_ν(Rx)/Y_ν(Rx))|_x=z = 0.

	Then, using the relations <cit.> and <cit.>, we formally derive
	
    dz/dν = 2z/(Y_ν^2(z)/Y_ν^2(Rz)-1)(Y_ν^2(z)/Y_ν^2(Rz)∫_0^∞ K_0(2Rz sinh t) e^-2ν t  dt - ∫_0^∞ K_0(2z sinh t) e^-2ν t  dt).

	Finally, recalling that J_ν(z)=J_ν(Rz)=0, R>1, and that J_ν^2(x) + Y_ν^2(x) decreases with respect to x>0 <cit.>, we have
	
    Y_ν^2(z)/Y_ν^2(Rz) = J_ν^2(z) + Y_ν^2(z)/J_ν^2(Rz) + Y_ν^2(Rz) > 1,

	and hence (<ref>) follows.
	The proof of (<ref>) in the case Y_ν(z)=Y_ν(Rz)=0 can be handled in much the same way as above; see also <cit.>.



Acknowledgements.
This research has been supported by the project 18-03253S of the Grant Agency of the Czech Republic, and by the project LO1506 of the Czech Ministry of Education, Youth and Sports.
The author thanks the anonymous colleague who pointed out the validity of equation (<ref>).
Moreover, the author is grateful to the anonymous referee for constructive remarks and valuable suggestions which helped to improve the manuscript.



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