Boundary value problems for a special Helfrich functional for surfaces of revolution: existence and asymptotic behaviour

The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor ε≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ge 0$$\end{document}. We collect several results concerning the existence of solutions to a Dirichlet boundary value problem for Helfrich surfaces of revolution and cover some specific regimes of boundary conditions and weight factors ε≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ge 0$$\end{document}. These results are obtained with the help of different techniques like an energy method, gluing techniques and the use of the implicit function theorem close to Helfrich cylinders. In particular, concerning the regime of boundary values, where a catenoid exists as a global minimiser of the area functional, existence of minimisers of the Helfrich functional is established for all weight factors ε≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ge 0$$\end{document}. For the singular limit of weight factors ε↗∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varepsilon \nearrow \infty $$\end{document} they converge uniformly to the catenoid which minimises the surface area in the class of surfaces of revolution.


Introduction
The Helfrich energy for a sufficiently smooth (two dimensional) surface S ⊂ R 3 (with or without boundary), introduced by Helfrich in [27] and Canham in [8], is defined by Here the integration is done with respect to the usual 2-dimensional area measure dS, H is the mean curvature of S, i. e. the mean value of the principal curvatures, K is the Gaussian curvature, γ ∈ R is a constant bending rigidity, ε ≥ 0 the weight factor of the area functional and H 0 ∈ R a given spontaneous curvature. For simplicity we will always assume H 0 = 0 so that the first term in the above energy reduces to the well-known Willmore functional for a given boundary value α > 0 and for profile curves u that are even, i. e. u(x) = u(−x). The precise setting and a derivation of (1) will be given in Sect. 2. First existence results for (1) have been obtained in [43] and [18]. The goal of this paper is to extend these results in several directions including a description of the qualitative behaviour of solutions when possible. Depending on the two parameters α and ε, Fig. 1 gives an overview of domains on which existence of solutions to the Helfrich problem (1) is guaranteed. Let us describe these domains in more detail and relate them to the various sections of this paper. As a starting point we consider in Sect. 3 the Helfrich boundary value problem as a regular perturbation of the Willmore boundary value problem for small ε. Note however that a direct application of the techniques developed in [12] for the Willmore functional is not possible because they are heavily based on its conformal invariance and on explicitly known Willmore minimisers. Nevertheless, by applying the direct method and utilising an energy gap of the Willmore energy to 4π we find in Theorem 2 solutions of (1) for any α > 0 and ε ≥ 0 not too large, see the blue regions in Fig. 1. In this case we solely give a quantitative existence proof with no qualitative description of the minimisers.
In order to develop some intuition for what happens when ε increases we then concentrate in Sect. 4 on the local picture around a Helfrich cylinder, i.e. u(x) ≡ α, which is an explicit solution and the unique Helfrich minimiser in the case ε = ε α := 1 4α 2 (see [43,Lemma 4.1]; red curve in Fig. 1). Linearising (1) for fixed α at the corresponding Helfrich cylinder, the implicit function theorem yields the existence of a smooth family (u ε ) ε≈ε α of solutions of (1), see Theorem 3; orange domain in Fig. 1. As a first step in order to understand whether and how this local branch is part of a possibly existing family of solutions (u ε ) ε≥0 we consider the rate of change function rc α := ∂u ε ∂ε | ε=ε α , for which we derive an explicit formula. A careful analysis of the resulting expression shows that rc α is negative, see Theorem 4, which gives rise to the (open) conjecture that the family (u ε ) ε∈[0,ε α ] exists and is decreasing with respect to ε. Furthermore, we find that the shape of rc α strongly depends on the boundary value α. More precisely, there exists a critical value α crit ≈ 0.18008 such that for α > α crit the function rc α is strictly increasing on [0, 1] (see Theorem 5), while for α < α crit and α 0 an increasing number of oscillations around a negative value show up. This interesting phenomenon of "overshoot" and subsequent oscillations around an expected "attracting state" has often been observed in numerical experiments and to our knowledge this is the first time where an analytical proof is given. It indicates that for small α > 0, understanding existence, qualitative properties and asymptotic behaviour for the full range of ε ∈ [0, ∞) remains an interesting and challenging open problem.
Next, in Sect. 5 we consider the case of larger values of ε. This means that the area functional as part of the Helfrich energy is given a bigger weight suggesting that minimisers of the area functional in the class of surfaces of revolution might be useful when minimising the Helfrich funtional in this case. It is well-known that there exists a threshold α m ≈ 1.805 such that for α > α m the minimiser is given by a catenoid while for α < α m it is a so-called Goldschmidt solution. Using this observation in the case α ≥ α m we are able to decrease the energy of minimising sequences for the Helfrich functional by gluing in suitable catenoids leading to the existence of minimisers, see Theorem 7 and the purple region in Fig. 1. At the same time this process gives some qualitative information about these minimisers. In addition, we prove in Corollary 3 that the sequence of minimisers converges to the areaminimising catenoid in the singular limit ε ∞. So far, there is only numerical evidence that for 0 < α < α m minimisers may exist for any ε ≥ 0 and converge in some singular sense to the Goldschmidt minimal surface as ε ∞. In Sect. 6 we finally summarise our results and describe some open problems. At the end we add three appendices. Appendix A provides a proof for the "well-known" fact, which catenoid minimises the area functional. Appendix B proves a divergence form of the Helfrich equation and Appendix C collects the basic estimates to prove the theorems of Sect. 4.

Geometric background for surfaces of revolution
For a sufficiently smooth (two dimensional) surface S ⊂ R 3 (with or without boundary) we consider the Helfrich functional (see e.g. [27,34]) Here dS denotes integration with respect to the surface area measure, while are the mean curvature and the Gauss curvature of S respectively. Furthermore, γ ∈ R as well as ε ≥ 0 are given constants. In what follows we consider surfaces of revolution S which arise when the sufficiently smooth profile curve u : [−1, 1] → (0, ∞) is rotated around the x-axis. Expressing H and K in terms of u yields Since dS = u(x) 1 + u (x) 2 dx dθ we have for the surface area while the Willmore functional is given by In what follows we shall consider for α > 0 the following class of admissible functions: (5) so that in our setting the Helfrich functional H = H ε : N α → R takes the form Using the fact that In particular we have for u ∈ N α that Let us next consider the Euler-Lagrange equation for H ε and fix u ∈ N α ∩ C 4 ([−1, 1]). Then we calculate for the first variation of A at u in direction ϕ ∈ H 2 0 (−1, 1) Combining (9) and (10) we see that if u ∈ N α ∩ C 4 ([−1, 1]) is a critical point of the energy H ε then it is a solution of Helfrich equation Using the calculations in [12,Section 2.1] in order to express the left hand side in terms of u we obtain that u is a solution of the following Dirichlet problem Note that (12) is a fourth order quasilinear equation, which is elliptic, but not uniformly elliptic. Let us briefly refer to three particular solutions of (12) in the class of symmetric, positive profile curves: in the case ε = 0; iii. cylinders: u(x) = c > 0 in the case ε = 1 4c 2 . Catenoids and spheres have been successfully employed in the construction of symmetric Willmore surfaces (see e.g. [12], [14]) as well as in the analysis of their asymptotic shape for small values of α, see [26].

Existence of minimisers via energy bounds
Proof a) Using Hölder's inequality and (7) we have Since u (0) = u (1) = 0 we calculate with the help of (7) and the required bound follows by solving for |u which implies the lower bound on u. A similar idea was used in [14,Lemma 4.9]. The upper bound is immediate.
Proof Lemma 1 a) implies that and hence which yields (14).
since H ε (v) < π(4+ε) and ε ≤ 4. Thus there exist δ > 0, k 0 ∈ N such that W (u k ) ≤ 4π −δ for all k ≥ k 0 . Lemma 1 then implies that (u k ) k∈N is bounded in C 1 ([−1, 1]) and that 0 < ≤ u k (x) ≤ 1 for a suitable > 0. Then one sees from (7) that a bound for W (u k ) also yields an L 2 -bound for u k k∈N . Arguing as in the proof of [12, Theorem 3.9] we obtain a subsequence, again denoted by (u k ) k∈N , and u ∈ N α such that and it is easily seen that u is a minimiser of H ε . The fact that u belongs to C ∞ ([−1, 1]) can be shown by a straightforward adaptation of the corresponding argument in the proof of [12,Theorem 3.9].
In order to apply Theorem 1 we look for suitable functions v ∈ N α such that H ε (v) < π(4 + ε). For the simple choice v ≡ α we find that As a second useful comparison function we define (see [23,Lemma 8.6]) is chosen in such a way that This condition means that the normal to the cosh-function through the point (x 0 , v(x 0 )) intersects the x-axis in the origin. One may observe that is strictly increasing on [0, 1], strictly negative for x = 1 2 and strictly positive for x = 1. The choice of x 0 and r ensures that v ∈ N α and that r ≤ √ 1 + α 2 . Since H = 0 for x 0 ≤ |x| ≤ 1, we obtain with the help of (7) that From the above calculations we infer the following existence result: The functional H ε attains a minimum on N α , if one of the following conditions is satisfied: Proof The result follows from (15), (16) and Theorem 1.

Perturbation of Helfrich cylinders
Throughout this section we fix for a given α > 0 ε α := 1 4α 2 as the parameter where the Helfrich cylinder

Recalling (12) and taking into account that
Proof We apply the implicit function theorem as it can be found in [17,Theorem 15.1].
In view of (17) we obtain for L : We claim that L is injective. To see this, consider the boundary value problem The roots of the characteristic equation for the differential equation in (18) are given by ± 1+i so that the functions > 0. Using the above expressions for ϕ j we find that and a calculation shows that det(M) = −4 sinh 2 (β) cosh 2 (β) + 4 sin 2 (β) cos 2 (β) < 0.
Hence b 1 = · · · = b 4 = 0, which yields ϕ ≡ 0, so that L is injective. Furthermore, since L allows for an elliptic theory, i. e. L is Fredholm of index 0, we infer that L is invertible with bounded inverse. Hence we can apply the implicit function theorem to obtain solutions u ε ∈ N α ∩ C 4 ([−1, 1]) for ε close to ε 0 . By arguing again as in the proof of Theorem 1 we also have u ε ∈ C ∞ ([−1, 1]).
To study the behaviour of u ε for ε close to ε α we introduce Since d dε F(ε, u ε ) |ε=ε α = 0 we find with the help of (17) that giving rise to the following boundary value problem for rc α : This problem is solved by (cf. Fig. 2) We have: The statement then follows since equality only holds at x = ±1.
Theorem 5 Let a c ∈ (π, 3 2 π) denote the smallest, strictly positive solution of the equation tanh(x) = tan(x) (see Lemma 6 in Appendix B) and α crit = 1 It follows from Lemma 7 in Appendix C with a = β that for every x ∈ (0, 1)

Remark 1
The end of the proof of Lemma 7 shows that for α ∈ (0, α crit ) and α 0, we observe an increasing number of sign changes of rc α .

Existence of minimisers via gluing techniques
We start with a discussion of symmetric positive profile curves satisfying The corresponding solutions are the catenaries with boundary values c cosh( 1 c ) and surface area where c 0 ≈ 0.8336 is the positive solution of the equation if α = α 0 and no solution for α < α 0 . Although presumably folklore and asymptotically obvious for α → ∞, we could not locate an easy reference for the following statement which is quite important in what follows: For the reader's convenience we give a proof in Appendix A.
In what follows we shall write In particular it follows from (24) that In order to make the role of v α in the minimisation of A more precise it is convenient to relax the class of profile curves and consider for a given boundary value α > 0 where the area functional is now given by In order to formulate the main result concerning the minimisation of A over C α we introduce for α > 0 the Goldschmidt solution γ α as a C 0,1 parametrisation of the polygon P − Q − Q + P + , where P ± = (±1, α), Q ± = (±1, 0). As a geometric object the Goldschmidt solution corresponds to two disks with radius α and centers Q − , Q + and its surface area is given by We use the above result in order to prove the following lemma.
for all x ∈ (0, 1] and H ε (v) ≤ H ε (u). If u has finitely many critical points, then the same holds for v. In particular we have that v(x) > v α (x) for all x ∈ (−1, 1). Clearly, Suppose thatĉ = c α . Then, in view of (26) and (29) we have that v α (x) ≥ u(x) for x ∈ [x 0 , 1] and hence v α (1) ≤ u (1) = 0, a contradiction, so thatĉ > c α . Assume next that x 1 ∈ {x 0 , 1}. Using again (26), (29) as well as (28) we infer that which is in either case a contradiction toĉ > c α . As a result we deduce that Fig. 3) then belongs to N α . Since the catenoid given by wĉ has zero mean curvature we have and Theorem 6 implies that wĉ /x 1 is a global minimum for the area functional in the class Cα . We see from the construction that v has finitely many critical points if this was the case for u. Finally, the inequality v(x) > v α (x), x ∈ (−1, 1) easily follows by integration.
and in particular v is positive on and therefore recalling (26) Combining the above inequality with the fact that z → 4εz Furthermore, in view of the definition of v and the fact that If we insert (33) and (34) into (8) a] has less critical points than u. If v (x 1 ) < 0 for some x 1 ∈ (0, a) we can repeat the above procedure on [−a, a] until we obtain a function with the desired properties in finitely many steps. (32) from below. There the condition on α and ε is given by
Let us next investigate the behaviour of minimisers as ε → ∞.
In order to identify u we claim that u is a minimiser of A in the class To see this, let v ∈ C α and fix 0 < δ ≤ 1 2 min [−1,1] v. Since v − α ∈ H 1 0 (−1, 1) there exists ζ δ ∈ C ∞ 0 (−1, 1) (i.e. smooth and compactly supported in (−1, 1)), which is even and Sending k → ∞ we infer with the help of the sequential weak lower semicontinuity of A in H 1 (−1, 1) that Since δ can be chosen arbitrarily small we deduce that A (u) = min v∈C α A (v) and hence Note that the above relation is first obtained for all even ϕ ∈ C ∞ 0 (−1, 1) and then extended to arbitrary ϕ ∈ C ∞ 0 (−1, 1) using a splitting into an even and an odd part. Hence u is a weak solution of (22) and it is not difficult to see that u ∈ C 2 ([−1, 1]), so that u ≡ w c 1 (α) or u ≡ w c 2 (α) . Recalling that A (w c 1 (α) ) < A (w c 2 (α) ) we deduce that u = w c 1 (α) = v α and hence (u k ) k∈N converges uniformly to v α . Furthermore, recalling that 0 This implies that u k → v α in L p (−1, 1) for all 1 ≤ p < ∞ since (u k ) k∈N is uniformly bounded. A standard argument then shows that the whole sequence converges to v α in W 1, p (−1, 1) for every 1 ≤ p < ∞.

Summary and outlook
In this article we studied the Dirichlet problem (12,13) for Helfrich surfaces of revolution depending on the parameters α > 0 for the boundary value and the weight factor ε ≥ 0. Fig. 1 in the introduction gives an overview over the existence results for solutions and in particular minimisers of the Helfrich functional.
According to Corollary 2 we have the existence of minimisers of H ε in N α ∩C ∞ ([−1, 1]) for all α ≥ α m and for all ε ≥ 0. On the other hand, with Theorem 2(ii) we can ensure that for every α > 0 and ε ≥ 0 less than a sufficiently small ε 0 (α) there exists a Helfrich minimiser. Our hope was to be able to prove existence of minimisers also for α < α m and for all ε ≥ 0. This turned out to be a hard task (too hard for us) because minimisers tend to develop oscillations being created around x = 0 and therefore gluing techniques do not seem to work. A first attempt to study this domain of parameters was done in Sect. 4. In particular Theorem 5, the analysis in Appendix C and Fig. 2 show that for α < α crit ≈ 0.18008 oscillatory behaviour of the rate of change function occurs, leading us to anticipate a similar behaviour for minimisers in the vicinity of the Helfrich cylinders. We conjecture that for any α < α m there will be someε α such that for ε >ε α (except possibly ε α ) all minimisers show oscillatory behaviour around the cylinder x → 1 2 √ ε . Theorem 5 and Lemma 7 indicate that for α ∈ (0, α crit ) one may expect that evenε α < ε α . This expectation is supported by numerical experiments.
The benefit of using gluing techniques like those we used in Sect. 5 is that we obtain qualitative results for the minimisers. In Theorem 7 we found that for α > α m and ε >ε α the minimisers lie below the Helfrich cylinder, i. e. its boundary value at α. We expect this to be true for all α > 0 and ε > ε α . On the other hand we do not know anything for the regime 0 < ε < ε α other than the linearisation results from Sect. 4 which indicate that minimisers start to grow as ε decreases. Thus we expect for these minimisers to lie above the Helfrich cylinder and possibly below the Willmore minimisers.
As for α we conclude from the strict monotonicity of the second derivatives and from g (c 0 ) = 0 that We may now prove (25) and take any α > α 0 . As above we choose Inequality (40) yields Since α( . ) is strictly increasing on (c 0 , ∞), this yields On the other hand, g( . ) is strictly decreasing on (c 0 , ∞), and we find by means of (41) Since the Goldschmidt contribution is the same for c 1 (α) and c 2 (α), we come up with as claimed. Finally one should note that g(c) < 0 ⇔ c > c m , which shows that in this regime, the "small" catenoid w c 1 (α) has smaller area than the Goldschmidt solution as well as the "large" catenoid w c 2 (α) .

Appendix B : Divergence form of the Helfrich equation
We note that the form of M [u] in the case ε = 0 is derived in [13,Theorem 3] and is a consequence of the invariance of the Willmore functional with respect to translations. This means that the Helfrich equation can be written in divergence form. For the Willmore equation this was observed and exploited by Rusu [40], Rivière [39] and many others.
Then we have Proof A straightforward calculation shows that d dx Combining the above relations and taking into account that

Appendix C : Estimating the oscillations
In order to prove the auxiliary results used in Sect. 4 we will give a description of the oscillatory behaviour of the functions x → A cosh(ax) cos(ax) − B sinh(ax) sin(ax) , where A, B and a are given constants.

Proposition 1 Consider constants a = 0 and
Proof Without loss of generality we assume that a = 1 since h is symmetric and the statement is invariant under linear transformations. Since the case where A = 0 or B = 0 is quite simple we may assume that A = 0 and B = 0. Further we can restrict ourselves to the case A > 0; otherwise −h is considered. Setting A = 1 would also be admissible but we like to keep the "A" to highlight the connection to B in the following discussion. Because h (x) = −2 A sinh(x) sin(x) − 2B cosh(x) cos(x) we see that h has also a strict local minimum in 0 iff B < 0 and a strict local maximum in 0 iff B ≥ 0. In the first case h must have a further extremum before its first zero. First of all we rewrite with E(x) := A 2 cosh 2 (x) + B 2 sinh 2 (x) 1 2 and ϕ is the smooth angular map defined modulo 2π by We choose ϕ such that it is uniquely determined by ϕ(0) = 0, i. e. ϕ(x) ∈ (−π/2, π/2). If B > 0 then ϕ(x) ∈ [0, π/2), if B < 0 then ϕ(x) ∈ (−π/2, 0], and if B = 0, then ϕ(x) ≡ 0. For the derivative of ϕ we compute which yields . (42) Putting K = B A and y = tanh(x) we apply Lemma 5 to η(y) := 1 K ϕ (x). Observing that d dx (x + ϕ(x)) = 1 + K η(y) we see that d dx (x + ϕ(x)) is either always positive or negative first and then positive. Because x + ϕ(x) > − π 2 , Id +ϕ has to become strictly increasing before the first zero of h. Denoting 0 < x 1 < . . . < x k < . . . the zeroes of h this shows: In particular Moreover, | cos(x + ϕ(x))| attains in each interval (x k , x k+1 ) the value 1.
Next we also rewrite the derivative of h 1 2 and the smooth function ψ determined by Here we choose ψ(0) = π 2 in case −A < B, ψ(0) = 0 if −B = A and ψ(0) = − π 2 else, such that ψ(x) ∈ (−π, π). Similar to ϕ we have Let 0 = y 0 < y 1 < . . . < y k < . . . denote the zeroes of h . Now, since E and F are strictly increasing, the proposition follows if we are able to show the claim: Between two consecutive zeroes of h in (0, ∞) there is exactly one zero of h and vice versa. (D) Claims (C) and (D) yield the proof of the proposition, because: • Case B ≥ 0, 0 is a local maximum of h: We shall see that 0 = y 0 < x 1 < . . . < y k < x k+1 < y k+1 < . . . and the above claims yield: The strict inequalities above are due to the strict monotonicity of E and to the facts that h(x k ) = 0 and that | cos( . )| = 1 once on each [x k , x k+1 ]. • Case B < 0, 0 is a local minimum of h: We shall see that 0 = y 0 < y 1 < x 1 < . . . < y k < x k < y k+1 < . . . and the above claims yield: |h(x)|. Note that the local extremum at x = 0 has always the smallest absolute value among all local extrema.
To prove the claim we need to discuss several cases (see Fig. 5): (42) and (43) with Lemma 5 by setting K = B A , resp. K = B−A A+B , and y = tanh(x) we see that ϕ and ψ are strictly increasing with This immediately implies for all x > 0 Since Id + ψ is strictly increasing we have that y k + ψ(y k ) = (k + 1/2)π and further Hence, again by strict monotonicity of Id + ψ which proves Claim (D). Case 0 ≤ B < A: Applying Lemma 5 once again we see that ϕ is still strictly increasing, but ψ is strictly decreasing with This implies for all x > 0 that |ψ(x) − ϕ(x)| < π 2 . Moreover, since B < A and (44) still holds. Obviously Id + ϕ is strictly increasing. But as B−A A+B ≥ −1 we also get from Lemma 5 that Id + ψ is strictly increasing so that y k + ψ(y k ) = (k + 1/2)π as above. Like in the previous case we obtain again the validity of Claim (D). Case −A < B < 0: ϕ and ψ are strictly decreasing with Similar to the first case this immediately implies (44). The strict monotonicity of Id + ϕ follows from Lemma 5 like in the previous step as B A > −1. From Lemma 5 with B−A A+B < −1 we also know that Id + ψ possesses a zero at x * < tanh −1 1/ √ 2 ≈ 0.8814 so that for x > x * it is strictly increasing and decreasing for x ∈ (0, x * ). In the latter situation we have so that neither h nor h have a zero in (0, x * ]. Hence y 1 > x * and For k ∈ N we obtain using the monotonicity behaviour of Id + ϕ and Id + ψ: We conclude that 0 = y 0 < y 1 < x 1 < . . . < y k < x k < y k+1 < . . ., which is again Claim (D). Case B ≤ −A < 0: We only prove the claim for B < −A. The case B = −A follows from a similar discussion. Here ϕ is strictly decreasing and ψ is strictly increasing with We first would like to establish the separation property (44). Obviously |ϕ(x) − ψ(x)| < π 2 holds for all x > 0. Unfortunately ϕ and ψ will intersect in a point x * > 0 so that (44) will only hold for x > x * . For x > 0 we consider the "distance" between ϕ and ψ given by If we substitute y = tanh(x) we find that G(x) 2 = 0 if whose positive solution is given by y * = A(A+B) B(A−B) 1 2 . Recalling that B < −A < 0 we find Thus ϕ and ψ will intersect in the point x * = tanh −1 (y * ) < tanh −1 1/ √ 2 ≈ 0.8814. From the strict monotonicity of ϕ and ψ we then get that ψ(x) < ϕ(x) for x < x * and ψ(x) > ϕ(x) for x > x * . Since ψ < 0 and ϕ < 0 we find that x 1 > π 2 , y 1 > π 2 and As in the previous case we see that Id + ϕ is strictly increasing on [π/2, ∞) and Id + ψ on [0, ∞). This yields for k ∈ N: We conclude that 0 = y 0 < y 1 < x 1 < . . . < y k < x k < y k+1 < . . ., so that again Claim (D) holds true.
In the last step we used that sin(x) > 0, sin(a) > 0. Case a ∈ (π, a c ]: Here f (a) ≥ f (a c ) = 1. We consider first x ∈ (π, a). Thanks to Lemma 6 we start with Since in this case sin(x) < 0, sin(a) < 0, we end up again with (46). For x ∈ (0, π), the starting point is But since now sin(x) > 0, sin(a) < 0, (46) follows again. For x = π, the claim is obvious. Case a > a c : According to Lemma 6 and the definition of a c , f (0, a) = R so that we always find x 1 , x 2 ∈ (0, a) with tanh(x 1 ) tan(x 1 ) > tanh(a) tan(a) > tanh(x 2 ) tan(x 2 ) and both x 1 , x 2 in the same interval (kπ, (k + 1)π). So the sign of sin(x 1 ) and sin(x 2 ) coincides. Depending on the sign of sin(a) one chooses x 0 = x 1 or x 0 = x 2 respectively and ends up in each case with a point x 0 ∈ (0, a) for which (46) is violated, while in the other point (46) is satisfied.