Asymptotic analysis of Emden–Fowler type equation with an application to power flow models

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider.


Introduction
Many problems in mathematical physics, astrophysics and chemistry can be modeled by an Emden-Fowler type equation of the form d dt t ρ du dt ± t σ h(u) = 0, (1.1) where ρ, σ are real numbers, the function u : R → R is twice differentiable and h : R → R is some given function of u. For example, choosing h(u) = u n for n ∈ R, ρ = 1, σ = 0 and plus sign in (1.1), is an important equation in the study of thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics [5,7]. Another example is known as Liouville's equation, which has been studied extensively in mathematics [9]. This equation can be reduced to an Emden-Fowler type equation with h(u) = e u , ρ = 1, σ = 0 and plus sign [7]. For more information on different applications of Emden-Fowler type equations, we refer the reader to [17].
In this paper, we study the Emden-Fowler type equation where h(u) = u −1 , ρ = 0, σ = 0, with the minus sign in (1.1), and initial conditions u(0) = k −1/2 , u (0) = k −1/2 w for w ≥ 0. For a positive constant k > 0, we consider the change of variables u = k −1/2 f , with resulting equation This specific Emden-Fowler type equation (1.2) arises in a queuing model [6], modeling the queue of consumers (e.g. electric vehicles (EVs)) connected to the power grid. The distribution of electric power to consumers leads to a resource allocation problem which must be solved subject to a

INTRODUCTION
constraint on the voltages in the network. These voltages are modeled by a power flow model known as the Distflow model; see Section 2 for background. The Distflow model equations are given by a discrete version of the nonlinear differential equation (1.2) and can be described as In this paper, we study the asymptotic behavior and associated properties of the solution of (1.2) using differential and integral calculus, and show its numerical validation, i.e., we show that the solutions of (1.2) have asymptotic behavior f (t) ∼ t (2k ln(t)) 1/2 , t → ∞, (1.4) which can be used in the study of any of the aforementioned resource allocation problems. It is natural to expect that the discrete version (1.3) of the Emden-Fowler type equation has the asymptotic behavior of the form (1.4) as well. However, to show (1.5) below, is considerably more challenging than in the continuous case, and this is the main technical challenge addressed in this work. We show the asymptotic behavior of the discrete recursion, as in (1.3) to be V j ∼ j (2k ln(j)) 1/2 , j → ∞. (1.5) There is a huge number of papers that deal with various properties of solutions of Emden-Fowler differential equations (1.1) and especially in the case where h(u) = u n or h(u) = exp(nu) for n ≥ 0. In this setting, for the asymptotic properties of solutions of an Emden-Fowler equation, we refer to [5], [17] and [11]. To the best of our knowledge, [14] is the only work that discusses asymptotic behavior in the case n = −1, however not the same asymptotic behavior as we study in this paper. More precisely, the authors of [14] study the more general Emden-Fowler type equation with h(u) = u n , n ∈ R, ρ + σ = 0 and minus sign in (1.1). In [14], the more general equation appears in the context of the theory of diffusion and reaction governing the concentration u of a substance disappearing by an isothermal reaction at each point t of a slab of catalyst. When such an equation is normalized so that u(t) is the concentration as a fraction of the concentration outside of the slab and t the distance from the central plane as a fraction of the half thickness of the slab, the parameter √ k may be interpreted as the ratio of the characteristic reaction rate to the characteristic diffusion rate. This ratio is known in the chemical engineering literature as the Thiele modulus. In this context, it is natural to keep the range of t finite and solve for the Thiele modulus as a function of the concentration of the substance u. Therefore, [14] studies the more general Emden-Fowler type equation for u as a function of √ k and study asymptotic properties of the solution as k → ∞. However, here we solve an Emden-Fowler equation for the special case n = −1 and for any given Thiele modulus k, and study what happens to the concentration u(t) as t goes to infinity, rather than k to infinity.
Although the literature devoted to continuous Emden-Fowler equations and generalizations is very rich, there are not many papers related to the discrete Emden-Fowler equation (1.3) or to more general second-order non-linear discrete equations of Emden-Fowler type within the following meaning. Let j 0 be a natural number and let N(j 0 ) denote the set of all natural numbers greater than or equal to a fixed integer j 0 , that is, Then, a second-order non-linear discrete equation of Emden-Fowler type is studied, where u : N(j 0 ) → R is an unknown solution, ∆u(j) := u(j + 1) − u(j) is its first-order forward difference, ∆ 2 u(j) := ∆(∆u(j)) = u(j + 2) − 2u(j + 1) + u(j) is its second-order forward difference, and α, m are real numbers. A function u * : N(j 0 ) → R is called a solution of (1.6) if the equality holds for every j ∈ N(j 0 ). The work done in this area focuses on finding conditions that guarantee the existence of a solution of such discrete equations. In [8], the authors consider the special case of (1.6) where α = −2, write it as a system of two difference equations, and prove a general theorem for this that gives sufficient conditions that guarantee the existence of at least one solution.
In [1,10], the authors replace the term j α in (1.6) by p(j), where the function p(j) satisfies some technical conditions, and find conditions that guarantee the existence of a non-oscillatory solution.
In [2,15], the authors find conditions under which the nonlinear discrete equation in (1.6) with m of the form p/q where p and q are integers such that the difference p − q is odd, has solutions with asymptotic behavior when j → ∞ that is similar to a power-type function, that is, for constants a ± and s defined in terms of α and m. However, we study the case m = −1 and this does not meet the condition that m is of the form p/q where p and q are integers such that the difference p − q is odd.
The paper is structured as follows. In Section 2, we present the application that motivated our study of particular equations in (1.2) and (1.3). We present the main results in two separate sections. In Section 3, we present the asymptotic behavior and associated properties of the continuous solution of the differential equation in (1.2), while in Section 4, we present the asymptotic behavior of the discrete recursion in (1.3). The proofs of the main results in the continuous case, except for the results of Section 3.1, and discrete case can be found in Sections 5 and 6, respectively. We finish the paper with a conclusion in Section 7. In the appendices, we gather the proofs for the results in Section 3.1.

Background on motivational application
Equation (1.2) emerges in the process of charging electric vehicles (EVs) by considering their random arrivals, their stochastic demand for energy at charging stations, and the characteristics of the electricity distribution network. This process can be modeled as a queue, with EVs representing jobs, and charging stations classified as servers, constrained by the physical limitations of the distribution network [3,6].
An electric grid is a connected network that transfers electricity from producers to consumers. It consists of generating stations that produce electric power, high voltage transmission lines that carry power from distant sources to demand centers, and distribution lines that connect individual customers, e.g., houses, charging stations, etc. We focus on a network that connects a generator to charging stations with only distribution lines. Such a network is called a distribution network.
In a distribution network, distribution lines have an impedance, which results to voltage loss during transportation. Controlling the voltage loss ensures that every customer receives safe and reliable energy [12]. Therefore, an important constraint in a distribution network is the requirement of keeping voltage drops on a line under control.
In our setting, we assume that the distribution network, consisting of one generator, several charging stations and distribution lines with the same physical properties, has a line topology.
The generator that produces electricity is called the root node. Charging stations consume power and are called the load nodes. Thus, we represent the distribution network by a graph (here, a line) with a root node, load nodes, and edges representing the distribution lines. Furthermore, we assume that EVs arrive at the same rate at each charging station.
In order to model the power flow in the network, we use an approximation of the alternating current (AC) power flow equations [16]. These power flow equations characterize the steady-state relationship between power injections at each node, the voltage magnitudes, and phase angles that are necessary to transmit power from generators to load nodes. We study a load flow model known as the branch flow model or the Distflow model [4,13]. Due to the specific choice for the network as a line, the same arrival rate at all charging stations, distribution lines with the same physical properties, and the voltage drop constraint, the power flow model has a recursive structure, that is, the voltages at nodes j = 0, . . . , N − 1, are given by recursion (1.3). Here, N is the root node, and V 0 = 1 is chosen as normalization. This recursion leads to real-valued voltages and ignores line reactances and reactive power, which is a reasonable assumption in distribution networks. We refer to [6] for more detail.

Main results of continuous Emden-Fowler type equation
In this section, we study the asymptotic behavior of the solution f of (1.2). To do so, we present in Lemma 3.1 the solution of a more general differential equation. Namely, we consider a more general initial condition f (0) = y > 0.
The solution f presented in Lemma 3.1 allows us to study the asymptotic behavior of f 0 (x), i.e., the solution of the differential equation in Lemma 3.1 where k = 1, y = 1 and w = 0, or in other words, the solution of the differential equation f (x) = 1/f (x) with initial conditions f (0) = 1 and f (0) = 0; see Theorem 3.1. We can then derive the asymptotic behavior of f ; see Corollary 3.1.
The following theorem provides the limiting behavior of f 0 (x), i.e., the solution of Equation (1.2) where k = 1, y = 1 and w = 0.
Theorem 3.1. Let f 0 (x) be the solution of (1.2) for k = 1, y = 1 and w = 0. The limiting behavior of the function f 0 (x) as x → ∞ is given by, We first derive an implicit solution to Equation (1.2) where k = 1, y = 1 and w = 0. Namely, we derive f 0 (x) in terms of a function U (x); cf. Lemma 3.1. We show, using Lemma 3.2, that we can derive an approximation of U (x) by iterating the following equation: We can then use this approximation of U (x) in the implicit solution of the differential equation to derive the asymptotic behavior of Theorem 3.1. The proofs of Theorem 3.1 and Lemma 3.2 can be found in Section 5. We now give the necessary lemmas for the proof of Theorem 3.1.

4)
and where the constants a, b, c are given by Notice that we do not find an elementary closed-form solution of the function f 0 (x), since f 0 (x) is given in terms of U (x), given implicitly by (3.4). For x ≥ 0, the left-hand side of (3.4) is equal to 1 2 √ πerfi(U (x)) where erfi(z) is the imaginary error function, defined by erfi(z) = −i erf(iz), (3.8) where erf(w) = 2 √ π w 0 exp(−v 2 )dv is the well-known error function.
Lemma 3.2. For y ≥ 0, we have the inequalities Now, we present the asymptotic behavior of the solution f of (1.2).
Corollary 3.1. The limiting behavior of the function f (t), defined in Equation (3.2), is given by Proof of Corollary 3.1. In order to derive a limit result of the exact solution of (1.2), i.e. for (3.2) with initial conditions f (0) = 1 and f (0) = w, we use the limiting behavior of the function f 0 (x) and the definitions of a, b and c as in (3.5)-(3.7). Denote v = ln(z). Then, by Theorem 3.1, we have In what follows, we carefully examine the quantities czv Therefore, using that cb = √ k, we get , t > exp(1), (3.13) and , t > exp(1). (3.14) Putting the results in (3.13) and (3.14) together in (3.12), yields

Associated properties of the ratio between f and its first order approximation
In this section, we study associated properties of the ratio between f (t) and its first order approximation. Using only the first term of the asymptotic expansion of (3.11), we define The reason for studying this ratio, and in particular the role of k, is twofold: (1) the useful insights that we get for (the proof of) the asymptotic behavior in the discrete case in Section 4, and (2) the applicability of Equation (1.2) in our motivational application, in cases where the parameter k in (1.2) is small.
Considering the practical application for charging electric vehicles, the ratio of normalized voltages where the tolerance ∆ is small (of the order 10 −1 ), due to the voltage drop constraint. Therefore, the parameter k, comprising given charging rates and resistances at all stations, is normally small (of the order 10 −3 ).
Furthermore, to match the initial conditions V 0 = 1 and V 1 = 1 + k of the discrete recursion with the initial conditions of the continuous analog, we demand f (0) = 1 and f (1) = 1 + k. However, notice that in our continuous analog described by (1.2), we have, next to the initial condition f (0) = 1, the initial condition f (0) = w, while nothing is assumed about the value f (1). The question arises whether it is possible to connect the conditions f (0) = w and f (1) = 1 + k. To do so, we use an alternative representation of f given in Lemma A.1. Then, using this representation, we show the existence and uniqueness of w ≥ 0 for every k such that the solution of (1. 2) satisfies The importance of the role of the parameter k becomes immediate from the comparison of the functions f (t) and g(t) in Theorem 3.2.
In what follows, we start with introducing notation for the proof of Theorem 3.2, and give a sketch of the proof. The theorem is proven in Appendix A.
Define the auxiliary function ψ : − k ln(2k ln(t)), (3.16) and notice (also for the proof in Lemma 4.3) that the function ψ(t) is strictly decreasing from +∞ at t = 1 to 0 at t = ∞. This follows easily from the definition of ψ in (3.16).
Denote the unique solution t > 1 of the equation where w comes from the initial condition f (0) = w ≥ 0. Additionally, define where the second line is a consequence of Lemma A.1 with y = 1. The proof of Theorem 3.2 centers about the unique solution t 0 (k) of (3.17). First, from (3.18), we notice that max t≥1 is only a function of the parameter k. In Lemma A.4, we show F (t 0 (k), k) is a strictly decreasing function of k. To prove Lemma A.4, we make use of additional Lemma A.5. Then, in Lemmas A.6 and A.7, we show that F (t 0 (k), k) is positive for small k and negative for large k, respectively. This allows us to conclude that F (t 0 (k), k) ≤ 0 is equivalent to k ≥ k c . In summary, to prove Theorem 3.2, we show Furthermore, in Lemma A.3, we show that F (t, k) has only one extreme point, and in particular that this extreme point is a maximum and that this is attained at the point t 0 (k). Thus, in the case where 0 < k < k c , we are left with Necessary Lemmas A.3-A.7 to prove Theorem 3.2 are stated and proven in Appendix A.
A comparison of the approximation g(t), i.e. for (3.15), to the exact solution f (t) of (1.2) where w is such that f (1) = 1 + k, for three values of k, is given in Figure 1. However, in the setting where k is small, the result in Theorem 3.2, case (b) leaves two practical questions; how small the ratio f (t)/g(t) can be when t 1 (k) ≤ t ≤ t 2 (k) and how large the ratio f (t)/g(t) can be when t ≥ t 2 (k). These practical questions are covered in Theorem 3.3.
3. Let f (t) be given by (3.2) with initial conditions f (0) = 1, f (0) = w such that f (1) = 1 + k, and let g(t) be given by (3.15). Then, for 0 < k < k c , we have The proof exploits properties of Theorem 3.1 and Theorem 3.2, such as exact representations (3.5)-(3.7) and actual values such as the one for k c , but most importantly, we use numerical results to compute bounds for the quantity f 0 (x)/g(x), where f 0 (x) is given in (3.3) and g(x) is given in (3.15). The proofs of Theorem 3.3, and supporting Lemmas A.8 and A.9 can be found in Appendix A.

Main results of discrete Emden-Fowler type equation
In this section, we present the asymptotic behavior of the discrete recursion (1.3). Thus, we consider the sequence V j , j = 0, 1, . . . defined in (1.3) and we let The proof of Theorem 4.1 relies on the following observations: there always exists a point n ∈ {1, 2, . . .} such that either V j ≥ W j for all j ≥ n or V j ≤ W j for all j ≥ n, and the existence of such a point implies in either case the desired asymptotic behavior of the sequence V j .
for j ≥ n 0 (k), where n 0 (k) is appropriately chosen. Then, Equation (4.2) implies that there exists either a point n ≥ n 0 (k) such that V n ≥ W n or not. If there exists a point n ≥ n 0 (k) such that V n ≥ W n , then we show in Lemma 4.3 that V j ≥ W j for all j ≥ n. If not, we have that V j < W j for all j ≥ n 0 (k).
Then, we are left to show that the existence of such a point implies the desired asymptotic behavior of V j . This is done in Lemma 4.4.
We now give the necessary lemmas to prove Theorem 4.1.

4)
for some constant C. 1. There is n ≥ n 0 (k) such that V n ≥ W n .
2. There is n ≥ n 0 (k) such that V j ≥ W j for all j ≥ n.
In any case, we can distinguish between two cases: there exists either a point n ≥ n 0 (k) such that V n ≥ W n or not, i.e., 1. There is n ≥ n 0 (k) such that V n ≥ W n , 2. V j < W j for all j ≥ n 0 (k). By Lemma 4.3, we have, on the one hand, that the existence of a point n ≥ n 0 (k) such that V n ≥ W n , implies that V j ≥ W j for all j ≥ n and on the other hand, that the non-existence of n ≥ n 0 (k) such that V n ≥ W n , implies that V j < W j for all j ≥ n 0 (k).
This situation exactly fits the framework of Lemma 4.4.
We consider the two cases above. First, assume that (1) holds. Then by Lemma 4.3, we have V j ≥ W j for all j ≥ n. From V j ≥ W j , for all j ≥ n, we have that Lemma 4.4, item (1) holds, and so Second, assume that (2) holds, so that V j < W j for all j > n 0 (k). Then, Lemma 4.4, item (2) holds, and so Hence, any of the two cases yields Although we do not provide associated properties of the asymptotic behavior of V j as j → ∞ as we did for the asymptotic behavior of f (x) as x → ∞, we compare the behavior of V j with the discrete counterpart of g(t), i.e. W j , for j = 1, . . . , 100 in Figure 2.

Proofs for Section 3
The main result in Section 3, i.e., Theorem 3.1 follows from Lemmas 3.1 and 3.2. In this section, we provide the proofs of both Theorem 3.1 and Lemma 3.2. For the proof of Lemma 3.1 we refer to [6].

Proof of Theorem 3.1
Proof of Theorem 3.1.
We consider for x ≥ 0 the equation The function h z (y) is concave in y ≥ 0 since where the first inequality follows from z > exp(1) − 1 and the second inequality follows from ln(1 + z 2 ) < z 2 , z > 0. Therefore, the equation y = h z (y) has for any z > exp (1)  Observe that Indeed, we have from (5.1) and the first inequality in (3.9) and so U ≤ y LB follows from increasingness of the function y ≥ 0 → (exp(y 2 ) − 1)/2y. In addition to the upper bound on y in (5.5), we also have the lower bound Indeed, from (5.1) and the second inequality in (3.9), where the inequality in (5.8) follows from − ln(w) ≥ (1 − w) 2 with w = 2 z ∈ (0, 1]. We have from (5.6) that When we use (5.7) in (3.10) with y = U , we see that .
(5.10) From (5.9) and (5.10), we then find that Observe that (5.11) coincides with (5.2) when we take y = U and replace the right-hand side x , we find that

Proof of Lemma 3.2
Proof of Lemma 3.2. We require the inequalities (3.9) and (3.10). The inequalities in (3.9) follow from expanding the three functions in (3.9) as a series involving odd powers y 2l+1 , l = 0, 1, . . . , of y and comparing coefficients, i.e., As to the inequality in (3.10), we use partial integration according to as follows from expanding the two functions in (5.14) as a series involving odd powers y 2l+1 , l = 0, 1, . . . , of y and comparing coefficients. Then (3.10) follows from (5.13)-(5.14) upon deleting the y 2 in the numerator at the right-hand side of (5.14).

Proofs for Section 4
The main result in Section 4 follows from Lemmas 4.1-4.4. The proof of each Lemma can be found in 6.1-6.4, respectively.

Proof of Lemma 4.2
In this section, we prove a lower bound for the first order finite differences of V j that is similar to the upper bound we obtained in (4.3). This result follows from Lemmas 6.1 and 6.2.
In more detail, the proof of Lemma 4.2 consists of algebraic manipulations of (1.3), but the key in the proof is the use of Lemma 6.2 in these manipulations, which, in turn, builds on technical results established in Lemma 6.1. We first state Lemmas 6.1 and 6.2.
Lemma 6.1. Let V j , j = 0, 1, . . . , be as in (1.3). Then, Both Lemmas 6.1 and 6.2 are proven later in this section. Here, we discuss the efficacy of Lemma 6.2 by numerical validation. We approximate, The efficacy of the approximation (6.6) of (6.5) is illustrated for the cases k = 0.001, k = 0.01 and k = 0.1 in Figure 3. For these cases, the approximation already yields relative errors smaller than 0.5% for j ≥ 10.
Having Lemmas 6.1 and 6.2 at our disposal, we are now ready to give the proof of Lemma 4.2.
Proof of Lemma 4.2. In order to relate the recursion in (1.3) to (4.3), we write (1.3) as and multiply both sides of (6.7) by Figure 3: Illustration of efficacy of the approximation (6.6) of (6.5) by showing the quotient of (6.6) and (6.5), for three values of k.
to obtain Summing this over j = 1, 2, . . . , n, we get We proceed with rewriting Equation (6.8) to an expression that is similar to the one we obtained for the sequence W j , j = 1, 2, . . . , N in Equation (6.1) using Lemma 6.2. Then, we have We observe a telescoping sum in the right-hand side of (6.9), so we have Furthermore, we introduce the following notation: . Thus, we rewrite (6.9) to Recall that we want to derive a lower bound for the first order finite differences V n+1 − V n . In order to do so, we use that V n+1 ≥ V n (see [6,Lemma 5.1]). Thus, as desired.
To complete the proof of Lemma 4.2, we are left to prove Lemmas 6.1 and 6.2. This is done in Sections 6.2.1 and 6.2.2, respectively.

Proof of Lemma 6.1
Proof of Lemma 6.1. The properties of the sequence V j , j = 0, 1, . . . are given in the following way.

This is a direct consequence of the inequalities in items
Hence,

Proof of Lemma 6.2
Proof of Lemma 6.2. We show the asymptotic behavior of Vj+1−Vj Vj as j → ∞. Let, for j = 1, 2, . . ., Then, Indeed, from Lemma 6.1, items 1 and 3, Here it has been used that the function y −1 ln(y), y ≥ 1, has a global maximum at y = exp(1) that equals exp(−1). The other inequalities follow by the increasingness of the sequence V j , j = 0, 1, . . . (see [6,Lemma 5.1]). Furthermore, we have where the bounds in (6.11) assure convergence of the infinite series. Since 0 < Y j < X j , we have Thus, we get that, In the last line, we used Lemma 6.1, item (4).

Proof of Lemma 4.3
Proof of Lemma 4.3. We establish the (non-trivial) implication from (1) to (2). Assume there is n ≥ n 0 (k) such that V n ≥ W n . We claim that V j ≥ W j for all j ≥ n. Indeed, when there is a n 2 > n such that V n2 < W n2 , we let n 3 := max{j : n ≤ j ≤ n 2 , V j ≥ W j }. Then V n3 ≥ W n3 and V j < W j for n 3 < j ≤ n 2 . However, since ψ(j) is strictly decreasing and n 3 + 1 > n 0 (k), we have which implies V n3+1 ≥ W n3+1 . This contradicts the definition of n 3 . Since the choice of n 2 is arbitrary, we have that V j ≥ W j for all j ≥ n. The implication from (2) to (1) is immediate.

Conclusion
Continuous and discrete Emden-Fowler type equations appear in many fields such as mathematical physics, astrophysics and chemistry, but also in electrical engineering, and more specifically under a popular power flow model. The specific Emden-Fowler equation we study, appears as a discrete recursion that governs the voltages on a line network and as a continuous approximation of these voltages. We show that the asymptotic behavior of the solution of the continuous Emden-Fowler equation (1.2), i.e. the approximation of the discrete recursion, and the asymptotic behavior of the solution of its discrete counterpart (1.3), are the same.
A.2 Proof of Lemma A.2 Proof. Again, we rely on the representation of f in (A.14). Thus the condition f (1) = 1 + k can be written as The left-hand side of (A.6) decreases in w ≥ 0 from a value greater than √ 2 to 0 as w increases from w = 0 to w = ∞. Indeed, as to w = 0 we consider This implies that That the left-hand side of (A.6) decreases strictly in w ≥ 0, to the value 0 at w = ∞, is obvious. We conclude that for any k > 0 there is a unique w > 0 such that (A.6) holds.

A.3 Proof of Theorem 3.2
Proof. From the definition of F in (3.19), it follows that F (t, k) = 0 if and only if f (t) = g(t).
Furthermore, we have By Lemma A.3, we have, for any k, max t≥1 F (t, k) = F (t 0 (k), k) and by Lemma A.4, we have that F (t 0 (k), k) is a strictly decreasing function of k. Notice that, by (3.19), we can alternatively write, Thus, by Lemma A.6, we have on the one hand, for small k, that F (t 0 (k), k) > 0, and by Lemma A.7, we have on the other hand, for large k, that F (t 0 (k), k) ≤ 0. Therefore, we conclude that F (t 0 (k), k) ≤ 0 is equivalent to k ≥ k c .

A.4 Proof of Lemma A.3
Lemma A.3. Let F (t, k) be given as in (3.19). Then, for any k, where t 0 (k) is given by (3.17).
Proof. To find, for a given k > 0, the maximum of F (t, k) over t ≥ 1, we compute from (3.19) Since ψ(t) is strictly decreasing in t > 1, while W 2 does not depend on t, we have from (A.9) that which completes the proof.
Next, we consider for a fixed t > 0 the identity (A.4). For any s > 1, the integrand w 2 (k) + 2k ln(s) decreases strictly in k > 0, and hence f (t) = f (t; k) increases strictly in k > 0, since t > 0 is fixed. As a consequence, we conclude from (A.13) that w(k)/k strictly increases in k > 0 since 1/f (u) strictly decreases in k > 0 for any u ∈ (0, 1).
A.6 Proof of Lemma A.6 Lemma A.6. Let F (t, k) be given as in (3.19). Then, for small k, we have that F (t 0 (k), k) > 0.
Proof. We have for t > 0, where ξ t is a number between 0 and t. Since f (1) = 1 + k and f (ξ t ) ≥ 1 > 0, it follows that w ≤ k. Therefore, On the other hand √ k for k > 0, we find that k < 0.05 is small enough. We conclude from (3.19) that F (t 0 (k), k) ≥ F 1 √ k , k > 0 when k is small.

A.7 Proof of Lemma A.7
Lemma A.7. Let F (t, k) be given as in (3.19). Then, for large k, we have that F (t 0 (k), k) ≤ 0.
For the second and third factor, we notice, from (3.5)-(3.7) and t ≥ 2/k, that However, the right-hand side of (A.24) is equal to 1 when k ≥ 1, and equal to ln .
Together with (A.21) this gives the desired result. Now, we turn to the proof of inequality (3.21). We follow the same approach as in the proof of inequality (3.20). Thus, we consider each factor of the right-hand side of (A.22) separately. The right-hand side of (A.22) is now to be considered for t ≥ t 2 (k), and so it is important to have specific information about t 2 (k). We claim that t 2 (k) ≥ exp(e 2−kc /2k). This claim is proven in Lemma A.9 below.
For the second factor, we notice, since bc = √ k, that we have Now, from (3.5) and (3.7), where in the last line it has been used that w/k is an increasing function of k; see Lemma A.5.