Asymptotics of Landau constants with optimal error bounds
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan 411201, Hunan, China
Institut Franco-Chinois de l’Energie Nucléaire, Sun Yat-sen University, GuangZhou 510275, China
Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, China
Abstract: We study the asymptotic expansion for the Landau constants
where , is Euler’s constant, and are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative . Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form
for all , , and .
The results are proved by approximating the coefficients with the Gauss hypergeometric functions involved, and by using the second order difference equation satisfied by , as well as an integral representation of the constants .
MSC2010: 39A60; 41A60; 41A17; 33C05
Keywords: Landau constants; second-order linear difference equation; asymptotic expansion; sharper bound.
1 Introduction and statement of results
A century ago, it was shown by Landau  that if a function is analytic in the unit disc, such that , with the Maclaurin expansion
then it holds
for , and the equal sign can be attained for each . The constants are termed Landau’s constants; see, e.g., Watson .
Efforts have been made to approximate these constants from the very beginning. Indeed, Landau himself  has worked out the large- behavior
see also Watson .
Since then, the approximation of goes to two related directions. One is to find sharper bounds of for all positive integers , and the other is to obtain large- asymptotic approximations for the constants.
1.1 Sharper bounds
Many authors have worked on the sharp bounds of . For example, in 1982, Brutman  obtains
The result is improved in 1991 by Falaleev  to give
where , is the Euler constant ([13, (5.2.3)]).
Inequalities of this type are revisited in a 2002 paper  of Alzer. In that paper, the problem is turned into the following: to find the largest and smallest such that
The answer is that and , appealing to the complete monotonicity of .
It seems possible to obtain sharper bounds involving terms of higher and higher orders. Accordingly, difficulties may arise. The case by case process of taking more and more terms might be endless.
1.2 Asymptotic approximations
Most of the above inequalities can be used to derive asymptotic approximations for . Such approximations can also be obtained by employing integral representations, generating functions and relations with hypergeometric functions; see, e.g., . Indeed, back to Watson , a formula of asymptotic nature is derived by using a certain integral representation:
for large and positive integer . Theoretically, an asymptotic expansion can be extracted from (1.3) by substituting the large- expansions of and into it. In fact, Waston obtains
of which (1.2) is an extended version.
We skip to some very recent progress in this direction. In the manuscript , Ismail, Li and Rahman derive a complete asymptotic expansion for the Landau constants , using the asymptotic sequence . The approach is based on a formula of Ramanujan, which connects the Landau constants with a hypergeometric function.
Several relevant papers are worth mentioning. In , Nemes and Nemes derive full asymptotic expansions using a formula in . They also conjecture a symmetry property of the coefficients in the expansion. The conjecture has been proved by G. Nemes himself in .
(Nemes) Let . The Landau constants have the following asymptotic expansions
as , where the coefficients are certain computable constants that satisfy for every .
As an important special case, Nemes  has further proved that
where the coefficients are positive rational numbers.
The argument in  is based on an integral representation of involving a Gauss hypergeometric function in the integrand. While in , the authors of the present paper study this asymptotic problem by using an entirely different approach, starting from an obvious observation that the Landau constants satisfy a difference equation
as can be seen from the explicit formula (1.1), where .
By applying the theory of Wong and Li for second-order linear difference equations  to (1.6), the general expansion in (1.4) is obtained, and the conjecture of  is also confirmed. An advantage of this approach, compared with the previous ones, is that all coefficients in the expansion are given iteratively in an explicit manner.
1.3 A question and numerical evidences
As pointed out in , the case corresponding to (1.5) is numerically efficient since all odd terms in the expansion vanish. We will find that this expansion in terms of is even more special, both from asymptotic and sharper bound points of view.
From (1.5), as suggested by the alternating signs and by numerical calculations, there is a natural question as follows:
Is the error due to truncation of (1.5) bounded in absolute value by, and of the same sign as, the first neglected term? Or, more precisely, do we have the following?
where , and
Recalling that are positive, it is readily seen that a positive answer to (1.7) is equivalent to
for all and .
The question reminds us of an earlier work of Shivakumar and Wong , where an asymptotic expansion is obtained for the Lebesgue constants associated with the polynomial interpolation at the zeros of the Chebyshev polynomials, and the error in stopping the series at any time is shown to have the sign as, and is in absolute value less than, the first term neglected. Similar discussion can be found in, e.g., Olver [12, p.285], on the Euler-Maclaurin formula.
1.4 Statement of results
In the present paper, we will justify (1.9). In fact, we will prove the following theorem.
The above theorem has direct applications both in asymptotics and sharp bounds. In asymptotic point of view, we can obtain error bounds which in a sense are optimal. To be precise, we have the following.
The error due to truncation of (1.5) is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative . That is,
for and , where .
The error bound in (1.11) is the first neglected term in the asymptotic expansion, and hence is optimal and can not be improved. The inequalities in (1.11) can be derived from Theorem 1 by noticing that , as can be seen from (1.8).
Another application of Theorem 1 is the construction of sharp bounds up to arbitrary orders.
For , it holds
for all , , and .
The inequalities in (1.12) are understood as sharp bounds on both sides up to arbitrary orders. In a sense, the bounds are optimal and can not be improved.
2 Proof of Theorem 1
The proof is based on the difference equation (1.6) and an approximation of the coefficients . To justify Theorem 1, several lemmas are stated, and all, except one, are proved in the present section. While the validity of Lemma 2 is the objective of the next section.
2.1 The coefficients in (1.5)
Write the difference equation (1.6) in the symmetrical form
For , the coefficients in expansion (2.2) fulfill
where for ,
In addition, it holds
Part of this lemma ( (2.3) and (2.7) ) has been proved in Nemes’ recent paper . Part of it, namely (2.3), and an equivalent form of (2.4), has been proved in our earlier paper . Following Wong and Li , (2.3) and (2.4) can be justified by substituting (2.2) into (2.1), expanding both sides in formal power series of , and equalizing the coefficients of the same powers.
It is readily seen that all for and , and for .
2.2 Analysis of
with the error term being given in (1.8), and .
There are several facts worth mentioning. It is readily seen from (1.8) that satisfies the difference equation (2.1), and can then be removed from in (2.8). If we write , then the logarithmic singularity at is also cancelled in . Therefore, each is an analytic function in for . Hence the asymptotic expansion for , in descending powers of , is actually a convergent Taylor expansion in ,
for , with the leading coefficient ; cf. (2.4).
For later use, we estimate the ratio of the consecutive coefficients . To this end, we introduce a sequence of positive constants
We shall use the following lemma and leave its proof to Section 3 below.
Now we proceed to analyze (sometimes denoted by , understanding that ), so as to show that for . More precisely, we prove a much stronger result, as follows:
Proof: The lemma can be proved by using induction with respect to . Initially, we have
In view of the fact that ; cf. Table 1, it is readily verified that
for . Thus (2.13) holds for .
Assume that for , it holds for . From (2.10), we can write
for . To show that (2.13) is valid for , it suffices to show that
for since the validity of (2.13) for is trivial. This is equivalent to show that
For and , we have
The last inequality holds since .
2.3 Proof of Theorem 1
Now Lemma 3 implies that for all and all .
To show that for all , we note first that as . Hence for large enough. Now assume that (1.10) is not true. Then there exists a finite defined as
so that is a positive integer and , while . For simplicity, we denote and . From (2.8) we have
The later terms on the right-hand side are non-negative, hence we obtain
which implies that
Using (2.8) again for , we have
A combination of the previous two inequalities gives
3 Proof of Lemma 2
The idea is simple: to approximate the coefficients , and then to work out the ratio . Yet the procedure is complicated.
A brief outline of the proof is as follows: In Section 3.1, we bring in an ordinary differential equation (3.10) with a specific analytic solution , of which are coefficients of the Maclaurin expansion. The function is then extended, in Section 3.2, and via the hypergeometric functions, to a function analytic in the cut-strip . An integral representation is then obtained by using the Cauchy integral formula, and the integration path is deformed based on the analytic continuation procedure. In Section 3.3, the integral is spilt, approximated, and estimated, and hence bounds for on both sides are established in (3.25) for all . Eventually, in Section 3.4, an upper bound for is obtained for all non-negative integer .
3.1 Differential equation
for , where , and
The idea now is to approximate , and then to estimate the ratio .
Taking in (3.1), we have
Summing up gives
In view of (3.2), it is readily verified by summing up the series that
Also we have, for ,
The existence of defined above and the validity of (3.3) can be justified by showing that for all positive integers , and being positive constants. Indeed, from (3.2) it is readily seen that for . Now we assume that for , where is small enough such that . Then, by using (3.1) we have
Hence we have by induction.
Applying the operator to both sides of (3.6), we see that solves the second order differential equation
in a neighborhood of , with initial conditions and .
In the next few steps we derive a representation of for later use. First, substituting
into equation (3.8) yields
in a neighborhood of , with and .
turns the equation into the hypergeometric equation
Taking the initial conditions into account, it is easily verified that
3.2 Analytic continuation
Well-known formulas for hypergeometric functions include
for as ; cf. [13, (15.6.1)], which extends the hypergeometric function to a single-valued analytic function in the cut-plane.
Now we proceed to consider (3.10) with complex variable . It is worth noting that the solution we seek is an even function. So we restrict ourselves to its analytic continuation on the right-half plane . To this aim, we define
We see that maps the strip duplicately and analytically onto the cut-plane . The same is true for . Hence from (3.12) we have the analytic continuation for . Moreover, from (3.14) it is readily seen that the function , defined and analytic in the disjoint strips, satisfies
Careful treatment should be brought in here, since there is a logarithmic singularity of at the boundary point , as we will see later, and as can be seen from the equation (3.10):