Abstract
In this paper, using the polynomial approximation and the continued fraction approximation, we present some sharp inequalities for the sequence \((1+1/n)^n\) and some applications to Carleman’s inequality. For demonstrating the superiority of our new inequalities over the classical one, some proofs and numerical computations are provided.
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1 Introduction
There has been considerable discussion concerning the following well-known double inequalities:
Since these are often used to improve inequalities of Hardy–Carleman type, there has been considerable interest in extending these inequalities in the recent past. See for example [2, 10, 13–15].
These inequalities (1.1) are equivalent to
Mortici and Hu [9] presented the best form approximation of (1.2) as follows:
Based on (1.3), double inequalities
hold for every real number \(x\in [1,\infty )\), where
In the asymptotic theory, there are many methods to obtain better approximations. First, the polynomial approximation is a very useful method to give superior increasing approximations as for example the Stirling series [1]:
Recently, the polynomial approximation method was used by Lu [3] to provide some more general convergent sequences for Euler’s constant. Using the polynomial approximation, Lu et al. [5] also obtained the extension of Windschitl’s formula. Second, the continued fraction approximation is also a very useful method to give superior increasing approximations. For example, Mortici [8] provided a new continued fraction approximation starting from the Nemes’ formula as follows:
where
Recently, the continued fraction approximation was used by Lu and Wang [4] to provide a new asymptotic expansion for the gamma function. Lu et al. [6] also obtained some new continued fraction approximations of Euler’s constant.
It is their works that motivated our study. In this paper, we give some polynomial and continued fraction approximations for the constant e in Sect. 2.
To obtain the main results in this paper, we need the following lemma which is very useful for constructing asymptotic expansions:
Lemma 1
If \((x_n)_{n\ge 1}\) is convergent to zero and the limit
exists for \(s > 1\), then
Lemma 1 was first proved by Mortici in [7]. From Lemma 1, we can see that the speed of convergence of the sequence \((x_n)_{n\ge 1}\) increases with the value s satisfying (1.5).
The rest of the paper is organized as follows: In Sect. 2, the main results and their proofs are provided. In Sect. 3, we give some comparisons to demonstrate the superiority of inequalities (2.10) and (2.11) over the inequalities (1.4) in Mortici and Hu [9]. Finally, in Sect. 4, some applications to Carleman’s inequality are presented.
2 Main results
Theorem 1
For (1.3), using the polynomial approximation, we have
where
Proof
Let \((x_i)_{i\ge 1}\) be a polynomial sequence which converges to \(\frac{1}{e}\left( 1+\frac{1}{n}\right) ^n\), where
To measure the accuracy of this approximation, we define a sequence \((t_i)_{i\ge 1}\),
Then, \(x_i\) converges to \(\frac{1}{e}\left( 1+\frac{1}{n}\right) ^n\) is equivalent to \(t_i\) converges to 0. Using (2.2) and (2.3), we have
From Lemma 1, we know that the speed of convergence \((t_i)_{i\ge 1}\) is even higher as the value s satisfying (1.5). Thus, using Lemma 1, we have the following:
-
(i)
If \(a_1\not =-2^{-1}\), then the rate of the sequence \(t_{1}(n)\) is \(n^{-1}\), since
$$\begin{aligned} \lim _{n\rightarrow \infty }nt_{1}(n)=\frac{-1-2a_1}{2}\ne 0. \end{aligned}$$ -
(ii)
If \(a_1=-2^{-1}\), then from (2.4), we have
$$\begin{aligned} t_{1}(n)-t_{1}(n+1)=\frac{11}{12n^3}, \end{aligned}$$
and the rate of convergence of the sequence \(t_1(n)\) is \(n^{-2}\), since
We know that the fastest possible sequence \(t_1(n)\) is obtained only for \(a_1=-2^{-1}\).
Using the same method, we have
\(\square \)
Theorem 2
For (1.3), using the continued fraction approximation, we have
where
Proof
Let \((y_i)_{i\ge 1}\) be a continued fraction sequence which converges to \(\frac{1}{e}\left( 1+\frac{1}{n}\right) ^n\), where
To measure the accuracy of this approximation, we define a sequence \((s_i)_{i\ge 1}\),
Then, \(y_i\) converges to \(\frac{1}{e}\left( 1+\frac{1}{n}\right) ^n\) is equivalent to \(s_i\) converges to 0. Using (2.6) and (2.7), we have
It is easy to see that the fastest possible sequence \(s_1(n)\) is obtained only for \(b_1=a_1=-2^{-1}\).
Using (2.6) and (2.7) again, we have
From Lemma 1, we have the following:
-
(i)
If \(b_2\not =11/12\), then the rate of the sequence \(s_2(n)\) is \(n^{-2}\), since
$$\begin{aligned} \lim _{n\rightarrow \infty }n^2 s_2(n)=\frac{11-12b_2}{24}\ne 0. \end{aligned}$$ -
(ii)
If \(b_2=11/12\), then from (2.9), we have
$$\begin{aligned} s_2(n)-s_{2}(n+1)=-\frac{5}{96n^4}, \end{aligned}$$
and the rate of convergence of the sequence \(s_2(n)\) is \(n^{-3}\), since
We know that the fastest possible sequence \(s_2(n)\) is obtained only for \(b_2=11/12\).
Using the same method, we have
Using Theorem 1, we obtain the following inequalities.
Theorem 3
For every real number \(x\in [1,\infty )\), the following inequalities hold:
where
Proof
The proof of inequalities (2.10) is equivalent to \(f_1>0\) and \(g_1<0\), as \(x\in [1,\infty )\), where
By some calculations, we have
where
Evidently, we have \(f_1''(x)>0\), \(g_1''(x)<0\) for \(x\ge 1\). Thus, \(g_1\) is strictly concave, and \(f_1\) is strictly convex. Combining \(f_1(\infty )=g_1(\infty )=0\), we obtain \(g_1<0\) and \(f_1>0\) on \([1,\infty )\). The proof of inequalities (2.10) is complete. \(\square \)
Using Theorem 2, we obtain the following inequalities.
Theorem 4
For every real number \(x\in [1,\infty )\), the following inequalities hold:
where
Proof
The proof of inequalities (4.3) is equivalent to \(f_2>0\) and \(g_2<0\), as \(x\in [1,\infty )\), where
By some calculations, we have
where
Evidently, we have \(f_2''(x)>0\), \(g_2''(x)<0\) for \(x\ge 1\). Thus, \(g_2\) is strictly concave, and \(f_2\) is strictly convex. Combining \(f_2(\infty )=g_2(\infty )=0\), we obtain \(g_2<0\) and \(f_2>0\) on \([1,\infty )\). The proof of inequalities (2.10) is complete. \(\square \)
3 Comparisons
In this section, we give some comparisons to demonstrate the superiority of inequalities (2.10) and (2.11) over the inequalities (1.4) in Mortici and Hu [9].
First, comparing (2.10) with (1.4), we have
Then, \(u_1(x)>u_0(x)\) for \(x\in [3,\infty )\).
Then, \(v_1(x)<v_0(x)\) for \(x\in [3,\infty )\). Thus, the inequalities (2.10) in Theorem 3 are more accurate than the inequalities (1.4) in Mortici and Hu [9].
Next, comparing (4.3) with (1.4), we have
where
Then, \(u_2(x)>u_0(x)\) for \(x\in (0,\infty )\).
where
Then, \(v_2(x)<v_0(x)\) for \(x\in (0,\infty )\). Thus, the inequalities (2.11) in Theorem 4 are more accurate than the inequalities (1.4) in Mortici and Hu [9].
Finally, comparing (2.10) with (4.3), we have
where
By some simulations, we obtain the following figures.
We see that for \(x\in [100,\infty )\), \(u_1(x)>u_2(x)\) and \(v_1(x)<v_2(x)\). So, the inequalities (2.10) in Theorem 3 are more accurate than the inequalities (2.11) in Theorem 4 for \(x\in [100,\infty )\). And for \(x\in (0,35]\), \(u_2(x)>u_1(x)\) and \(v_2(x)<v_1(x)\). So we can see that for \(x\in (0,35]\), the inequalities (4.3) in Theorem 4 are more accurate than the inequalities (2.10) in Theorem 3 (Fig. 1).
Furthermore, some numerical computations are given to demonstrate the superiority of our new double inequalities over the classical ones again. Let \(E(n)=\frac{1}{e}\left( 1+\frac{1}{n}\right) ^n\). Combining Theorems 3 and 4, we have Tables 1 and 2.
4 Applications to Carleman’s inequality
If \(\sum a_n\) is a convergent series of nonnegative reals, then the following inequality
holds. Now, it is known as the Carleman’s inequality which was firstly discovered by Torsten Carleman.
The Carleman’s inequality appeared in many problems from pure and applied analysis. Up to now, many researchers have made great efforts to improve it. For example, using AM-GM inequality
where \(c_1, c_2,\cdots , c_n>0\), P\(\acute{o}\)lya [11, 12] obtained the following inequality:
Using \((1+1/n)^n<e\), the Carleman’s inequality (4.1) holds.
Almost all improvements stated in the recent past used upper bounds for \((1+1/n)^n\), stronger than \((1+1/n)^n<e\). For example, Mortici and Hu [9] used the double inequalities (1.4) to establish the following improvements of the Carleman’s inequality:
and
where
and \(a_n>0\) such that \(\sum a_n<\infty \).
Using the same idea, combining Theorems 3 and 4, we establish the following improvements of the Carleman’s inequality.
Theorem 5
Let \(a_n>0\) such that \(\sum a_n<\infty \). Then
and
Combining the comparisons in Sect. 4, it is easy to see that our upper bounds in (4.4) and (4.5) are sharper than ones in (4.2) and (4.3), respectively.
References
Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Nation Bureau of Standards, Applied Mathematical Series 55, 9th printing. Dover, New York (1972)
Hu, Y.: A strengthened Carleman’s inequality. Commun. Math. Anal. 1(2), 115–119 (2006)
Lu, D.: Some new convergent sequences and inequalities of Euler’s constant. J. Math. Anal. Appl. 419, 541–552 (2014)
Lu, D., Wang, X.: A new asymptotic expansion and some inequalities for the Gamma function. J. Math. Anal. Appl. 140, 314–323 (2014)
Lu, D., Song, L., Ma, C.: A generated approximation of the gamma function related to Windschitl’s formula. J. Number Theory. 140, 215–225 (2014)
Lu, D., Song, L., Yu, Y.: Some new continued fraction approximation of Euler’s constant. J. Number Theory. 147, 69–80 (2014)
Mortici, C.: Product approximations via asymptotic integration. Am. Math. Mon. 117(5), 434–441 (2010)
Mortici, C.: A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402, 405–410 (2013)
Mortici, C., Hu, Y.: On some convergences to the constant e and improvements of Carleman’s inequality. Carpathian J. Math. 31(2), 243–249 (2015)
Ping, Y., Sun, G.: A strengthened Carleman’s inequality. J. Math. Anal. Appl. 240, 290–293 (1999)
Pólya, G.: Proof of an inequality. Proc. Lond. Math. Soc. 24(2), 55 (1925)
Pólya, G.: With, or without motivation? Am. Math. Mon. 56, 684–691 (1949)
Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. I. Springer, New York (1972)
Yang, X.: On Carleman’s inequality. J. Math. Anal. Appl. 253, 691–694 (2001)
Yang, B., Debnath, L.: Some inequalities involving the constant e and an application to Carleman’s inequality. J. Math. Anal. Appl. 223, 347–353 (1998)
Acknowledgments
Computations made in this paper were performed using Maple software.
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The research of the first author was supported by the National Natural Science Foundation of China (Grant No. 11571058) and the Fundamental Research Funds for the Central Universities (Grant No. DUT15LK19). The second author was supported by the National Natural Science Foundation of China (Grant No. 11371077).
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Lu, D., Song, L. & Liu, Z. Some new approximations and inequalities of the sequence \((1+1/n)^n\) and improvements of Carleman’s inequality. Ramanujan J 43, 69–82 (2017). https://doi.org/10.1007/s11139-015-9765-x
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DOI: https://doi.org/10.1007/s11139-015-9765-x