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Sharp inequalities related to the constant e
Journal of Inequalities and Applications volume 2014, Article number: 382 (2014)
Abstract
The aim of this work is to extend the results obtained by Batir and Cancan in (Int. J. Math. Educ. Sci. Technol. 40(8):1101-1109, 2009).
MSC:26A09, 33B10, 26D99.
1 Introduction and preliminary results
Batir and Cancan [[1], Theorem 2.5] presented the following sharp inequalities:
where and . The proof is based on the fact that the function
is strictly increasing on , while inequalities (1) follow from . Such an approach of the problem does not offer good results in the left-hand side inequality (1), when n approaches infinity. As we wish to see (1) as an accurate approximation of the form
we are interested in finding which gives the best such approximation for large values of n. Moreover, numerical computations show us that for large values of n, the expression gets closer to the right-hand side of (1). This fact suggests us that the best approximation (2) is obtained when tends to , as . For , we deduce
but a better result is
The rigorous argument is the following theorem, which is also an improvement of the Batir and Cancan inequality (1).
Theorem 1 For every real number , we have
Proof Let
and
We have , and
where
Evidently, f is concave, g is convex on , with , so and on . The proof is completed. □
The same remarks we make on Batir and Cancan’s assertion [[1], Theorem 2.6], which is proven to have some computation errors, since the expression can be approximated for large values of n as
but a better result is
as we can see from the following.
Theorem 2 For every real number , we have
Proof Let
We have , and
where
Evidently, v is concave, u is convex, with , so and on . The proof is completed. □
2 Some extensions
In this section we discuss the problem of approximating , , in the form
More precisely, we propose as an open problem the following approximation formula:
Particular values and in (5) lead again to (3) and (4).
The special case is treated at the final part of this section.
By now, we proved the approximation formula (5) for
where is the unique real root of .
This assertion is sustained by the following three theorems.
Theorem 3 Let . Then there exists (depending on a) such that for every real number , the following inequalities hold true:
Theorem 4 Let . Then there exists (depending on a) such that, for every real number , the following inequalities hold true:
Theorem 5 Let . Then there exists (depending on a) such that, for every real number , the following inequalities hold true:
Inequalities (6)-(8) are closely related to the functions
and
We have
where
and
where
Proofs of Theorems 3 and 4 For , the leading coefficients of the polynomials P and Q are negative. We are in a position to consider (depending on a) such that and , for every . By (9) and (10), s is concave and t is convex. But , so and on .
Now inequalities and , for every are (6) and (7) and we are done. □
Proof of Theorem 5 For , the leading coefficients of polynomials P and Q are positive. We are in a position to consider (depending on a) such that and , for every . By (9) and (10), now s is convex and t is concave. But , so and on .
Now inequalities and , for every are (6) and (7) and we are done. □
The special case provides the approximation formula
It is sustained by the following.
Theorem 6 For every real number , the following inequalities hold:
Proof Let us define the functions
and
We have
and
where
Evidently, b is concave, c is convex, with , so and on . The proof is completed. □
In case , the entire asymptotic representation
can be constructed. In this sense, we write (12) as
where
By using the Maclaurin expansion of , with , we deduce that
Now the coefficients in (12) can be inductively obtained by equating the following relation:
The first coefficients are , (see (11)), then , , ….
Further research in the problem of approximating the constant e can be found in [1–4].
Finally, we leave as an open problem the approximation formula (5) for values of other than those discussed in this paper.
References
Batir N, Cancan M: Sharp inequalities involving the constant e and the sequence . Int. J. Math. Educ. Sci. Technol. 2009,40(8):1101-1109. 10.1080/00207390902912902
Agarwal RP, Sen SK: Pi, Epsilon, Phi with MATLAB: Random and Rational Sequences with Scope in Supercomputing Era. Cambridge Scientific Publishers, Cambridge; 2011.
Mortici C: Refinements of some bounds related to the constant e . Miskolc Math. Notes 2011,12(1):105-111.
Mortici C: A quicker convergence toward the gamma constant with the logarithm term involving the constant e . Carpath. J. Math. 2010,26(1):86-91.
Acknowledgements
The authors would like to thank the Editor and the referees for the comments that improved the initial form of this manuscript. The work of Cristinel Mortici was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI project number PN-II-ID-PCE-2011-3-0087. Cristinel Mortici contributed to this work during his visit at National Technical University of Athens, Greece. He would like to thank Prof. Themistocles M Rassias for hospitality.
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The paper is a joint work of all the authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.
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Hu, Y., Mortici, C. Sharp inequalities related to the constant e. J Inequal Appl 2014, 382 (2014). https://doi.org/10.1186/1029-242X-2014-382
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DOI: https://doi.org/10.1186/1029-242X-2014-382