Some Refinements and Generalizations of I. Schur Type Inequalities
Recently, extensive researches on estimating the value of e have been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help of Maple and automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.
It is well known that and are, respectively, monotone increasing and monotone decreasing, and both of them converge to the constant . In fact, extensive researches for the estimated value of have been studied [1–4], and the methods for estimating the value of are of benefit to the improvements of the Hardy inequality, Carleman inequality, Gamma function inequality, and so forth [5–13], which is an essential motivation for this work. Klambauer and Schur have reached the following conclusion.
Lemma 1 (see ). Both and are monotone decreasing sequences if and only if .
Recently, I. Schur has obtained the so-called I. Schur inequality as follows: where and .
It can also solve the problem proposed by Klambauer in : “Is the contained in a quarter of the interval of ?” Therefore, here is included in the interval length of
In , Xu and Yang have obtained a series of improved forms of the I. Schur inequalities, from which there is one conclusion which can be drawn that the necessary and sufficient condition for is .
Many new estimated values on have been obtained when researchers study the improvements of the Carleman inequality; see [24–26] for details. We mainly analyze the structural characteristics of I. Schur type inequalities and find out their upper and lower bounds are closely related to the mean value sequences of and . At last, we generalized a series of conclusions on the developments of I. Schur inequality via introducing variable parameters.
The rest of work is organized as follows. In Section 2, upper and lower bounds for a series of the I. Schur type inequalities are improved. Meanwhile, novel I. Schur-like inequalities are obtained by means of the variable parameter method. Some numerical examples are given to show the reliability of the approximation of new upper and lower bounds in Section 3. Finally, the paper closes with conclusions in Section 4.
2. Main Results and Proofs
2.1. Improvements Based on Computation of Mean Value
In this section, we consider the arithmetic mean value , geometric mean value , logarithmic mean value , harmonic mean value , and antilogarithmic mean of and , respectively; they are where is a monotone increasing sequence, see  for a discussion of this issue, which means that the left-hand side of I. Schur inequality is valid. A lower bound for I. Schur inequality can be obtained via the monotone convergence theorem [27, pp. 87-88]. Moreover, it has been confirmed that and are monotone decreasing in : both and are monotone decreasing sequences, and we obtained two inequalities as follows: The first inequality has verified the rationality of the right-hand side of I. Schur inequality, and the second inequality has put forward a method for sharpening the upper bound of I. Schur inequality because of . Besides, the monotonicity of the remaining mean value sequences should also be checked, and the conclusions can be drawn as follows.
Proposition 2. is a monotone decreasing sequence.
Remark 3. We can also continue to construct geometric mean value sequence . By Lemma 1 and , we can obtain that is a monotone decreasing sequence; thus .
Consider , times of geometric mean value operation by combining with Lemma 1 and , and we have , which is a monotone increasing sequence for each .
Theorem 4. is a monotone increasing sequence, and the following inequality holds for .
Proof. We consider the monotonicity of , and when , and hence is monotone decreasing. It is known that is monotone increasing, and . According to monotonous property of the composite functions, is a monotone decreasing sequence; thus Take ; then Based on this conclusion, it is not hard to note that it is an improvement of upper bound for I. Schur inequality. There is another conclusion shown as follows (see ).
Proposition 5. The double inequality for is valid in the sense that the maximum and minimum in (11) are best possible.
Then we study the necessary and sufficient condition for the validness of the class of and obtain the following.
Proposition 6. The inequality holds if and only if , and is the best constant.
In fact, the proof of this theorem can be introduced via the conclusions in  as follows: Use the left-hand side of (12) and change into , ; then and we can also obtain . The conclusion on the best optimality of can be found in . It needs to be mentioned that the authors have made a mistake when they cited this inequality in ; for instance, let ; then , but . It means that the right-hand side of the inequality may not be valid. It is found that they mistake for after checking the original paper . In fact, we can prove , and the constant is the best possible.
Theorem 7. The sequence is monotone decreasing, and the inequality holds for .
Proof. We consider , ; then where . By , we obtain (). Therefore, is monotone decreasing, and we can straightforwardly prove the desired inequality of the theorem with .
Theorem 8. The sequence is monotone decreasing, and the following inequality holds for .
Proof. We consider , , then , and where . According to , we obtain (). Therefore, is monotone decreasing, and we can straightforwardly prove the desired inequality with .
In fact, the previous theorems can be viewed as the different improvements of original I. Schur inequality. The above improved inequalities follow the motivation of researching mean value sequences of the upper and lower bounds for original inequalities; the authors study the monotonicity of the sequences constructed by the mean value of upper and lower bounds for I. Schur inequality and its relationship with . The arithmetic mean value should be replaced with the “weighted” mean value , ; then we have the following.
Theorem 9. For and , the sequence is monotone decreasing, and the inequality holds.
Proof. We consider ; then Now rewrite numerator of (20) and denote . This is a quadratic function of , and its discriminant is We note that two roots of are It is not hard to prove that is strictly monotone decreasing, and and is strictly monotone increasing; we also have Hence, the inequality always holds for and . By using , thus . It shows that sequence is monotone increasing under this condition, and the inequality in this theorem can be proved by using .
2.2. Further Discussions Based on Introducing the Parameters
Here we study the monotonicity of two new sequences and their relationships with by introducing real parameter . Two sequences are, respectively, and , where , and it is easy to obtain . Next, we consider their arithmetic mean value sequence, geometric mean value sequence, and harmonic mean value sequence, respectively; consider
Theorem 10. For and , the sequences are monotone decreasing, and the inequality holds.
Proof. We consider , ; then Denote the numerator of in (26) as where . Take ; then and It is found that if and , then . Furthermore, it implies that then is monotone increasing, and has been verified, such that , and are monotone decreasing; hence can be proved because of .
Theorem 11. For and , the sequences are monotone decreasing, and the inequality holds.
Proof. We consider , ; then Denote the numerator of (31) as , which is a quadratic function of . And its discriminant is . Then two roots of are We note that is strict monotone decreasing and , . According to the characteristic of parabola curve, always holds for and ; then from , . It shows that the sequences are monotone decreasing.
Theorem 12. If , then are monotone increasing, and the following inequality holds for .
Proof. We consider , ; then
Denote numerator of (35) as
where . Since , that is, , we have
It implies that the inequality always holds for . Then is monotone decreasing, and according to , we have . Hence, are monotone increasing sequences, and we prove that
A remaining issue of Theorem 12 can be described as follows: when , are the monotone increasing or monotone decreasing? With the help of the Bottema (available at the site of http://old.irgoc.org/Soft/ShowSoft.asp?SoftID=15, and the practical implementing Maple codes also are available by the authors’ email) (developed by Yang and Xia and based on the Maple programme; see  and references therein), we can find that the inequality holds for , . Moreover, it can be also shown in Algorithm 1.
Then we can conclude that , , is monotone decreasing under the previous discussion in the proof of the Theorem 12. The further conclusions need to be studied and proven by analytical methods in future work.
3. Numerical Examples
In this section, we will display some new upper and lower bounds of the I. Schur inequality by using the Matlab 2011b in a personal computer. We give some figures to show their variation trend; the symbols here mean the corresponding real continuous function for , , , , and , respectively.
Form Figure 1, we can note that the original upper bound of the I. Schur inequality is not better than the other novel upper bounds ; the is the best upper bound among five upper bounds. Moreover, we note that the variation curves of , are greatly similar; they are both the other alternatives. This figure also shows that the improvement of the upper bound of the I. Schur inequality is beneficial.
Firstly, we define the corresponding real continuous function for , , , , and , respectively. From Figure 2, we can note that the original lower bound of the I. Schur inequality is better than the other novel lower bounds ; the is an alternative among other novel lower bounds. This figure shows that we improve the lower bound of the I. Schur inequality and it is not very successful. But we give some generalized upper and lower bounds for the , ; this work is meaningful in terms of the generalized type of the I. Schur inequality.
From Table 1, we find that can be viewed as a good lower bound of , in terms of approximating precision, especially for the case with . It is favorable to note that the reliability of the approximation of based on depends on the closeness between and 1.
In the previous sections, we have studied the monotonicity of the sequences and their variants based on various mean values of and , such as the arithmetic mean value, geometric mean value, logarithmic mean value, and harmonic mean value. Besides, we have given some extensions and remarks for known results in the recent literature. At the same time, we extend some ideas and conclusions in [26, 30–32]. In conclusion, it is not hard to find that the conclusions of previous generalized sequences with variable parameters are in terms of the I. Schur type inequality via the parameter techniques and are a kind of extensions of I. Schur type inequality.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Xian-Ming Gu carried out the main part of this paper. Ting-Zhu Huang, Wei-Ru Xu, Hou-Biao Li, Liang Li, and Xi-Le Zhao participated in the discussion and corrected the main theorem. All authors read and approved the final paper.
The authors would like to express their thankfulness to the referees for their suggestions while revising this paper. This research is supported by 973 Program (2013CB329404), NSFC (61370147, 61170311, 11101071, and 11301057), the Fundamental Research Funds for the Central Universities (ZYGX2013J106), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), and the Fundamental Research Funds for China Scholarship Council, and the project was sponsored by OATF, UESTC.
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