Some Refinements and Generalizations of I. Schur Type Inequalities

Abstract

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.

1. Introduction

It is well known that 𝑥_𝑛=(1+(1/𝑛))^𝑛 and 𝑦_𝑛=(1+(1/𝑛))^𝑛+1 are, respectively, monotone increasing and monotone decreasing, and both of them converge to the constant 𝑒. In fact, extensive researches for the estimated value of 𝑒 have been studied [1–4], and the methods for estimating the value of 𝑒 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 [14]). Both 𝑠_𝑛=(1+(1/𝑛))^𝑛+𝛼 and 𝑡_𝑛=(1+(1/𝑛))^𝑛+1(1+(𝛼/𝑛)) are monotone decreasing sequences if and only if 𝛼≥(1/2).

In fact, it is not hard to prove that 1𝑒<1+𝑛^𝑛+(1/2)=11+𝑛11+𝑛^𝑛, (1) which has been proved by different ways; refer to [14–17].

Besides, Fischer and Qi had further studied this issue (see [18–20]) and they demonstrated that 𝑥_𝑛 is a monotone increasing sequence if and only if 𝛼≤2ln3−3ln22ln2−ln3=0.409…. (2)

Moreover, Alzer and Qi have obtained the necessary and sufficient conditions for the monotonicity of generalized types of 𝑏_𝑛=(1+(𝛼/𝑛))^𝑛+𝛽 and 𝐹_𝛼,𝛽(𝑥)=(1+(𝛼/𝑥))^𝑥+𝛽; see [21, 22] for details.

Recently, I. Schur has obtained the so-called I. Schur inequality as follows: 𝑝_𝑛<𝑒<𝑞_𝑛, (3) where 𝑝_𝑛=𝑥_𝑛(1+(1/(2𝑛+1))) and 𝑞_𝑛=𝑥_𝑛(1+(1/2𝑛)).

It can also solve the problem proposed by Klambauer in [15]: “Is the 𝑒 contained in a quarter of the interval of (1+(1/𝑛))^𝑛<𝑒<(1+(1/𝑛))^𝑛+1?” Therefore, here 𝑒 is included in the interval length of 1𝐸=1+−12𝑛1+=12𝑛+1()≤12𝑛2𝑛+16,𝑛=1,2,…. (4)

In [23], 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 (1+(1/𝑛))^𝑛(1+(1/𝑎𝑛))<𝑒<(1+(1/𝑛))^𝑛(1+(1/2𝑛)) is 𝑎>(2/(𝑒−2)).

Many new estimated values on 𝑒 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 𝑥_𝑛 and 𝑦_𝑛. 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 𝐴_𝑛, geometric mean value 𝐺_𝑛, logarithmic mean value 𝐿_𝑛, harmonic mean value 𝐻_𝑛, and antilogarithmic mean Ω_𝑛 of 𝑥_𝑛 and 𝑦_𝑛, respectively; they are 𝐴_𝑛=𝑥_𝑛+𝑦_𝑛2=(1+(1/𝑛))^𝑛+(1+(1/𝑛))^𝑛+12=𝑥_𝑛11+,𝐺2𝑛_𝑛=𝑥_𝑛𝑦_𝑛^1/2=11+𝑛^𝑛11+𝑛^𝑛+11/2=𝑥_𝑛11+𝑛^1/2,𝐻_𝑛=2𝑥_𝑛𝑦_𝑛𝑥_𝑛+𝑦_𝑛=𝑥_𝑛11+,𝐿2𝑛+1_𝑛=𝑦_𝑛−𝑥_𝑛ln𝑦_𝑛−ln𝑥_𝑛=(1+(1/𝑛))^𝑛+1−(1+(1/𝑛))^𝑛ln(1+(1/𝑛))^𝑛+1−ln(1+(1/𝑛))^𝑛=𝑥_𝑛((,Ω𝑛ln1+1/𝑛))_𝑛=𝑥_𝑛𝑦_𝑛ln𝑦_𝑛−ln𝑥_𝑛𝑦_𝑛−𝑥_𝑛=𝑥_𝑛11+𝑛ln𝑥_𝑛=𝑥_𝑛()1𝑛+1ln1+𝑛, (5) where 𝐻_𝑛 is a monotone increasing sequence, see [14] 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 𝐴_𝑛 and Ω_𝑛 are monotone decreasing in [28]: both 𝐴_𝑛=𝑥_𝑛(1+(1/2𝑛)) and Ω_𝑛=𝑥_𝑛(𝑛+1)ln(1+(1/𝑛)) are monotone decreasing sequences, and we obtained two inequalities as follows: 𝑒<𝑥_𝑛11+2𝑛,𝑒<𝑥_𝑛()1𝑛+1ln1+𝑛,𝑛=1,2,…. (6) 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 (1+(1/2𝑛))>(𝑛+1)ln(1+(1/𝑛)),𝑛∈𝑁^∗. Besides, the monotonicity of the remaining mean value sequences 𝐺_𝑛,𝐿_𝑛 should also be checked, and the conclusions can be drawn as follows.

Proposition 2. 𝐺_𝑛=𝑥_𝑛(1+(1/𝑛))^1/2(𝑛∈𝑁^∗) is a monotone decreasing sequence.

Remark 3. (1) We can also continue to construct geometric mean value sequence 𝑄_𝑛=((1+(1/𝑛))^2𝑛(1+(1/𝑛))^1/2)^1/2=𝑥_𝑛(1+(1/𝑛))^1/4. By Lemma 1 and (1/4)=0.25<((2ln3−3ln2)/(2ln2−ln3))=0.409…, we can obtain that 𝑄_𝑛=𝑥_𝑛(1+(1/𝑛))^1/4 is a monotone decreasing sequence; thus 𝑥_𝑛(1+(1/𝑛))^1/4<𝑒.
(2) Consider 𝑚≥2,𝑚∈𝑁^∗, times of geometric mean value operation by combining with Lemma 1 and (1/2^𝑚)<((2ln3−3ln2)/(2ln2−ln3))=0.409…, and we have 𝑄_𝑛=𝑥_𝑛(1+(1/𝑛))^1/2𝑚<𝑒, which is a monotone increasing sequence for each 𝑚.

Theorem 4. 𝐿_𝑛=𝑥_𝑛/(𝑛ln(1+(1/𝑛))) is a monotone increasing sequence, and the following inequality 𝐿_𝑛=𝑥_𝑛𝑛ln(1+(1/𝑛))>𝑒 (7) holds for 𝑛∈𝑁^∗.

Proof. We consider the monotonicity of 𝑓(𝑥)=(𝑥/ln𝑥),(1<𝑥<𝑒), and 𝑓^′(𝑥)=ln𝑥−1(ln𝑥)², (8) when 𝑥∈(1,𝑒), 𝑓^′(𝑥)<0 and hence 𝑓(𝑥) is monotone decreasing. It is known that 𝑥_𝑛=(1+(1/𝑛))^𝑛 is monotone increasing, and 𝑥_𝑛=(1+(1/𝑛))^𝑛∈(1,𝑒). According to monotonous property of the composite functions, 𝐿_𝑛=𝑥_𝑛/(𝑛ln(1+(1/𝑛))) is a monotone decreasing sequence; thus 𝑓(𝑥)=𝑥(𝑒)=𝑒ln𝑥>𝑓ln𝑒=𝑒. (9) Take 𝑥=(1+(1/𝑛))^𝑛∈(1,𝑒); then 𝐿_𝑛=𝑥_𝑛𝑛ln(1+(1/𝑛))>𝑒. (10) 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 [14]).

Proposition 5. The double inequality 11+𝑛^𝑛+𝛼1≤𝑒≤1+𝑛^𝑛+𝛽 (11) for 𝑛≥1 is valid in the sense that the maximum 𝛼=(1/ln2)−1 and minimum 𝛽=1/2 in (11) are best possible.

Then we study the necessary and sufficient condition for the validness of the class of (1+(1/(2𝑛+𝛽)))𝑥_𝑛≤𝑒 and obtain the following.

Proposition 6. The inequality (1+(1/(2𝑛+𝛽)))𝑥_𝑛≤𝑒 holds if and only if 𝛽≥(5/6), and 𝛽=5/6 is the best constant.

In fact, the proof of this theorem can be introduced via the conclusions in [14] as follows: 6𝑒112𝑥+11<𝑒−1+𝑥^𝑥<7𝑒().14𝑥+12𝑥≥1 (12) Use the left-hand side of (12) and change 𝑥 into 𝑛, 𝑛∈𝑁^∗; then 11+𝑛^𝑛<𝑒−6𝑒=12𝑛+1112𝑛+512𝑛+11𝑒,𝑛∈𝑁^∗ (13) and we can also obtain (1+(1/(2𝑛+(5/6))))𝑥_𝑛≤𝑒. The conclusion on the best optimality of 𝛽 can be found in [13]. It needs to be mentioned that the authors have made a mistake when they cited this inequality in [14]; for instance, let 𝑥=(1/10); then 𝑒−(1+(1/𝑥))^𝑥=1.447300213…, but (7𝑒/(14𝑥+12))=1.419997970…. It means that the right-hand side of the inequality may not be valid. It is found that they mistake 𝑥≥1 for 𝑥>0 after checking the original paper [8]. In fact, we can prove 𝑒−(1+(1/𝑥))^𝑥<(𝑒/(2𝑥+𝛼)),𝑥>0, and the constant 𝛼=𝑒/(𝑒−1) is the best possible.

Theorem 7. The sequence 𝑆_𝑛=𝑝_𝑛+𝑞_𝑛2=𝑥_𝑛8𝑛²+8𝑛+18𝑛²+4𝑛 (14) is monotone decreasing, and the inequality 𝑥_𝑛(1+(4𝑛+1)/(8𝑛²+4𝑛))>𝑒 holds for 𝑛∈𝑁^∗.

Proof. We consider 𝑔(𝑥)=ln(8𝑥²+8𝑥+1)−ln(8𝑥²+4𝑥)+𝑥ln(𝑥+1)−𝑥ln𝑥, 𝑥>0; then 𝑔^′(𝑥)=16𝑥+88𝑥²−+8𝑥+116𝑥+48𝑥²()1+4𝑥+ln𝑥+1−ln𝑥−,𝑔𝑥+1_′′(𝑥)=168𝑥²−()+8𝑥+116𝑥+8²8𝑥²+8𝑥+1²−168𝑥²+()+4𝑥16𝑥+4²8𝑥²+4𝑥²−1𝑥(𝑥+1)²=128𝑥⁶+384𝑥⁵+488𝑥⁴+336𝑥³+125𝑥²×+21𝑥+18𝑥²+8𝑥+1²(2𝑥+1)²(𝑥+1)²𝑥²⁻¹>0, (15) where 𝑥>0. By lim_𝑥→∞𝑔^′(𝑥)=0, we obtain 𝑔^′(𝑥)<0 (𝑥>0). Therefore, 𝑆_𝑛 is monotone decreasing, and we can straightforwardly prove the desired inequality of the theorem with lim_𝑛→∞𝑆_𝑛=𝑒.

Theorem 8. The sequence 𝑇_𝑛=2𝑝_𝑛𝑞_𝑛𝑝_𝑛+𝑞_𝑛=𝑥_𝑛8𝑛²+12𝑛+48𝑛²+8𝑛+1 (16) is monotone decreasing, and the following inequality 𝑥_𝑛8𝑛²+12𝑛+48𝑛²+8𝑛+1=𝑥_𝑛1+4𝑛+38𝑛²+8𝑛+1>𝑒 (17) holds for 𝑛∈𝑁^∗.

Proof. We consider ℎ(𝑥)=ln(8𝑥²+12𝑥+4)−ln(8𝑥²+8𝑥+1)+𝑥ln(𝑥+1)−𝑥ln𝑥, 𝑥≥1, then ℎ^′(𝑥)=((16𝑥+12)/(8𝑥²+12𝑥+4))−((16𝑥+8)/(8𝑥²+8𝑥+1))+ln(𝑥+1)+(𝑥/(𝑥+1))−ln𝑥−1, and ℎ^′′(𝑥)=168𝑥²−()+12𝑥+416𝑥+12²8𝑥²+12𝑥+4²−168𝑥²+()+8𝑥+116𝑥+8²8𝑥²+8𝑥+1²+2−𝑥𝑥+1(𝑥+1)²−1𝑥=128𝑥⁴+256𝑥³+152𝑥²+24𝑥−1𝑥(𝑥+1)8𝑥²+8𝑥+1²(2𝑥+1)²>0, (18) where 𝑥≥1. According to lim_𝑥→∞ℎ^′(𝑥)=0, we obtain ℎ^′(𝑥)<0 (𝑥≥1). Therefore, 𝑇_𝑛 is monotone decreasing, and we can straightforwardly prove the desired inequality with lim_𝑛→∞𝑇_𝑛=𝑒.

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 𝑒. The arithmetic mean value should be replaced with the “weighted” mean value 𝑊_𝑛=(𝑥_𝑛+𝜆𝑦_𝑛)/(1+𝜆)=[1+𝜆/(𝜆+1)𝑛]𝑥_𝑛, 𝜆∈ℝ; then we have the following.

Theorem 9. For 𝜆∈[(−1−√3)/4,(−1+√3)/4] and 𝑛∈𝑁^∗, the sequence 𝑊_𝑛 is monotone decreasing, and the inequality 𝑥_𝑛[1+𝜆/(𝜆+1)𝑛]<𝑒 holds.

Proof. We consider 𝑑(𝑥)=ln(𝑥+𝜆𝑥+𝜆)−ln(𝑥+𝜆𝑥)+𝑥ln(𝑥+1)−𝑥ln𝑥(𝑥>0); then 𝑑^′(𝑥)=1+𝜆−𝑥+𝜆𝑥+𝜆1+𝜆()+𝑥𝑥+𝜆𝑥+ln𝑥+1𝑑𝑥+1−ln𝑥−1,_′′(𝑥)()=−1+𝜆²(𝑥+𝜆𝑥+𝜆)²+(1+𝜆)²(𝑥+𝜆𝑥)²+2−𝑥𝑥+1(𝑥+1)²−1𝑥=2𝜆𝑥²+3𝜆²𝑥²+3𝜆²𝑥+𝑥³𝜆²−𝑥³+𝜆²×()+2𝜆𝑥𝑥+1²𝑥²(𝑥+𝜆𝑥+𝜆)²⁻¹. (19)(20) Now rewrite numerator of (20) and denote 𝐼_𝑥(𝜆)=(3𝑥²+3𝑥+𝑥³+1)𝜆²+(2𝑥+2𝑥²)𝜆−𝑥³. This is a quadratic function of 𝜆, and its discriminant is Δ=2𝑥+2𝑥²²+43𝑥²+3𝑥+𝑥³+1𝑥³>0. (21) We note that two roots of 𝐼_𝑥(𝜆) are 𝜆₁√=−1+1+𝑥²+𝑥𝑥(𝑥+1)²,𝜆₂=−1+√1+𝑥²+𝑥𝑥(𝑥+1)². (22) It is not hard to prove that {−((1+√1+𝑥²+𝑥)𝑥/(𝑥+1)²)}^∞_𝑥=1 is strictly monotone decreasing, and √−1<−1+1+𝑥²+𝑥𝑥(𝑥+1)²√≤−1+34=−0.6830127…, (23) and {(−1+√1+𝑥²+𝑥)𝑥/(𝑥+1)²}^∞_𝑥=1 is strictly monotone increasing; we also have −1+√34√=0.1830127…≤−1+1+𝑥²+𝑥𝑥(𝑥+1)²<1. (24) Hence, the inequality 𝐼_𝑥(𝜆)≤0 always holds for 𝑥≥1 and 𝜆∈[(−1−√3)/4,(−1+√3)/4]. By using lim_𝑥→∞𝑑^′(𝑥)=0, thus 𝑑^′(𝑥)≥0. It shows that sequence 𝑊_𝑛=[1+𝜆/(𝜆+1)𝑛]𝑥_𝑛 is monotone increasing under this condition, and the inequality in this theorem can be proved by using lim_𝑛→∞𝑊_𝑛=𝑒.

2.2. Further Discussions Based on Introducing the Parameters

Here we study the monotonicity of two new sequences and their relationships with 𝑒 by introducing real parameter 𝜆. Two sequences are, respectively, 𝑖_𝑛=(1+(𝜆/𝑛))^𝑛 and 𝑗_𝑛=(1+(𝜆/𝑛))^𝑛+1, where 𝜆>0, and it is easy to obtain lim_𝑛→∞𝑖_𝑛=lim_𝑛→∞𝑗_𝑛=𝑒^𝜆. Next, we consider their arithmetic mean value sequence, geometric mean value sequence, and harmonic mean value sequence, respectively; consider 𝐴₁()=𝑖𝑛,𝜆^𝑛+𝑗_𝑛2=𝜆1+𝜆2𝑛1+𝑛^𝑛,𝐺₁(𝑛,𝜆)=𝑖_𝑛𝑗_𝑛^1/2=𝜆1+𝑛^1/2𝜆1+𝑛^𝑛,𝐻₁(𝑛,𝜆)=2𝑖_𝑛𝑗_𝑛𝑖_𝑛+𝑗_𝑛=𝜆1+𝜆2𝑛+𝜆1+𝑛^𝑛. (25)

Theorem 10. For 0<𝜆≤1 and 𝑛∈𝑁^∗, the sequences 𝐴₁(𝑛,𝜆)=(1+(𝜆/2𝑛))(1+(𝜆/𝑛))^𝑛 are monotone decreasing, and the inequality 𝑒^𝜆<(1+(𝜆/2𝑛))(1+(𝜆/𝑛))^𝑛 holds.

Proof. We consider 𝑀(𝑥)=𝑥ln(𝑥+𝜆)−𝑥ln𝑥+ln(2𝑥+𝜆)−ln(2𝑥), 𝑥>0; then 𝑀^′(𝑥)()+𝑥=ln𝑥+𝜆2𝑥+𝜆−ln𝑥−1+−12𝑥+𝜆𝑥,𝑀^′′(𝑥)1=2−𝑥𝑥+𝜆(𝑥+𝜆)²−1𝑥−4(2𝑥+𝜆)²+1𝑥²=−𝜆4𝑥³𝜆+4𝑥²𝜆²+𝑥𝜆³−4𝑥³−9𝑥²𝜆−6𝑥𝜆²−𝜆³×𝑥²(𝑥+𝜆)²(2𝑥+𝜆)²⁻¹. (26) Denote the numerator of 𝑀^′′(𝑥) in (26) as 𝑁(𝑥)∶=−𝜆4𝑥³𝜆+4𝑥²𝜆²+𝑥𝜆³−4𝑥³−9𝑥²𝜆−6𝑥𝜆²−𝜆³∶=𝜆𝑌(𝑥), (27) where 𝑌(𝑥)∶=4𝑥³−4𝑥³𝜆+9𝑥²𝜆−4𝑥²𝜆²+6𝑥𝜆²−𝑥𝜆³+𝜆³. Take ∀𝑝≥0; then 𝜆∶=1/(𝑝+1)∈(0,1] and 𝑌(𝑥)∶=4𝑥³−4𝑥³+𝑝+19𝑥²−𝑝+14𝑥²𝑝+1²+6𝑥𝑝+1²−𝑥𝑝+1³+1𝑝+1³=4𝑥³𝑝³+8𝑥³𝑝²+4𝑥³𝑝+9𝑥²𝑝²+14𝑥²𝑝+5𝑥²+6𝑥𝑝+5𝑥+1×𝑝+1³⁻¹. (28) It is found that if 0<𝜆≤1 and 𝑥>0, then 𝑌(𝑥)≥0. Furthermore, it implies that 𝑁(𝑥)=𝜆𝑌(𝑥)≥0; (29) then 𝑀^′(𝑥) is monotone increasing, and lim_𝑥→∞𝑀^′(𝑥)=0 has been verified, such that 𝑀^′(𝑥)<0, and 𝐴₁(𝑛,𝜆)=(1+(𝜆/2𝑛))(1+(𝜆/𝑛))^𝑛 are monotone decreasing; hence 𝑒^𝜆<(1+(𝜆/2𝑛))(1+(𝜆/𝑛))^𝑛 can be proved because of lim_𝑛→∞𝐴₁(𝑛,𝜆)=𝑒^𝜆.

Theorem 11. For 0<𝜆≤1 and 𝑛∈𝑁^∗, the sequences 𝐺₁(𝑛,𝜆)=(1+(𝜆/𝑛))^1/2(1+(𝜆/𝑛))^𝑛 are monotone decreasing, and the inequality (1+(𝜆/𝑛))^1/2(1+(𝜆/𝑛))^𝑛>𝑒^𝜆 holds.

Proof. We consider 𝑓₁=𝑥ln(1+(𝜆/𝑥))+(1/2)ln(1+(𝜆/𝑥)), 𝑥≥1; then 𝑓^′₁(𝑥)𝜆=ln1+𝑥−𝜆𝑥((−𝜆1+𝜆/𝑥))2𝑥²((,𝑓1+𝜆/𝑥))_1′′(𝑥)=−𝜆²𝑥³(1+(1/𝑥))²+𝜆𝑥³((−𝜆1+𝜆/𝑥))₂2𝑥⁴(1+(𝜆/𝑥))²=−𝜆(2𝜆𝑥−2𝑥−𝜆)2𝑥²(𝑥+𝜆)²=(−2𝑥+1)𝜆²+2𝜆𝑥2𝑥²(𝑥+𝜆)². (30)(31) Denote the numerator of (31) as 𝛿(𝜆)=(−2𝑥+1)𝜆²+2𝑥𝜆, which is a quadratic function of 𝜆. And its discriminant is Δ₁=4𝑥²>0. Then two roots of 𝛿(𝜆) are 𝜆₁=0,𝜆₂=2𝑥.2𝑥−1 (32) We note that {2𝑥/(2𝑥−1)}^∞_𝑥=1 is strict monotone decreasing and 1<(2𝑥/(2𝑥−1))≤2, 𝑥≥1. According to the characteristic of parabola curve, 𝛿(𝜆)=(−2𝑥+1)𝜆²+2𝑥𝜆>0 always holds for 0<𝜆≤1 and 𝑥≥1; then from lim_𝑥→∞𝑓^′₁(𝑥)=0, 𝑓^′₁(𝑥)≤0. It shows that the sequences 𝐺₁(𝑛,𝜆) are monotone decreasing.

Theorem 12. If 𝜆≥1, then 𝐻₁(𝑛,𝜆)=(2𝑖_𝑛𝑗_𝑛)/(𝑖_𝑛+𝑗_𝑛)=(1+𝜆/(2𝑛+𝜆))(1+(𝜆/𝑛))^𝑛 are monotone increasing, and the following inequality 𝜆1+𝜆2𝑛+𝜆1+𝑛^𝑛<𝑒^𝜆 (33) holds for 𝑛∈𝑁^∗.

Proof. We consider 𝜑(𝑥)=ln(2𝑥+2𝜆)−ln(2𝑥+𝜆)+𝑥ln(𝑥+𝜆)−𝑥ln𝑥, 𝑥>0; then 𝜑^′(𝑥)=2−22𝑥+2𝜆()+𝑥2𝑥+𝜆+ln𝑥+𝜆𝜑𝑥+𝜆−ln𝑥−1,_′′(𝑥)=−4(2𝑥+2𝜆)²+4(2𝑥+𝜆)²+2−𝑥𝑥+𝜆(𝑥+𝜆)²−1𝑥=−𝜆−4𝑥²−3𝜆𝑥+4𝜆𝑥²+4𝑥𝜆²+𝜆³×(𝑥+𝜆)²(2𝑥+𝜆)²𝑥⁻¹. (34)(35) Denote numerator of (35) as 𝛾(𝑥)=−𝜆−4𝑥²−3𝜆𝑥+4𝜆𝑥²+4𝑥𝜆²+𝜆³∶=−𝜆𝑍(𝑥), (36) where 𝑍(𝑥)∶=−4𝑥²−3𝜆𝑥+4𝜆𝑥²+4𝑥𝜆²+𝜆³. Since 𝜏+1∶=𝜆≥1, that is, 𝜏≥0, we have 𝑍(𝑥)∶=−4𝑥²−3𝑥(𝜏+1)+4𝑥²()()𝜏+1+4𝑥𝜏+1²+(𝜏+1)³=4𝑥²𝜏+(𝜏+1)(4𝜏+1)𝑥+(𝜏+1)³()≥0.𝑥>0,𝜏≥0 (37) It implies that the inequality 𝛾(𝑥)=−𝜆𝑍(𝑥)≤0 always holds for 𝑥>0,𝜆≥1. Then 𝜑^′(𝑥) is monotone decreasing, and according to lim_𝑥→∞𝜑^′(𝑥)=0, we have 𝜑^′(𝑥)>0. Hence, 𝐻₁(𝑛,𝜆)=(1+𝜆/(2𝑛+𝜆))(1+(𝜆/𝑛))^𝑛 are monotone increasing sequences, and we prove that 𝑒^𝜆>𝜆1+𝜆2𝑛+𝜆1+𝑛^𝑛. (38)
A remaining issue of Theorem 12 can be described as follows: when 𝜆<1, are the 𝐻₁(𝑛,𝜆) 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 [29] and references therein), we can find that the inequality 𝛾(𝑥)≥0 holds for 0<𝜆≤(4/5), 𝑥≥1. Moreover, it can be also shown in Algorithm 1.
Then we can conclude that 𝐻₁(𝑛,𝜆), 𝑛=1,2,…, 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.

Input: (i)    read “D:/Program Files/bottema2009”; (ii)    ineq:= -lambda*(-4*x∧2-3*x*lambda+4*x∧2*lambda+4*x*lambda∧2+lambda∧3); (iii) xprove(ineq >= 0, [lambda <= 4/5, x >= 1]); Output:              Found border curves…         (−4𝑥2−3𝑥𝜆+4𝑥2𝜆+4𝑥𝜆2+𝜆3)(5𝜆−4)(𝑥−1)𝑥𝜆             Start to project curves…, 3.307                 [𝑥,𝜆]               do 1-th partition…            Start to find the sample points., 3.307              in 1-dimensional space…             finished in 1-dimensional space.              in 2-dimensional space…             finished in 2-dimensional space.            number(s) of sample points:, 1, 3.322                 [𝑥,𝜆]                 12,2                  −1               The inequality holds.                  true

Algorithm 1 

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 𝐿₀--𝐿₄ here mean the corresponding real continuous function for 𝐿₀=𝑥_𝑛(1+(1/2𝑛)), 𝐿₁=𝑥_𝑛(𝑛+1)ln(1+(1/𝑛)), 𝐿₂=(𝑥_𝑛/(𝑛ln(1+(1/𝑛)))), 𝐿₃=𝑥_𝑛(1+((4𝑛+1)/(8𝑛²+4𝑛))), and 𝐿₄=𝑥_𝑛(1+((4𝑛+3)/(8𝑛²+8𝑛+1))), respectively.

Form Figure 1, we can note that the original upper bound 𝐿₀ of the I. Schur inequality is not better than the other novel upper bounds 𝐿₁--𝐿₄; the 𝐿₁ is the best upper bound among five upper bounds. Moreover, we note that the variation curves of 𝐿₃, 𝐿₄ 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.

Figure 1 

The variation trend of the different upper bounds value of the I. Schur inequality.

Firstly, we define the corresponding real continuous function for 𝑇₁=𝑥_𝑛(1+(1/(2𝑛+1))),   𝑇₂=𝑥_𝑛(1+(1/𝑛))^1/8, 𝑇₃=𝑥_𝑛(1+(1/𝑛))^1/4,   𝑇₄=𝑥_𝑛(1+((1/100)/(1+(1/100))𝑛)), and 𝑇₅=𝑥_𝑛(1+(−1/3)/(1−(1/3))𝑛), respectively. From Figure 2, we can note that the original lower bound 𝑇₁ of the I. Schur inequality is better than the other novel lower bounds 𝑇₂--𝑇₅; the 𝑇₃ 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 𝑒^𝜆, 𝜆∈(0,1); this work is meaningful in terms of the generalized type of the I. Schur inequality.

Figure 2 

The variation trend of the different lower bounds value of the I. Schur inequality.

From Table 1, we find that 𝐻₁(𝑛,𝜆) can be viewed as a good lower bound of 𝑒^𝜆, 𝜆≥1 in terms of approximating precision, especially for the case with 𝜆=1. It is favorable to note that the reliability of the approximation of 𝑒^𝜆 based on 𝐻₁(𝑁,𝜆) depends on the closeness between 𝜆 and 1.

Table 1 

Using 𝐻₁(𝑁,𝜆) to approximate the 𝑒^𝜆, when 𝜆=1,3/2,2,3,4 with different 𝑛.

4. Conclusions

In the previous sections, we have studied the monotonicity of the sequences and their variants based on various mean values of 𝑥_𝑛 and 𝑦_𝑛, 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.

Authors’ Contribution

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.

Acknowledgments

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.