Journal of Function Spaces
Volume 2016 (2016), Article ID 4165601, 5 pages
http://dx.doi.org/10.1155/2016/4165601

Research Article

A Sharp Lower Bound for Toader-Qi Mean with Applications

Zhen-Hang Yang and Yu-Ming Chu

School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

Received 28 October 2015; Revised 10 December 2015; Accepted 24 December 2015

Academic Editor: Kehe Zhu

Copyright © 2016 Zhen-Hang Yang and Yu-Ming Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove that the inequality holds for all with if and only if , where , , and are, respectively, the Toader-Qi and -order logarithmic means of and . As applications, we find two fine inequalities chains for certain bivariate means.

1. Introduction

Let and with . Then the Toader-Qi mean [1–3] and -order logarithmic mean are defined by

respectively. In particular, is the classical logarithmic mean of and .

It is well-known that the -order logarithmic mean is continuous and strictly increasing with respect to for fixed with . Recently, the Toader-Qi and -order logarithmic means have been the subject of intensive research. In particular, many remarkable inequalities for the Toader-Qi and -order logarithmic means can be found in the literature [2–7].

In [2], Qi et al. proved that the identity

and the inequalities

hold for all with , where

is the modified Bessel function of the first kind [8] and , and are, respectively, the classical arithmetic, geometric, and identric means of and .

In [3], Yang proved that the double inequalities

and conjectured that the inequalities

hold for all with . Inequality (8) was proved by Yang et al. in [9].

Let and . Then from (1)–(3) we clearly see that

The main purpose of this paper is to give a positive answer to the conjecture given by (9). As applications, we present two fine inequalities chains for certain bivariate means and a lower bound for the kernel function of the Szász-Mirakjan-Durrmeyer operator.

2. Lemmas

In order to prove our main result we need several lemmas, which we present in this section.

Lemma 1 (see [10]). The double inequality

holds for all and , where is the classical Euler gamma function.

Lemma 2 (see [3]). Let be defined by (5). Then the identity

holds for all .

Lemma 3 (see [3]). The Wallis ratio

is strictly decreasing and log-convex with respect to all integers .

Lemma 4. The identity

holds for all and .

Proof. Let be the number of combinations of objects taken at a time. Then from the well-known binomial theorem we have

Equation (15) leads to

Lemma 5. Let with and

Then

for all .

Proof. Let be defined by (13). Then it follows from Lemmas 1 and 3 together with (17) and that

for all and .

3. Main Result

Theorem 6. The inequality

holds for all with if and only if .

Proof. Since both the Toader-Qi mean and -order logarithmic mean are symmetric and homogeneous and and is strictly increasing with respect to for all with , without loss of generality, we assume that and . Let . Then it follows from (10) that inequality (20) is equivalent to

for all .
If inequality (21) holds for all . Then (5) and (21) lead to

which gives .
Next, we only need to prove that inequality (21) holds for and all ; that is

It follows from (5) and Lemma 2 that

where is defined as in (17).
Note that

Let

Then simple computations lead to

From Lemmas 4 and 5 together with (24)–(26), we have

for all .
Therefore, inequality (23) follows from (27) and (28).

Remark 7. Theorem 6 gives a positive answer to the conjecture given by (9).

Remark 8. It follows from (23) that the inequality

holds for all .

4. Applications

For , the Toader mean [1] and arithmetic-geometric mean [11] are, respectively, defined by

where and are given by

Let and be the -order Toader and -order identric means of and , respectively. Then Theorem 6 leads to two fine inequalities chains for certain bivariate means.

Theorem 9. The inequalities

hold for all with .

Proof. The following inequalities can be found in the literature [3, 4, 7, 12–14]:

for all with .
It follows from (35) that

for all with .
Therefore, inequality (32) follows easily from (7), (8), (33), (34), (36), and Theorem 6.

Remark 10. Let and . Then simple computations lead to

Note that

Inequalities (37) and (38) imply that there exist small enough and large enough such that

for all with and

for all with .
Let , , , , and . Then the kernel function of the Szász-Mirakjan-Durrmeyer operator [15]

is given by

Berdysheva [16] proved that is completely monotonic with respect to for fixed and

for all .

From Remark 8 and (42), we get a lower bound for the kernel function immediately.

Corollary 11. The inequality

holds for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grant 61374086 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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