/ / Article

Research Article | Open Access

Volume 2016 |Article ID 4165601 | 5 pages | https://doi.org/10.1155/2016/4165601

# A Sharp Lower Bound for Toader-Qi Mean with Applications

Revised10 Dec 2015
Accepted24 Dec 2015
Published17 Jan 2016

#### 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  and -order logarithmic mean are defined byrespectively. 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 .

In , Qi et al. proved that the identityand the inequalitieshold for all with , whereis the modified Bessel function of the first kind  and , and are, respectively, the classical arithmetic, geometric, and identric means of and .

In , Yang proved that the double inequalities and conjectured that the inequalitieshold for all with . Inequality (8) was proved by Yang et al. in .

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 ). The double inequality holds for all and , where is the classical Euler gamma function.

Lemma 2 (see ). Let be defined by (5). Then the identity holds for all .

Lemma 3 (see ). The Wallis ratiois strictly decreasing and log-convex with respect to all integers .

Lemma 4. The identityholds for all and .

Proof. Let be the number of combinations of objects taken at a time. Then from the well-known binomial theorem we haveEquation (15) leads to

Lemma 5. Let with andThenfor 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 inequalityholds 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 tofor 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 isIt follows from (5) and Lemma 2 thatwhere is defined as in (17).
Note thatLetThen simple computations lead toFrom Lemmas 4 and 5 together with (24)–(26), we havefor 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  and arithmetic-geometric mean  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 inequalitieshold for all with .

Proof. The following inequalities can be found in the literature [3, 4, 7, 1214]:for all with .
It follows from (35) thatfor all with .
Therefore, inequality (32) follows easily from (7), (8), (33), (34), (36), and Theorem 6.

Remark 10. Let and . Then simple computations lead toNote thatInequalities (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  is given byBerdysheva  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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