#### 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 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 [2–7].

In [2], Qi et al. proved that the identityand the inequalitieshold for all with , whereis 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 inequalitieshold 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 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 [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 inequalitieshold 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) 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 [15] is given byBerdysheva [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.