Research Article | Open Access

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

**Academic Editor:**Kehe Zhu

#### 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.

#### References

- G. Toader, “Some mean values related to the arithmetic-geometric mean,”
*Journal of Mathematical Analysis and Applications*, vol. 218, no. 2, pp. 358–368, 1998. View at: Publisher Site | Google Scholar - F. Qi, X.-T. Shi, F.-F. Liu, and Z.-H. Yang, “A double inequality for an integral mean in terms of the exponential and logarithmic means,” http://www.researchgate.net/publication/278968439. View at: Publisher Site | Google Scholar
- Z.-H. Yang, “Some sharp inequalities for the Toader-Qi mean,” http://arxiv.org/abs/1507.05430. View at: Google Scholar
- J. M. Borwein and P. B. Borwein, “Inequalities for compound mean iterations with logarithmic asymptotes,”
*Journal of Mathematical Analysis and Applications*, vol. 177, no. 2, pp. 572–582, 1993. View at: Publisher Site | Google Scholar - Z.-H. Yang, “A new proof of inequalities for Gauss compound mean,”
*International Journal of Mathematical Analysis*, vol. 4, no. 21-24, pp. 1013–1018, 2010. View at: Google Scholar | Zentralblatt MATH - Y.-M. Chu, B.-Y. Long, W.-M. Gong, and Y.-Q. Song, “Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means,”
*Journal of Inequalities and Applications*, vol. 2013, article 10, 13 pages, 2013. View at: Publisher Site | Google Scholar - Z.-H. Yang, Y.-Q. Song, and Y.-M. Chu, “Sharp bounds for the arithmetic-geometric mean,”
*Journal of Inequalities and Applications*, vol. 2014, article 192, 13 pages, 2014. View at: Publisher Site | Google Scholar - M. Abramowitz and I. A. Stegun,
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, U. S. Government Printing Office, Washington, DC, USA, 1964. - Z.-H. Yang, Y.-M. Chu, and Y.-Q. Song, “Sharp bounds for Toader-Qi mean in terms of logarithmic and identric means,”
*Mathematical Inequalities & Applications*, In press. View at: Google Scholar - F. Qi, “Bounds for the ratio of two gamma functions,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 493058, 84 pages, 2010. View at: Publisher Site | Google Scholar - J. M. Borwein and P. B. Borwein,
*Pi and the AGM*, John Wiley & Sons, New York, NY, USA, 1987. - H. Alzer, “Ungleichungen für Mittelwerte,”
*Archiv der Mathematik*, vol. 47, no. 5, pp. 422–426, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Sándor, “On the identric and logarithmic means,”
*Aequationes Mathematicae*, vol. 40, no. 1, pp. 261–270, 1990. View at: Publisher Site | Google Scholar - B. C. Carlson and M. Vuorinen, “Inequality of the AGM and the logarithmic mean,”
*SIAM Review*, vol. 33, no. 4, pp. 655–655, 1991. View at: Publisher Site | Google Scholar - S. M. Mazhar and V. Totik, “Approximation by modified Szász operators,”
*Acta Universitatis Szegediensis (Szeged)*, vol. 49, no. 1–4, pp. 257–269, 1985. View at: Google Scholar - E. E. Berdysheva, “Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions,”
*Journal of Approximation Theory*, vol. 149, no. 2, pp. 131–150, 2007. View at: Publisher Site | Google Scholar

#### Copyright

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.