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 Correspondence should be addressed to Yu-Ming Chu; chuyuming2005@126.com Received 28 October 2015; Revised 10 December 2015; Accepted 24 December 2015 Academic Editor: Kehe Zhu Copyright © 2016 Z.-H. Yang and Y.-M. 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. We prove that the inequality TQ(a, b) > Lp(a, b) holds for all a, b > 0 with a ̸ = b if and only if p ≤ 3/2, where TQ(a, b) =

It is well-known that the -order logarithmic mean   (, ) is continuous and strictly increasing with respect to  ∈ R for fixed ,  > 0 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][3][4][5][6][7].
Let  >  > 0 and  = (log  − log )/2 > 0. 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.

Lemmas
In order to prove our main result we need several lemmas, which we present in this section.
Lemma 3 (see [3]).The Wallis ratio is strictly decreasing and log-convex with respect to all integers  ≥ 0.

Lemma 4. The identity
holds for all  ∈ R and  ∈ N.
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 ,  ∈ N with  ≤  and Then for all  ≥ 8.

Main Result
for all  > 0.
If inequality (21) holds for all  > 0. Then ( 5) and ( 21) lead to lim which gives  ≤ 3/2.Next, we only need to prove that inequality (21) holds for  = 3/2 and all  > 0; that is It follows from (5) and Lemma 2 that where  , is defined as in (17).
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  ̸ = 0.