A Sharp Lower Bound for Toader-Qi Mean with Applications
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.
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 , 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 .
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.
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 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 .
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 8. It follows from (23) that the inequality holds for all .
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  is given byBerdysheva  proved that is completely monotonic with respect to for fixed and for all .
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.
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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