Research Article | Open Access
Sharp Bounds for Toader Mean in terms of Arithmetic and Second Contraharmonic Means
We present the best possible parameters and such that double inequalities , hold for all with , where , and are the arithmetic, second contraharmonic, and Toader means of and , respectively.
For the Toader mean , second contraharmonic mean , and arithmetic mean of and are given by respectively, where is the complete elliptic integral of the second kind. The Toader mean is well known in mathematical literature for many years; it satisfies for all and , where stands for the symmetric complete elliptic integral of the second kind (see [2–4]); therefore it cannot be expressed in terms of the elementary transcendental functions.
Let , , and . Then the th power mean , th Gini mean , th Lehmer mean , and th generalized Seiffert mean are defined by respectively. It is well known that , , , and are continuous and strictly increasing with respect to and for fixed with , respectively.
Alzer and Qiu  presented a best possible upper power mean bound for the Toader mean as follows: for all with .
Chu and Wang  proved that double inequality holds for all with if and only if and .
Inequality (8) leads to for all with .
Let with be fixed and . Then it is not difficult to verify that is continuous and strictly increasing on . Note that
Motivated by inequalities (9) and (10), it is natural to ask what are the best possible parameters, and , such that double inequalities hold for all with ? The main purpose of this paper is to answer this question.
2. Main Results
In order to prove our main results we need some basic knowledge and two lemmas, which we present in this section.
For the complete elliptic integral of the first kind is defined by
We clearly see that and and satisfy formulas (see [17, Appendix E, p. 474-475])
Lemma 1 (see [17, Theorem 1.25]). Let , be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2 (see [17, Theorem 3.21]). (1) Function is strictly increasing from to .
(2) Function is strictly decreasing from to if .
Theorem 3. Double inequality holds for all with if and only if and ….
Proof. Since , , and are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let . Then (1) and (2) lead toWe clearly see that inequality (16) is equivalent toIt follows from (17) and (18) thatLetThen simple computations lead toFrom Lemmas 1 and 2 together with (21) and (22) we know that is strictly increasing on andTherefore, Theorem 3 follows from (19)–(21) and (23) together with the monotonicity of .
Theorem 4. Let , . Then double inequality holds for all with if and only if … and ….
Proof. Let and . We first prove thatfor all with .
Without loss of generality, we assume that . Let and . Then (2) leads toIt follows from (17) and (27) thatLetThen making use of Lemma 2 and simple computations lead toWe divide the proof into two cases.
Case 1. Consider . Then (34) becomesIt follows from Lemma 2(1) and (33) together with (36) that for all .
Therefore, inequality (25) follows easily from (28)–(31) and (37).
Case 2. Consider . Then (32), (34), and (35) lead toIt follows from Lemma 2(1), (33), (39), and (40) that there exists such that for and for . Then (30) leads to the conclusion that is strictly decreasing on and strictly increasing on .
Therefore, inequality (26) follows easily from (28), (29), (31), (38), and the piecewise monotonicity of .
Next, we prove that is the best possible parameter on such that inequality holds for all with .
Indeed, if , then (34) leads to and there exists such that for .
Equations (28)–(31) and inequality (42) imply that for .
Finally, we prove that is the best possible parameter on such that double inequality for all with .
In fact, if , then (32) leads to and there exists such thatfor .
Equations (28) and (29) together with inequality (45) imply that for .
Corollary 5. Double inequalities