Abstract
In the article, we provide several sharp bounds for the Toader mean by use of certain combinations of the arithmetic, quadratic, contraharmonic, and Gaussian arithmetic-geometric means.
1. Introduction
Let be a real number, and with . Then the complete elliptic integrals and [1–22] of the first and second kinds, geometric mean , logarithmic mean , arithmetic mean , quadratic mean , contraharmonic mean , second contraharmonic mean , th Hölder mean , and Toader mean [23–27] of and are given byandrespectively.
The classical Gaussian arithmetic-geometric mean of two positive real numbers and is defined by the common limit of the sequences and , which are given by
The well-known Gaussian identity [10] and (2) show thatfor all , where .
If , then it is well known that the function is strictly increasing on the interval and the inequalitiesare valid.
Recently, the bounds for the Toader mean and Gaussian arithmetic-geometric means have attracted the interest of many mathematicians. The following inequalitiesfor all with were established in [28–32].
Alzer and Qiu [12] and Barnard, Pearce, and Richards [33] proved that the double inequalities hold for all with .
In [23, 34], the authors proved that , , and are the best possible parameters such that the double inequalitieshold for all with .
Qian, Song, Zhang, and Chu [35] proved that the two-sided inequalities are valid for all with if and only if , , and if and .
Inequalities (5), (6), and (9) and the identity lead tofor all with .
Motivated by inequalities (12)–(14), in this article we deal with the optimality of the parameters and on the interval such that for all with .
2. Lemmas
It is well known that and satisfy the following formulas (see [10]):
Lemma 1 (See [10, Theorem 1.25]). Let with , be continuous real-valued functions and differentiable on with . Then both the functions and are (strictly) increasing (decreasing) on if the function is (strictly) increasing (decreasing) on .
Lemma 2. For the complete elliptic integrals and , we have the following monotonicity results:
(1) The function is strictly increasing from onto .
(2) The function is strictly increasing from onto and the function is strictly decreasing from onto .
(3) The function is strictly decreasing from onto if .
(4) The function is strictly increasing from onto .
(5) The function is strictly increasing from onto .
(6) The function is strictly increasing from onto .
(7) The function is strictly decreasing from onto .
(8) The function is strictly increasing from onto .
(9) The function is strictly decreasing from onto .
(10) The function is strictly decreasing from onto .
(11) The function is strictly increasing from onto .
Proof. Parts (1)–(6) can be found in [10, Theorem 3.21(1), (2), (7) and (8), and Exercise 3.43(4), (11) and (13)]. For part (7), it is not difficult to verify thatIt follows from parts (1) and (2) together with (18) thatfor .
Therefore, part (7) follows from (17) and (19).
For part (8), simple computations lead toFrom parts (4) and (5) together with (21) we clearly see thatfor .
Therefore, part (8) follows from (20) and (22).
For part (9), we clearly see thatDifferentiating and making use of part (5) we getfor .
Therefore, part (9) follows from (23) and (24).
For part (10), elaborated computation givesIt follows from parts (2), (4), and (5) together with (26) thatfor .
Therefore, part (10) follows from (25) and (27).
For part (11), it is not difficult to verify thatLetThen we clearly see that thatTherefore, part (11) follows easily from parts (1) and (2) together with (28)–(32).
3. Main Results
Theorem 3. The double inequality holds for all with if and only if and .
Proof. We clearly see that , and are symmetric and homogenous of degree 1. Without loss of generality, we assume that . Let . Then and (2) and (4) lead toIt follows from (34) and (35) together with thatLet and . Then elaborated computations lead toIt follows from Lemma 2(1) and (7) together with (38) that is strictly increasing on . Then from Lemma 1 and (37) we know that is strictly increasing on . Moreover,Therefore, Theorem 3 follows from (36) and (39) together with the monotonicity of .
Theorem 4. The double inequality holds for all with if and only if and .
Proof. Since , , and are symmetric and homogenous of degree 1, without loss of generality, we assume that . Let . Then , and (34) and (35) together with lead toLet and . Then simple computations lead toIt follows from Lemma 2(3) and (8) together with (43) that is strictly increasing on . Then from Lemma 1 and (42) we know that is strictly increasing on . Moreover,Therefore, Theorem 4 follows from (41) and (44) together with the monotonicity of .
Theorem 5. The double inequality holds for all with if and only if and .
Proof. Without loss of generality, we assume that . Let . Then it follows from (34), (35) and thatLet and . Then elaborated computations lead to
It follows from Lemma 2(1), (3), and (6) together with (48) that is strictly increasing on . Then from Lemma 1 and (47) we know that is strictly increasing on . Moreover,
Therefore, Theorem 5 follows from (46) and (49) together with the monotonicity of .
Theorem 6. The double inequality holds for all with if and only if and .
Proof. Without loss of generality, we assume that . Let . Then it follows from (34), (35) and thatLet and . Then simple computations lead toIt follows from Lemma 2(9)–(11) and (53) that is strictly increasing on . Then from Lemma 1 and (52) we know that is strictly increasing on . Moreover,Therefore, Theorem 6 follows from (51) and (54) together with the monotonicity of .
Let and . Then (2), (4) and Theorems 3–6 lead to Corollary 7 immediately.
Corollary 7. The double inequalities hold for all .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of China (Grant No. 61673169) and the Natural Science Foundation of Huzhou City (Grant No. 2018YZ07).