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Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral
We present the best possible parameters and such that the double inequalities , , hold for all with , where , , and are the arithmetic, quadratic, and Gauss arithmetic-geometric means of and , respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.
Let and . Then the elliptic elliptic integral of the first kind and second kind , Gaussian arithmetic-geometric mean , arithmetic mean , and quadratic mean are, respectively, given by
It is well known that the elliptic elliptic integrals and and the Gaussian arithmetic-geometric mean have many applications in mathematics, physics, mechanics, and engineering [4–9]. Recently, the bounds for the Gaussian arithmetic-geometric mean have attracted the attention of many researchers.
The inequalities for all and with can be found in the literature [10–12], where and are, respectively, the logarithmic and th generalized logarithmic means of and . The first inequality of (5) is due to Carlson and Vuorinen .
By using a variant of L’Hospital’s rule and representation theorems with elliptic integrals, Vamanamurthy and Vuorinen  proved, among other results, the inequalities for all with and , where is the identric mean of and .
By use of the homogeneity of the above means and a series representation of due to Gauss, Sándor  obtained, among other results, new proofs for inequalities (7), (8) and a counterpart of inequality (9): for all with , where is the geometric mean of and . Inequalities (9) and (12) show that lies between the arithmetic and geometric means of and . In , Sándor provided new proofs for inequalities (6) and (8), (9), (10), and (12) by using only elementary methods for recurrent sequences and found much stronger forms of these results.
Neuman and Sándor  gave the comparison of the Gaussian arithmetic-geometric mean and the Schwab-Borchardt mean.
Qiu and Vamanamurthy  presented that and are, respectively, the lower and upper bounds for with . Alzer and Qiu  proved that and are the best possible parameters such that the double inequality holds for all with .
Chu and Wang  proved that the double inequality holds for all with if and only if and , where and is the th Gini mean of and . In , Yang et al. proved that the inequalitieshold for all and with , where is the Stolarsky mean  of and .
Let with and . Then it is not difficult to verify that the function is continuous and strictly increasing on the interval . Note that
Inequalities (16) give us the motivation to deal with the best possible parameters and such that the double inequalities hold for all with .
In order to prove our main results we need several derivative formulas and particular values for and , which we present in this section.
and satisfy the formulas (see )where is the classical Euler gamma function.
Lemma 1 (see [24, Theorem 1.25]). Let , be continuous on and differentiable on and on . Then both functions are increasing (decreasing) on if is increasing (decreasing) on . If is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2 (see [24, Theorem 3.21(1), Theorem 3.21(7), and Exercises 3.43(32)]). The following statements are true:(1)The function is strictly increasing from onto .(2)The function is strictly decreasing from onto if .(3)The function is strictly increasing from onto .
Lemma 3. Let and be defined by Then there exists such that for and for .
Proof. From (21) we clearly see that can be rewritten as It follows from Lemma 2 and together with (22) that is strictly increasing on .
Numerical computations show that Therefore, Lemma 3 follows easily from (23) and the monotonicity of on the interval .
3. Main Results
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 (2) and (3) lead to Let Then simple computations give It follows from Lemmas 1, 2 and (27) and (28) that and is strictly increasing on the interval .
Note that Therefore, Theorem 4 follows easily from (26), (27), (29), and (30) and the monotonicity of on the interval .
Remark 5. The left side inequality of Theorem 4 for can be derived directly from the fact that and for all with .
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 (2) and (3) that Let Then simple computations lead to It follows from Lemmas 1, 2(1) and (2) together with (33) and (34) that and is strictly decreasing on the interval .
Note that Therefore, Theorem 6 follows easily from (32), (33), (35), and (36) and the monotonicity of on the interval .
Remark 7. The right side inequality of Theorem 6 for can be derived directly from the fact that and for all with .
Theorem 8. Let . Then the double inequality holds for all with if and only if and .
Proof. Without loss of generality, we assume that . Let and . Then (2) and (3) lead to wherewhere is defined by (21).
We divide the proof into four cases.
Case 1 (). Then (42) becomesIt follows from Lemma 3 and (43) that there exists such that is strictly decreasing on and strictly increasing on . Therefore, follows from (39), (41), and (44) together with the piecewise monotonicity of on the interval .
Case 2 (). Then we clearly see that Case 3 (). Then (42) leads to Equation (39) and inequality (47) imply that there exists small enough such that for all with .
Case 4 (). Then (40) leads to Note that Equations (39) and (49) together with inequality (50) imply that there exists small enough such that for all with .
Theorem 9. Let , , and . Then the double inequalities hold for all .
It follows from the inequality given in  that
Theorem 10. Let , , and . Then the double inequalities hold for all .
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 Grants 11371125, 61374086, and 11401191, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of the Zhejiang Broadcast and TV University under Grant XKT-15G17.
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