Abstract
In the article, we present several sharp upper and lower bounds for the complete elliptic integral of the first kind in terms of inverse trigonometric and inverse hyperbolic functions. As consequences, some sharp bounds for the Gaussian arithmetic-geometric mean in terms of other bivariate means are also given.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
For two distinct positive real numbers u and v, the Gaussian arithmetic–geometric mean AG(u, v) [1, 2], first Seiffert mean P(u, v), arithmetic mean A(u, v), Neuman–Sándor mean M(u, v) and second Seiffert mean T(u, v) are given by
respectively, where \(\sinh ^{-1}(\omega )=\log (\omega +\sqrt{1+\omega ^2})\) is the inverse hyperbolic sine function.
The Gaussian identity [3, Theorem 4.4] shows that
for all \(0<\tau <1\), where
is the complete elliptic integral of the first kind [4,5,6,7,8] and it is the special case of the Gaussian hypergeometric function [9,10,11,12,13,14,15]
where \((\alpha )_{0}=1\) for \(\alpha \ne 0\), \((\alpha )_{n}=\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +n-1)=\Gamma (\alpha +n)/\Gamma (\alpha )\) is the shifted factorial function and \(\Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}dt\ (x>0)\) is the gamma function [16,17,18]. Indeed,
Let \(r\in (0, 1)\). Then it is well known that the conformal modulus \(\mu (r)\) of the plane Grötzsch ring [19, 20] \(\{z\in {\mathbb {C}}| |z|<1\}\backslash [0, r]\) can be expressed by
which paly an important role in the geometric functions theory [21,22,23].
Recently, the bounds for the the complete elliptic integral \({\mathcal {K}}(\tau )\) have attracted the attention of many mathematicians [24,25,26,27,28,29,30], many applications for \({\mathcal {K}}(\tau )\) and its related special functions in mathematics, physics and engineering can be found in the literature [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51].
Anderson, Vamanamurthy and Vuorinen [52] provided the upper bound for \({\mathcal {K}}(\tau )\) in terms of the ratio of the inverse hyperbolic tangent function \(\tanh ^{-1}(\tau )=\log [(1+\tau )/(1-\tau )]/2\) with \(\tau \) as follows:
for all \(0<\tau <1\).
In [53], Yang, Qian, Chu and Zhang proved that the inequality
holds for all \(0<\tau <1\) if and only if \(p\ge \pi /2\).
Let \(u=1\) and \(v=(1-\tau )/(1+\tau )\in (0, 1)\). Then (1.1)–(1.4) lead to the identities
From (1.7)–(1.9) we clearly see that the well known inequalities
for all \(u,v>0\) with \(u\ne v\) is equivalent to
for all \(0<\tau <1\).
Inequalities (1.5), (1.6) and (1.10) give us the motivation to find the sharp bounds for \({\mathcal {K}}(\tau )\) in terms of the convex combinations of \(\tanh ^{-1}(\tau )/\tau \) and \(\arcsin (\tau )/\tau \), \(\tanh ^{-1}(\tau )/\tau \) and \(\sinh ^{-1}(\tau )/\tau \), and \(\tanh ^{-1}(\tau )/\tau \) and \(\arctan (\tau )/\tau \).
Our first result is Theorem 1.1 which states as follows.
Theorem 1.1
The double inequalities
hold for all \(0<\tau <1\) if and only if \(\alpha _{1}\le 1/2\), \(\alpha _{2}\le 2/\pi \), \(\alpha _{3}\le 2/\pi \), \(\beta _{1}\ge 2/\pi \), \(\beta _{2}\ge 5/6\) and \(\beta _{3}\ge 7/8\).
To further improve and refine the lower bound in (1.11) and the upper bounds in (1.12) and (1.13) we establish the following Theorem 1.2.
Theorem 1.2
The double inequalities
hold for all \(0<\tau <1\) if and only if \(\alpha _{4}\le 3/40\), \(\alpha _{5}\le 1/20\), \(\beta _{4}\ge 3(4-\pi )/(2\pi )=0.4098\ldots \) and \(\beta _{5}\ge 4(4-\pi )/(3\pi )=0.3643\ldots \).
2 Lemmas
In order to simplify the proofs of our Theorems 1.1 and 1.2, we need several lemmas which we present in this section.
First of all, we introduce the complete elliptic integral of the second kind [54,55,56,57,58,59,60]
and the elementary formulas [3, Appendix E] for \({\mathcal {K}}(\tau )\) and \({\mathcal {E}}(\tau )\) as follows
where and in what follows \(\tau ^{\prime }=\sqrt{1-\tau ^{2}}\).
Lemma 2.1
(See [3, Theorem 1.25]) Let \(-\infty<\lambda<\mu <\infty \), \(f,g: [\lambda , \mu ]\mapsto (-\infty , \infty )\) be continuous on \([\lambda , \mu ]\) and differentiable on \((\lambda , \mu )\) with \(g'(t)\ne 0\) on \((\lambda , \mu )\). If \(f^{\prime }(t)/g^{\prime }(t)\) is increasing (decreasing) on \((\lambda , \mu )\), then so are the functions
If \(f^{\prime }(t)/g^{\prime }(t)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
The following statements are true:
- (1):
The function \(\tau \mapsto \left[ {\mathcal {E}}(\tau )-\tau '^{2}\mathcal {K}(\tau )\right] /\tau ^2\) is strictly increasing from (0, 1) onto \((\pi /4, 1)\);
- (2):
The function \(\tau \mapsto \left[ {\mathcal {E}}(\tau )-\tau '^{2}\mathcal {K}(\tau )-\tau '^{2}({\mathcal {K}}(\tau )-{\mathcal {E}}(\tau ))\right] /\tau ^4\) is strictly increasing from (0, 1) onto \((3\pi /16, 1)\);
- (3):
The function \(\tau \mapsto \left[ {\mathcal {K}}(\tau )-\mathcal {E}(\tau )\right] /\tau ^2\) is strictly increasing from (0, 1) onto \((\pi /4, \infty )\);
- (4):
The function \(\tau \mapsto \left[ (1+\tau '^{2}){\mathcal {K}}(\tau )-2\mathcal {E}(\tau )\right] /\tau ^4\) is strictly increasing from (0, 1) onto \((\pi /16, \infty )\);
- (5):
The function \(\tau \mapsto \phi (\tau )={\mathcal {E}}(\tau )/(3+2\tau ^2)\) is strictly decreasing from (0, 1) onto \((1/5, \pi /6)\);
- (6):
The function \(\tau \mapsto \varphi (\tau )=[2{\mathcal {E}}(\tau )-\tau '^2\mathcal {K}(\tau )]/(3+2\tau ^2)\) is strictly decreasing from (0, 1) onto \((2/5, \pi /6)\);
- (7):
The function \(\tau \mapsto \psi (\tau )=(1+\tau ^2)[{\mathcal {E}}(\tau )-\mathcal {K}(\tau )]/\tau ^2\) is strictly decreasing from (0, 1) onto \((-\infty , -\pi /4)\).
Proof
Parts (1)–(4) can be found in [3, Theorem 3.21(1) and Exercise 3.43(10), (11) and (29)]. For part (5), it is not difficult to verify that
It follows from (2.2) and part (3) together with the monotonicity of \({\mathcal {E}}(\tau )\) that
for \(0<\tau <1\).
Therefore, part (5) follows from (2.1) and (2.3).
For part (6), simple computations lead to
From parts (1) and (3), (2.5) and the monotonicity of \({\mathcal {E}}(\tau )\) we clearly see that
for \(0<\tau <1\).
Therefore, part (6) follows from (2.4) and (2.6).
For part (7), elaborated computations give
It follows from parts (2) and (3) together with (2.8) that
for \(0<\tau <1\).
Therefore, part (7) follows from (2.7) and (2.9). \(\square \)
Lemma 2.3
The function
is strictly increasing from (0, 1) onto \((1/2, 2/\pi )\).
Proof
Let \(F_{1}(\tau )=2\tau {\mathcal {K}}(\tau )/\pi -\arcsin (\tau )\), \(F_{2}(\tau )=\tanh ^{-1}(\tau )-\arcsin (\tau )\), \(F_{3}(\tau )=2{\mathcal {E}}(\tau )/(\pi \sqrt{1-\tau ^2})-1\) and \(F_{4}(\tau )=1/\sqrt{1-\tau ^2}-1\). Then elaborated computations lead to
It follows from Lemma 2.2(1) and (2.12) that \(F'_{3}(\tau )/F'_{4}(\tau )\) is strictly increasing on (0,1). Then from (2.10) and (2.11) together with Lemma 2.1 we know that \(F(\tau )\) is strictly increasing on (0,1).
Note that
Therefore, Lemma 2.3 follows from (2.13) and the monotonicity of \(F(\tau )\). \(\square \)
Lemma 2.4
The function
is strictly decreasing from (0, 1) onto \((2/\pi , 5/6)\).
Proof
Let \(G_{1}(\tau )=2\tau {\mathcal {K}}(\tau )/\pi -\sinh ^{-1}(\tau )\), \(G_{2}(\tau )=\tanh ^{-1}(\tau )-\sinh ^{-1}(\tau )\), \(G_{3}(\tau )=2\sqrt{1+\tau ^2}{\mathcal {E}}(\tau )/(\pi \tau '^2)-1\), \(G_{4}(\tau )=\sqrt{1+\tau ^2}/\tau '^2-1\), \(G_{5}(\tau )=2[(1+3\tau ^2)\mathcal {E}(\tau )-\tau '^2(1+\tau ^2){\mathcal {K}}(\tau )]/\pi \) and \(G_{6}(\tau )=\tau ^2(3+\tau ^2)\). Then elaborated computations lead to
It follows from Lemma 2.2(5) and (6) together with (2.17) that \(G'_{5}(\tau )/G'_{6}(\tau )\) is strictly decreasing on (0,1). Then from (2.14)-(2.16) and Lemma 2.1 we know that \(G(\tau )\) is strictly decreasing on (0,1).
Note that
Therefore, Lemma 2.4 follows from (2.18) and the monotonicity of \(G(\tau )\). \(\square \)
Lemma 2.5
The function
is strictly decreasing from (0, 1) onto \((2/\pi , 7/8)\).
Proof
Let \(H_{1}(\tau )=2\tau {\mathcal {K}}(\tau )/\pi -\arctan (\tau )\), \(H_{2}(\tau )=\tanh ^{-1}(\tau )-\arctan (\tau )\), \(H_{3}(\tau )=2(1+\tau ^2) {\mathcal {E}}(\tau )/\pi -(1-\tau ^2)\) and \(H_{4}(\tau )=2\tau ^2\). Then simple computations lead to
It follows from Lemma 2.2(7) and (2.21) together with the monotonicity of \({\mathcal {E}}(\tau )\) that \(H'_{3}(\tau )/H'_{4}(\tau )\) is strictly decreasing on (0,1). Then from (2.19), (2.20) and Lemma 2.1 we know that \(H(\tau )\) is strictly decreasing on (0,1).
Note that
Therefore, Lemma 2.5 follows easily from (2.22) and the monotonicity of \(H(\tau )\). \(\square \)
Lemma 2.6
The function
is strictly increasing from (0, 1) onto \((3/40, 3(4-\pi )/(2\pi ))\).
Proof
Let \(I_{1}(\tau )=2\tau {\mathcal {K}}(\tau )/\pi -[\tanh ^{-1}(\tau )+\arcsin (\tau )]/2\), \(I_{2}(\tau )=\tanh ^{-1}(\tau )/3+\sinh ^{-1}(\tau )/6-\arcsin (\tau )/2\), \(I_{3}(\tau )=2{\mathcal {E}}(\tau )/\pi -(1+\tau ')/2\), \(I_{4}(\tau )=1/3+\tau '^2/(6\sqrt{1+\tau ^2})-\tau '/2\), \(I_{5}(\tau )=1/2-2\tau '[\mathcal {K}(\tau )-{\mathcal {E}}(\tau )]/(\pi \tau ^2)\) and \(I_{6}(\tau )=1/2-\tau '(3+\tau ^2)/[6(3+\tau ^2)^{3/2}]\). Then elaborated computations lead to
It is easy to verify that the function \(\tau \mapsto (1+\tau ^2)^{5/2}/(5-\tau ^2)\) is positive and strictly increasing on (0, 1). Then (2.26) and Lemma 2.2(4) lead to the conclusion that \(I'_{5}(\tau )/I'_{6}(\tau )\) is strictly increasing on (0,1). Thus, from (2.23), (2.24) and (2.25) together with Lemma 2.1 we know that \(I(\tau )\) is strictly increasing on (0,1).
Note that
Therefore, Lemma 2.6 follows from (2.27) and the monotonicity of \(I(\tau )\). \(\square \)
Lemma 2.7
The function
is strictly increasing from (0, 1) onto \((1/20, 4(4-\pi )/(3\pi ))\).
Proof
Let \(J_{1}(\tau )=2\tau {\mathcal {K}}(\tau )/\pi -[\tanh ^{-1}(\tau )+\arcsin (\tau )]/2\), \(J_{2}(\tau )=3\tanh ^{-1}(\tau )/8+\sinh ^{-1}(\tau )/8-\arcsin (\tau )/2\), \(J_{3}(\tau )=2{\mathcal {E}}(\tau )/\pi -(1+\tau ')/2\), \(J_{4}(\tau )=3/8+\tau '^2/[8(1+\tau ^2)]-\tau '/2\), \(J_{5}(\tau )=1-4\tau '[\mathcal {K}(\tau )-{\mathcal {E}}(\tau )]/(\pi \tau ^2)\) and \(J_{6}(\tau )=1-\tau '/(1+\tau ^2)^{2}\). Then simple computations lead to
It is not difficult to verify that the function \(\tau \mapsto (1+\tau ^2)^{3}/(5-3\tau ^2)\) is positive and strictly increasing on (0, 1). Then (2.31) and Lemma 2.2 (4) lead to the conclusion that \(J'_{5}(\tau )/J'_{6}(\tau )\) is strictly increasing on (0,1). Hence, from (2.28), (2.29) and (2.30) together with Lemma 2.1 we know that \(J(\tau )\) is strictly increasing on (0,1).
Note that
Therefore, Lemma 2.7 follows from (2.32) and the monotonicity of \(J(\tau )\). \(\square \)
3 Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1
We clearly see that inequalities (1.11)–(1.13) can be rewritten as
respectively. Therefore, Theorem 1.1 follows easily from (3.1)–(3.3) and Lemmas 2.3– 2.5. \(\square \)
Proof of Theorem 1.2
We clearly see that inequalities (1.14) and (1.15) are equivalent to
and
respectively. Therefore, Theorem 1.2 follows easily from (3.4) and (3.5) together with Lemmas 2.6 and 2.7. \(\square \)
Let \(0<\tau <1\), \(u=1\), \(v=(1-\tau )/(1+\tau )\) and
be the logarithmic mean of u and v. Then we clearly see that
From (1.7)–(1.9), Theorems 1.1 and 1.2, and (3.6) we get Corollaries 3.1 and 3.2 immediately.
Corollary 3.1
The double inequalities
hold for all \(0<\tau <1\).
Corollary 3.2
The double inequalities
hold for all \(u, v>0\) with \(u\ne v\) if and only if \(\alpha _{1}\le 1/2\), \(\alpha _{2}\le 2/\pi \), \(\alpha _{3}\le 2/\pi \), \(\alpha _{4}\le 3/40\), \(\alpha _{5}\le 1/20\), \(\beta _{1}\ge 2/\pi \), \(\beta _{2}\ge 5/6\), \(\beta _{3}\ge 7/8\), \(\beta _{4}\ge 3(4-\pi )/(2\pi )=0.4098\ldots \) and \(\beta _{5}\ge 4(4-\pi )/(3\pi )=0.3643\ldots \).
References
Borwein, J.M., Borwein, P.B.: Pi and AGM. Wiley, New York (1987)
Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, 7 (2019). Article ID 6082413
Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.K.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965)
Alzer, H.: Sharp inequalities for the complete elliptic integral of the first kind. Math. Proc. Camb. Philos. Soc. 124(2), 309–314 (1998)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018)
Yang, Z.-H., Chu, Y.-M.: A monotonicity property involving the generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017)
Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014)
Wang, M.-K., Chu, Y.-M., Jiang, Y.-P.: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679–691 (2016)
Huang, T.-R., Tan, S.-Y., Ma, X.-Y., Chu, Y.-M.: Monotonicity properties and bounds for the complete \(p\)-elliptic integrals. J. Inequal. Appl. 2018, 11 (2018). Article ID 239
Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeometric functions, with applications. J. Math. Anal. Appl. 474(2), 1306–1337 (2019)
Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019)
Wang, M.-K., Chu, Y.-M., Zhang, W.: Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. 49(3), 653–668 (2019)
Wang, M.-K., Zhang, W., Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. 39B(5), 1440–1450 (2019)
Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete \(p\)-elliptic integrals. J. Math. Anal. Appl. (2019). https://doi.org/10.1016/j.jmaa.2019.123388
Zhao, T.-H., Chu, Y.-M., Wang, H.: Logarithmically complete monotonicity properties relating to the gamma function. Abstr. Appl. Anal. 2011, 13 (2011). Article ID 896483
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, 17 (2017). Article ID 210
Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, 9 (2018). Article ID 118
Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality for the Grötzsch ring function. Math. Inequal. Appl. 14(4), 833–837 (2011)
Qiu, S.-L., Qiu, Y.-F., Wang, M.-K., Chu, Y.-M.: Hölder mean inequalities for the generalized Grötzsch ring and Hersch–Pfluger distortion functions. Math. Inequal. Appl. 15(1), 237–245 (2012)
Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Distortion functions for plane quasiconformal mappings. Israel J. Math. 62(1), 1–16 (1988)
Qiu, S.-L., Vuorinen, M.: Submultiplicative properties of the \(\phi _{K}\)-distortion function. Studia Math. 117(3), 225–242 (1996)
Wang, M.-K., Qiu, S.-L., Chu, Y.-M.: Infinite series formula for Hübner upper bound function with applications to Hersch–Pfluger distortion function. Math. Inequal. Appl. 21(3), 629–648 (2018)
Ullah, S.-Z., Khan, M.-A., Chu, Y.-M.: A note on generalized convex functions. J. Inequal. Appl. 2019, 10 (2019). Article ID 291
Wang, B., Lu-o, C.-L., Li, S.-H., Chu, Y.-M.: Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(1), (2020). https://doi.org/10.1007/s13398-019-00734-0
Abbas Baloch, I., Chu, Y.-M.: Petrović-type inequalities for harmonic h-convex function. J. Funct. Sp. 2020, 7 (2020). Article ID 3075390
Qiu, S.-L., Vamanamurthy, M.K.: Sharp estimates for complete elliptic integrals. SIAM J. Math. Anal. 27(3), 823–834 (1996)
Chu, Y.-M., Qiu, Y.-F., Wang, M.-K.: Hölder mean inequalities for the complete elliptic integrals. Integral Transforms Spec. Funct. 23(7), 521–527 (2012)
Wang, M.-K., Chu, Y.-M., Qiu, S.-L., Jiang, Y.-P.: Convexity of the complete elliptic integrals of the first kind with respect to Hölder means. J. Math. Anal. Appl. 388(2), 1141–1146 (2012)
Chu, Y.-M., Wang, M.-K., Qiu, Y.-F.: On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function. Abstr. Appl. Anal. 2011, 7 (2011). Article ID 697547
Huang, C.-X., Yang, Z.-C., Yi, T.-S., Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)
Zhou, X.-S.: Weighted sharp function estimate and boundedness for commutator associated with singular integral operator satisfying a variant of Hörmander’s condition. J. Math. Inequal. 9(2), 587–596 (2015)
Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Generalized Lyapunov–Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. Discret. Contin. Dyn. Syst. Ser. B 22(9), 3591–3614 (2017)
Hu, H.-J., Zhou, X.-F.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(1), 4763–4771 (2017)
Wang, W.-S.: On A-stable one-leg methods for solving nonlinear Volterra functional differential equations. Appl. Math. Comput. 314, 380–390 (2017)
Hu, H.-J., Liu, L.-Z.: Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition. Math. Notes 101(5–6), 830–840 (2017)
Duan, L., Huang, L.-H., Guo, Z.-Y., Fang, X.-W.: Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2), 233–245 (2017)
Duan, L., Huang, C.-X.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)
Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018)
Zhu, K.-X., Xie, Y.-Q., Zhou, F.: Pullback attractors for a damped semilinear wave equation with delays. Acta Math. Sin. 34(7), 1131–1150 (2018)
Huang, C.-X., Qiao, Y.-C., Huang, L.-H., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, 26 (2018). Article ID 186
Liu, Z.-Y., Wu, N.-C., Qin, X.-R., Zhang, Y.-L.: Trigonometric transform splitting methods for real symmetric Toeplitz systems. Comput. Math. Appl. 75(8), 2782–2794 (2018)
Duan, L., Fang, X.-W., Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018)
Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 323, 204–219 (2018)
Tan, Y.-X., Huang, C.-X., Sun, B., Wang, T.: Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)
Li, J., Ying, J.-Y., Xie, D.-X.: On the analysis and application of an ion size-modified Poisson–Boltzmann equation. Nonlinear Anal. Real World Appl. 47, 188–203 (2019)
Wang, J.-F., Chen, X.-Y., Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)
Wang, J.-F., Huang, C.-X., Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)
Tian, Z.-L., Liu, Y., Zhang, Y., Liu, Z.-Y., Tian, M.-Y.: The general inner–outer iteration method based on regular splittings for the PageRank problem. Appl. Math. Comput. 356, 479–501 (2019)
Qian, W.-M., He, Z.-Y., Zhang, H.-W., Chu, Y.-M.: Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean. J. Inequal. Appl. 2019, 13 (2019). Article ID 168
Huang, C.-X., Zhang, H., Huang, L.-H.: Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun. Pure Appl. Anal. 18(6), 3337–3349 (2019)
Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23(2), 512–524 (1992)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic–geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(1), 1714–1726 (2018)
Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer-Verlag, New York (1971)
Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)
Chu, Y.-M., Wang, M.-K., Jiang, Y.-P., Qiu, S.-L.: Concavity of the complete elliptic integrals of the second kind with respect to Hölder means. J. Math. Anal. Appl. 395(2), 637–642 (2012)
Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012)
Chu, Y.-M., Wang, M.-K., Qiu, S.-L., Jiang, Y.-P.: Bounds for complete elliptic integrals of the second kind with applications. Comput. Math. Appl. 63(7), 1177–1184 (2012)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, 13 (2017). Article ID 106
Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)
Acknowledgements
This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485) and the Key Project of the Scientific Research of Zhejiang Open University in 2019 (Grant no. XKT-19Z02).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Qian, WM., He, ZY. & Chu, YM. Approximation for the complete elliptic integral of the first kind. RACSAM 114, 57 (2020). https://doi.org/10.1007/s13398-020-00784-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00784-9
Keywords
- Complete elliptic integrals
- Inverse trigonometric function
- Inverse hyperbolic function
- Arithmetic-geometric mean