## Abstract

We establish the monotonicity and convexity properties for several special functions involving the generalized elliptic integrals, and present some new analytic inequalities.

### Similar content being viewed by others

## References

Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: US Government Printing Office, 1964

Wang M K, Chu Y M. Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math Sci, 2017,

**37B**(3): 607622Qiu S L, Ma X Y, Chu Y M. Sharp Landen transformation inequalities for hypergeometric functions, with applications. J Math Anal Appl, 2019,

**474**(2): 13061337Wang M K, Chu Y M, Zhang W. Monotonicity and inequalties involving zero-balanced hypergeometric function. Math Inequal Appl, 2019,

**22**(2): 601617Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York: John Wiley & Sons, 1997

Anderson G D, Qiu S L, Vamanamurthy M K, et al. Generalized elliptic integrals and modular equations. Pacific J Math, 2000,

**192**(1): 137Almkvist G, Berndt B. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipse, π, and the Ladies diary. Amer Math Monthly, 1988,

**95**(7): 585608Borwein J M, Borwein P B. Pi and the AGM. New York: John Wiley & Sons, 1987

Borwein J M, Borwein P B. Inequalities for compound mean iterations with logarithmic asymptotes. J Math Anal Appl, 1993,

**177**(2): 572582Barnard R W, Pearce K, Richards K C. A monotonicity property involving

_{3}F_{2}and comparisons of the classical approximations of elliptical arc length. SIAM J Math Anal, 2000,**32**(2): 403419Alzer H. Sharp inequalities for the complete elliptic integral of the first kind. Math Proc Cambridge Philos Soc, 1998,

**124**(2): 309314Wang M K, Chu Y M. Asymptotical bounds for complete elliptic integrals of the second kind. J Math Anal Appl, 2013,

**402**(1): 119126Anderson G D, Qiu S L, Vamanamurthy M K. Elliptic integral inequalities, with applications. Constr Approx, 1998,

**14**(2): 195207Chu Y M, Wang M K, Qiu S L. Optimal combinations bounds of root-square and arithmetic means for Toadr mean. Proc Indian Acad Sci Math Sci, 2012,

**122**(1): 4151Chu Y M, Wang M K. Optimal Lehmer mean bounds for the Toader mean. Results Math, 2012,

**61**(3/4): 223229Chu Y M, Qiu Y F, Wang M K. Hölder mean inequalities for the complete elliptic integrals. Intetral Transfroms Spec Funct, 2012,

**23**(7): 521527Qiu 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, 2012,

**15**(1): 237245Wang 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, 2018,

**21**(3): 629648Chu Y M, Cheng J F, Wang G D. Remarks on John disks. Acta Math Sci, 2009,

**29B**(1): 160168Chu Y M, Sun T C. The Schur harmonic convexity for a class of symmetric functions. Acta Math Sci, 2010,

**30B**(5): 15011506Huang 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 Differential Equations, 2014,

**256**(7): 21012114Huang C X, Guo S, Liu L Z. Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón-Zygmund kernel. J Math Inequal, 2014,

**8**(3): 453464Huang C X, Liu L Z. Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest Math, 2017,

**40**(3): 295312Yang Z H, Qian W M, Chu Y M. Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math Inequal Appl, 2018,

**21**(4): 11851199Duan 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, 2018,

**41**(5): 19541965Yang Z H, Chu Y M, Zhang W. High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl Math Comput, 2019,

**348**: 552564Wang 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, 2019,

**469**(1): 405427Wang M K, Li Y M, Chu Y M. Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J, 2018,

**46**(1): 189200Alzer H, Richards K C. A note on a function involving complete elliptic integrals: monotonicity, convexity, inequalities. Anal Math, 2015,

**41**(3): 133139Yang Z H, Chu Y M, Wang M K. Monotonicity criterion for the quotient of power series with applications. J Math Anal Appl, 2015,

**428**(1): 587604Pinelis I. On L’Hospital-type rules for monotonicity. J Inequal Pure Appl Math, 2006,

**7**(2): Article 40Huang T R, Tan S Y, Zhang X H. Monotonicity, convexity, inequalities for the generalized elliptic integrals. J Inequal Appl, 2017,

**2017**: Article 278

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

This research was supported by the Natural Science Foundation of China (11701176, 61673169, 11301127, 11626101, 11601485), and the Science and Technology Research Program of Zhejiang Educational Committee (Y201635325).

## Rights and permissions

## About this article

### Cite this article

Wang, M., Zhang, W. & Chu, Y. Monotonicity, Convexity and Inequalities Involving the Generalized Elliptic Integrals.
*Acta Math Sci* **39**, 1440–1450 (2019). https://doi.org/10.1007/s10473-019-0520-z

Received:

Revised:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10473-019-0520-z