Abstract
In this paper, we establish a necessary and sufficient condition for the convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means.
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Abramowitz, M., Stegun, I.A.: Handbook of Mathematical functions with Formulas, Graphs, and Mathematical Tables. U. S. Government Printing Office, Washington (1964)
Baricz, Á.: Convexity of the zero-balanced Gaussian hypergeometric functions with respect to Hölder means. J. Inequal. Pure Appl. Math. 8, 2 (2007)
Baricz, Á.: Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math. 26, 279–293 (2008)
Baricz, Á.: Geometrically concave univariate distributions. J. Math. Anal. Appl. 363(1), 182–196 (2010)
Borwein, D., Borwein, J., Fee, G., Girgensohn, R.: Refined convexity and special cases of the Blaschke-Santalo inequality. Math. Inequal. Appl. 4(4), 631–638 (2001)
Bullen, P.S.: Handbook of Means and Their Inequalities. Kluwer, Dordrecht (2003)
Chu, Y.-M., Long, B.-Y.: Bounds of the Neuman-Sándor mean using power and identric means. Abstr. Appl. Anal. 2013, (2013)
Chu, Y.-M., Zhao, T.-H.: Concavity of the error function with respect to Hölder means. Math. Inequal. Appl. 19(2), 589–595 (2016)
Chu, Y.-M., Qiu, Y.-F., Wang, M.-K.: Sharp power mean bounds for the combination of Seiffert and geometric means. Abstr. Appl. Anal. 2010, (2010)
Chu, Y.-M., Wang, S.-S., Zong, C.: Optimal lower power mean bounds for the convex combination of harmonic and logarithmic means. Abstr. Appl. Anal. 2011, (2011)
Chu, Y.-M., Qiu, Y.-F., Wang, M.-K.: Hölder mean inequalities for the complete elliptic integrals. Integr. Transforms Spec. Funct. 23(7), 521–527 (2012)
Chu, Y.-M., Yang, Z.-H., Wu, L.-M.: Sharp power mean bounds for Sándor mean. Abstr. Appl. Anal. 2015, (2015)
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 Höder means. J. Math. Anal. Appl. 395(2), 637–642 (2012)
Li, Y.-M., Long, B.-Y., Chu, Y.-M.: A best possible double inequality for power mean. J. Appl. Math. 2012, (2012)
Li, Y.-M., Long, B.-Y., Chu, Y.-M.: Sharp bounds by the power mean for the generalized Heronian mean. J. Inequal. Appl. 2012, (2012)
Li, Y.-M., Long, B.-Y., Chu, Y.-M., Gong, W.-M.: Optimal inequalities for power means. J. Appl. Math. 2012, (2012)
Neuman, E.: Inequalities involving modified Bessel functions of the first kind. J. Math. Anal. Appl. 171, 532–536 (1992)
Pólya, G., Szegő, G.: Problems and Theorems in Analysis I. Springer, Berlin (1998)
Qian, W.-M., Zhang, X.-H., Chu, Y.-M.: Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means. J. Math. Inequal. 11(1), 121–127 (2017)
Qian, W.-M., He, Z.-Y., Chu, Y.-M.: Approximation for the complete elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(2), 57 (2020). https://doi.org/10.1007/s13398-020-00784-9
Qiu, S.-L., Qiu, Y.-F., Wang, M.-K., Chu, Y.-M.: Hölder mean inequalities for the generalized the Grötzsch ring and Hersch-Pfluger distortion function. Math. Inequal. Appl. 15(1), 237–245 (2012)
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, 168 (2019)
Thiruvenkatachar, V.R., Nanjundiah, T.S.: Inequalities concerning Bessel functions and orthogonal polynomials. Proc. Indian Acad. Sci. Sect. A 33, 373–384 (1951)
Wang, M.-K., Chu, Y.-M.: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607–622 (2017)
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)
Wang, M.-K., Li, Y.-M., Chu, Y.-M.: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. 46(1), 189–200 (2018)
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. 480(2), 123388 (2019). https://doi.org/10.1016/j.jmaa.2019.123388
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., He, Z.-Y., Chu, Y.-M.: Sharp power mean inequalities for the generalized elliptic integral of the first kind. Comput. Methods Funct. Theory (2020). https://doi.org/10.1007/s40315-020-00298-w
Wang, M.-K., Chu, Y.-M., Qiu, Y.-F., Qiu, S.-L.: An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 24(6), 887–890 (2011)
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)
Wang, M.-K., Hong, M.-Y., Xu, Y.-F., Shen, Z.-H., Chu, Y.-M.: Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 14(1), 1–21 (2020)
Wang, B., Luo, 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), 7 (2020). https://doi.org/10.1007/s13398-019-00734-0
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1955)
Xia, W.-F., Janous, W., Chu, Y.-M.: The optimal convex combination bounds of arithmetic and harmonic means in terms of power mean. J. Math. Inequal. 6(2), 241–248 (2012)
Yang, Z.-H., Chu, Y.-M., Wang, M.-K.: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428(1), 587–604 (2015)
Yang, Z.-H., Zhang, W., Chu, Y.-M.: Sharp Gautschi inequality for parameter \(0 < p < 1\) with applications. Math. Inequal. Appl. 20(4), 1107–1120 (2017)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, (2017)
Yang, Z.-H., Qian, W.-M., Zhang, W., Chu, Y.-M.: Notes on the complete elliptic integral of the first kind. Math. Inequal. Appl. 23(1), 77–93 (2020)
Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: A note on generalized convex functions. J. Inequal. Appl. 2019, 291 (2019)
Zhou, L.-M., Qiu, S.-L., Wang, F.: Inequalities for the generalized elliptic integrals with respect to Hölder means. J. Math. Anal. Appl. 386(2), 641–646 (2012)
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This research was supported by the Natural Science Foundation of China (Grant Nos. 11971142, 61673169, 11871202, 11701176, 11626101, 11601485) and the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010012).
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Zhao, TH., Shi, L. & Chu, YM. Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. RACSAM 114, 96 (2020). https://doi.org/10.1007/s13398-020-00825-3
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DOI: https://doi.org/10.1007/s13398-020-00825-3