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Association of Jensen’s inequality for \(s\)-convex function with Csiszár divergence. (English) Zbl 1499.26087

Summary: In the article, we establish an inequality for Csiszár divergence associated with \(s\)-convex functions, present several inequalities for Kullback-Leibler, Renyi, Hellinger, Chi-square, Jeffery’s, and variational distance divergences by using particular \(s\)-convex functions in the Csiszár divergence. We also provide new bounds for Bhattacharyya divergence.

MSC:

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
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[1] Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, Boston (1992) · Zbl 0749.26004
[2] Udrişte, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic, Dordrecht (1994) · Zbl 0932.53003
[3] 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) · Zbl 1297.34084
[4] 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) · Zbl 1359.34091
[5] 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) · Zbl 1386.35159
[6] Wang, W.-S., Chen, Y.-Z.: Fast numerical valuation of options with jump under Merton’s model. J. Comput. Appl. Math. 318, 79-92 (2017) · Zbl 1355.91083
[7] 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) · Zbl 1373.42015
[8] Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Generalized Lyapunov-Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. Discrete Contin. Dyn. Syst. 22B(9), 3591-3614 (2017) · Zbl 1371.34107
[9] Hu, H.-J., Zou, X.-F.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763-4771 (2017) · Zbl 1372.34057
[10] Yang, C., Huang, L.-H.: New criteria on exponential synchronization and existence of periodic solutions of complex BAM networks with delays. J. Nonlinear Sci. Appl. 10(10), 5464-5482 (2017) · Zbl 1412.34169
[11] 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) · Zbl 1378.92077
[12] Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge-Kutta-Nyström methods. Appl. Math. Comput. 323, 204-219 (2018) · Zbl 1426.65203
[13] 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) · Zbl 1446.65033
[14] 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) · Zbl 1415.65078
[15] Liu, B.-W., Tian, X.-M., Yang, L.-S., Huang, C.-X.: Periodic solutions for a Nicholson’s blowflies model with nonlinear mortality and continuously distributed delays. Acta Math. Appl. Sin. 41(1), 98-109 (2018) · Zbl 1424.34236
[16] 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) · Zbl 1392.35046
[17] Zhang, Y.: On products of consecutive arithmetic progressions II. Acta Math. Hung. 156(1), 240-254 (2018) · Zbl 1424.11078
[18] 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) · Zbl 1429.34037
[19] 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) · Zbl 1412.78003
[20] Jiang, Y.-J., Xu, X.-J.: A monotone finite volume method for time fractional Fokker-Planck equations. Sci. China Math. 62(4), 783-794 (2019) · Zbl 1426.65139
[21] Lin, L., Liu, Z.-Y.: An alternating projected gradient algorithm for nonnegative matrix factorization. Appl. Math. Comput. 217(24), 9997-10002 (2011) · Zbl 1219.65036
[22] Liu, Z.-Y., Zhang, Y.-L., Santos, J., Ralha, R.: On computing complex square roots of real matrices. Appl. Math. Lett. 25(10), 1565-1568 (2012) · Zbl 1251.65060
[23] Wang, W.-S.: High order stable Runge-Kutta methods for nonlinear generalized pantograph equations on the geometric mesh. Appl. Math. Model. 39(1), 270-283 (2015) · Zbl 1429.65137
[24] Li, J., Liu, F., Fang, L., Turner, I.: A novel finite volume method for the Riesz space distributed-order advection-diffusion equation. Appl. Math. Model. 46, 536-553 (2017) · Zbl 1443.65162
[25] Tan, Y.-X., Jing, K.: Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations. Math. Methods Appl. Sci. 39, 2821-2839 (2016) · Zbl 1357.34118
[26] Li, J.-L., Sun, G.-Y., Zhang, R.-M.: The numerical solution of scattering by infinite rough interfaces based on the integral equation method. Comput. Math. Appl. 71(7), 1491-1502 (2016)
[27] Dai, Z.-F.: Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl. Math. Comput. 276, 297-300 (2016) · Zbl 1410.65234
[28] Dai, Z.-F., Chen, X.-H., Wen, F.-H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378-386 (2015) · Zbl 1410.90248
[29] Xie, D.-X., Li, J.: A new analysis of electrostatic free energy minimization and Poisson-Boltzmann equation for protein in ionic solvent. Nonlinear Anal., Real World Appl. 21, 185-196 (2015) · Zbl 1298.78036
[30] Tang, W.-S., Sun, Y.-J.: Construction of Runge-Kutta type methods for solving ordinary differential equations. Appl. Math. Comput. 234, 179-191 (2014) · Zbl 1298.65112
[31] Liu, Y.-C., Wu, J.: Fixed point theorems in piecewise continuous function spaces and applications to some nonlinear problems. Math. Methods Appl. Sci. 37(4), 508-517 (2014) · Zbl 1524.47065
[32] Li, X.-F., Tang, G.-J., Tang, B.-Q.: Stress field around a strike-slip fault in orthotropic elastic layers via a hypersingular integral equation. Comput. Math. Appl. 66(11), 2317-2326 (2013) · Zbl 1381.74098
[33] Jiang, Y.-J., Ma, J.-T.: Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. J. Comput. Appl. Math. 244, 115-124 (2013) · Zbl 1263.65134
[34] Dai, Z.-F.: Two modified HS type conjugate gradient methods for unconstrained optimization problems. Nonlinear Anal. 74(3), 927-936 (2011) · Zbl 1203.49049
[35] Yang, X.-S., Zhu, Q.-X., Huang, C.-X.: Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations. Nonlinear Anal., Real World Appl. 12(1), 93-105 (2011) · Zbl 1203.93125
[36] Zhou, W.-J., Zhang, L.: Global convergence of a regularized factorized quasi-Newton method for nonlinear least squares problems. Comput. Appl. Math. 29(2), 195-204 (2010) · Zbl 1201.90203
[37] Shi, H.-P., Zhang, H.-Q.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl. 361(2), 411-419 (2010) · Zbl 1178.35351
[38] Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite-Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414-1430 (2017) · Zbl 1378.26012
[39] Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex function. J. Funct. Spaces 2018, Article ID 6595921 (2018) · Zbl 1393.26030
[40] Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: Majorization theorems for strongly convex functions. J. Inequal. Appl. 2019, Article ID 58 (2019) · Zbl 1499.26207
[41] Zaheer Ullah, S., Adil Khan, M., Khan, Z.A., Chu, Y.-M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, Article ID 9487823 (2019) · Zbl 1429.26029
[42] Khurshid, Y., Adil Khan, M., Chu, Y.-M., Khan, Z.A.: Hermite-Hadamard-Fejér inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019) · Zbl 1416.26046
[43] Zhang, X.-M., Chu, Y.-M., Zhang, X.-H.: The Hermite-Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl. 2010, Article ID 507560 (2010) · Zbl 1187.26012
[44] Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite-Hadamard type in terms of GG- and GA-convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019) · Zbl 1416.26045
[45] Chu, Y.-M., Xia, W.-F., Zhao, T.-H.: Schur convexity for a class of symmetric functions. Sci. China Math. 53(2), 465-474 (2010) · Zbl 1230.05278
[46] Chu, Y.-M., Wang, G.-D., Zhang, X.-H.: Schur convexity and Hadamard’s inequality. Math. Inequal. Appl. 13(4), 725-731 (2010) · Zbl 1205.26030
[47] Chu, Y.-M., Wang, G.-D., Zhang, X.-H.: The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 284(5-6), 653-663 (2011) · Zbl 1221.26020
[48] Chu, Y.-M., Xia, W.-F., Zhang, X.-H.: The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivar. Anal. 105, 412-421 (2012) · Zbl 1241.05148
[49] Wu, S.-H., Chu, Y.-M.: Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters. J. Inequal. Appl. 2019, Article ID 57 (2019) · Zbl 1499.26228
[50] Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017) · Zbl 1360.26018
[51] Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018) · Zbl 1497.26030
[52] Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite-Hadamard inequalities and their applications. J. Funct. Spaces 2018, Article ID 6928130 (2018) · Zbl 1391.26054
[53] Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018) · Zbl 1400.26040
[54] Adil Khan, M., Wu, S.-H., Ullah, H., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019, Article ID 16 (2019) · Zbl 1499.51018
[55] Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012) · Zbl 1259.26035
[56] 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) · Zbl 1276.26057
[57] 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, Article ID 106 (2017) · Zbl 1360.26008
[58] Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017) · Zbl 1370.41056
[59] Qian, W.-M., Chu, Y.-M.: Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters. J. Inequal. Appl. 2017, Article ID 374 (2017) · Zbl 1374.26080
[60] Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler-Mascheroni constant. J. Inequal. Appl. 2018, Article ID 118 (2018) · Zbl 1497.11292
[61] 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, Article ID 239 (2018) · Zbl 1495.33001
[62] Zhao, T.-H., Wang, M.-K., Zhang, W., Chu, Y.-M.: Quadratic transformation inequalities for Gaussian hypergeometric function. J. Inequal. Appl. 2018, Article ID 251 (2018) · Zbl 1498.33007
[63] 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) · Zbl 1403.33012
[64] 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) · Zbl 1428.33037
[65] Zhao, T.-H., Zhou, B.-C., Wang, M.-K., Chu, Y.-M.: On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, Article ID 42 (2019) · Zbl 1499.26229
[66] 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) · Zbl 1414.33006
[67] Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601-617 (2019) · Zbl 1416.33007
[68] Chen, X.-S.: New convex functions in linear spaces and Jensen’s discrete inequality. J. Inequal. Appl. 2013, Article ID 472 (2013) · Zbl 1302.46061
[69] Csiszár, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hung. 2, 299-318 (1967) · Zbl 0157.25802
[70] Csiszár, I., Körner, J.: Information Theory. Academic Press, New York (1981) · Zbl 0568.94012
[71] Dragomir, S.S.: Some inequalities for the Csiszár ϕ-divergence when ϕ is and L-Lipschitzian function and applications. Ital. J. Pure Appl. Math. 15, 57-76 (2004) · Zbl 1170.94318
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