Integral Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are *s*-Convex

Affiliation(s)

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China.

Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, China.

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China.

Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, China.

ABSTRACT

In the paper, the authors find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.

In the paper, the authors find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.

KEYWORDS

Integral Inequality; Hermite-Hadamard’s Integral Inequality; s-Convex Function; Derivative; Mean

Integral Inequality; Hermite-Hadamard’s Integral Inequality; s-Convex Function; Derivative; Mean

Cite this paper

L. Chun and F. Qi, "Integral Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are*s*-Convex," *Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1680-1685. doi: 10.4236/am.2012.311232.

L. Chun and F. Qi, "Integral Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are

References

[1] S. S. Dragomir, J. Pecaric and L.-E. Persson, “Some Inequalities of Hadamard Type,” Soochow Journal of Mathematics, Vol. 21, No. 3, 1995, pp. 335-341.

[2] H. Hudzik and L. Maligranda, “Some Remarks on s-Convex Functions,” Aequationes Mathematicae, Vol. 48, No. 1, 1994, pp. 100-111.

[3] S. S. Dragomir and R. P. Agarwal, “Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula,” Applied Mathematics Letters, Vol. 11, No. 5, 1998, pp. 91-95. doi:10.1016/S0893-9659(98)00086-X

[4] C. E. M. Pearce and J. Pecaric, “Inequalities for Differentiable Mappings with Application to Special Means and Quadrature Formulae,” Applied Mathematics Letters, Vol. 13, No. 2, 2000, pp. 51-55. doi:10.1016/S0893-9659(99)00164-0

[5] U. S. Kirmaci, “Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Midpoint Formula,” Applied Mathematics and Computation, Vol. 147, No. 1, 2004, pp. 137-146. doi:10.1016/S0096-3003(02)00657-4

[6] U. S. Kirmaci, M. K. Bakula, M. E. Ozdemir and J. Pecaric, “Hadamard-Type Inequalities for s-Convex Functions,” Applied Mathematics and Computation, Vol. 193, No. 1, 2007, pp. 26-35. doi:10.1016/j.amc.2007.03.030

[7] M. Alomari and S. Hussain, “Two Inequalities of Simpson Type for Quasi-Convex Functions and Applications,” Applied Mathematics E-Notes, Vol. 11, 2011, pp. 110-117.

[8] R.-F. Bai, F. Qi and B.-Y. Xi, “Hermite-Hadamard Type Inequalities for the mand (α, m)-Logarithmically Convex Functions,” Filomat, Vol. 27, No. 1, 2013, 1-7.

[9] S.-P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard Type Inequalities for n-Time Differentiable (α, m)-Convex Functions,” Journal of Inequalities and Applications, 2013, in Press.

[10] W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard Inequality to n-Time Differentiable Functions Which Are s-Convex in the Second Sense,” Analysis (Munich), Vol. 32, No. 3, 2012, pp. 209-220. doi:10.1524/anly.2012.1161

[11] F. Qi, Z.-L. Wei and Q. Yang, “Generalizations and Refinements of Hermite-Hadamard’s Inequality,” The Rocky Mountain Journal of Mathematics, Vol. 35, No. 1, 2005, pp. 235-251. doi:10.1216/rmjm/1181069779

[12] S.-H. Wang, B.-Y. Xi and F. Qi, “On Hermite-Hadamard Type Inequalities for (α, m)-Convex Functions,” International Journal of Open Problems in Computer Science and Mathematics, Vol. 5, No. 4, 2012, in Press.

[13] S.-H. Wang, B.-Y. Xi and F. Qi, “Some New Inequalities of Hermite-Hadamard Type for n-Time Differentiable Functions Which Are m-Convex,” Analysis (Munich), Vol. 32, No. 3, 2012, pp. 247-262. doi:10.1524/anly.2012.1167

[14] B.-Y. Xi, R.-F. Bai and F. Qi, “Hermite-Hadamard Type Inequalities for the mand (α; m)-Geometrically Convex Functions,” Aequationes Mathematicae, 2012, in Press. doi:10.1007/s00010-011-0114-x

[15] B.-Y. Xi and F. Qi, “Some Hermite-Hadamard Type Inequalities for Differentiable Convex Functions and Applications,” Hacettepe Journal of Mathematics and Statistics, Vol. 42, 2013, in Press.

[16] B.-Y. Xi and F. Qi, “Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means,” Journal of Function Spaces and Applications, Vol. 2012, 2012, 14 pp. doi:10.1155/2012/980438

[17] T.-Y. Zhang, A.-P. Ji and F. Qi, “On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions,” Abstract and Applied Analysis, Vol. 2012, 2012, 15 pp. doi:10.1155/2012/560586

[18] S. S. Dragomir and C. E. M. Pearce, “Selected Topics on Hermite-Hadamard Type Inequalities and Applications,” RGMIA Monographs, Victoria University, Melbourne, 2000.

[19] C. P. Niculescu and L.-E. Persson, “Convex Functions and Their Applications: A Contemporary Approach (CMS Books in Mathematics),” Springer-Verlag, New York, 2005.

[1] S. S. Dragomir, J. Pecaric and L.-E. Persson, “Some Inequalities of Hadamard Type,” Soochow Journal of Mathematics, Vol. 21, No. 3, 1995, pp. 335-341.

[2] H. Hudzik and L. Maligranda, “Some Remarks on s-Convex Functions,” Aequationes Mathematicae, Vol. 48, No. 1, 1994, pp. 100-111.

[3] S. S. Dragomir and R. P. Agarwal, “Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula,” Applied Mathematics Letters, Vol. 11, No. 5, 1998, pp. 91-95. doi:10.1016/S0893-9659(98)00086-X

[4] C. E. M. Pearce and J. Pecaric, “Inequalities for Differentiable Mappings with Application to Special Means and Quadrature Formulae,” Applied Mathematics Letters, Vol. 13, No. 2, 2000, pp. 51-55. doi:10.1016/S0893-9659(99)00164-0

[5] U. S. Kirmaci, “Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Midpoint Formula,” Applied Mathematics and Computation, Vol. 147, No. 1, 2004, pp. 137-146. doi:10.1016/S0096-3003(02)00657-4

[6] U. S. Kirmaci, M. K. Bakula, M. E. Ozdemir and J. Pecaric, “Hadamard-Type Inequalities for s-Convex Functions,” Applied Mathematics and Computation, Vol. 193, No. 1, 2007, pp. 26-35. doi:10.1016/j.amc.2007.03.030

[7] M. Alomari and S. Hussain, “Two Inequalities of Simpson Type for Quasi-Convex Functions and Applications,” Applied Mathematics E-Notes, Vol. 11, 2011, pp. 110-117.

[8] R.-F. Bai, F. Qi and B.-Y. Xi, “Hermite-Hadamard Type Inequalities for the mand (α, m)-Logarithmically Convex Functions,” Filomat, Vol. 27, No. 1, 2013, 1-7.

[9] S.-P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard Type Inequalities for n-Time Differentiable (α, m)-Convex Functions,” Journal of Inequalities and Applications, 2013, in Press.

[10] W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard Inequality to n-Time Differentiable Functions Which Are s-Convex in the Second Sense,” Analysis (Munich), Vol. 32, No. 3, 2012, pp. 209-220. doi:10.1524/anly.2012.1161

[11] F. Qi, Z.-L. Wei and Q. Yang, “Generalizations and Refinements of Hermite-Hadamard’s Inequality,” The Rocky Mountain Journal of Mathematics, Vol. 35, No. 1, 2005, pp. 235-251. doi:10.1216/rmjm/1181069779

[12] S.-H. Wang, B.-Y. Xi and F. Qi, “On Hermite-Hadamard Type Inequalities for (α, m)-Convex Functions,” International Journal of Open Problems in Computer Science and Mathematics, Vol. 5, No. 4, 2012, in Press.

[13] S.-H. Wang, B.-Y. Xi and F. Qi, “Some New Inequalities of Hermite-Hadamard Type for n-Time Differentiable Functions Which Are m-Convex,” Analysis (Munich), Vol. 32, No. 3, 2012, pp. 247-262. doi:10.1524/anly.2012.1167

[14] B.-Y. Xi, R.-F. Bai and F. Qi, “Hermite-Hadamard Type Inequalities for the mand (α; m)-Geometrically Convex Functions,” Aequationes Mathematicae, 2012, in Press. doi:10.1007/s00010-011-0114-x

[15] B.-Y. Xi and F. Qi, “Some Hermite-Hadamard Type Inequalities for Differentiable Convex Functions and Applications,” Hacettepe Journal of Mathematics and Statistics, Vol. 42, 2013, in Press.

[16] B.-Y. Xi and F. Qi, “Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means,” Journal of Function Spaces and Applications, Vol. 2012, 2012, 14 pp. doi:10.1155/2012/980438

[17] T.-Y. Zhang, A.-P. Ji and F. Qi, “On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions,” Abstract and Applied Analysis, Vol. 2012, 2012, 15 pp. doi:10.1155/2012/560586

[18] S. S. Dragomir and C. E. M. Pearce, “Selected Topics on Hermite-Hadamard Type Inequalities and Applications,” RGMIA Monographs, Victoria University, Melbourne, 2000.

[19] C. P. Niculescu and L.-E. Persson, “Convex Functions and Their Applications: A Contemporary Approach (CMS Books in Mathematics),” Springer-Verlag, New York, 2005.