Authors: Bo-Yan Xi 1 and Feng Qi
View More View Less
  • 1 Inner Mongolia University for Nationalities College of Mathematics Tongliao City, Inner Mongolia Autonomous Region 028043 China
  • 2 Tianjin Polytechnic University Department of Mathematics, College of Science Tianjin City 300160 China
  • 3 Henan Polytechnic University Institute of Mathematics Jiaozuo City, Henan Province 454010 China
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

In the paper, the authors introduce a new concept of geometrically r-convex functions and establish some inequalities of Hermite-Hadamard type for this class of functions.

  • Alomari, M. and Darus, M., On The Hadamard’s inequality for log-convex functions on the coordinates, J. Inequal. Appl., 2009 (2009), Article ID 283147, 13 pages; Available online at http://dx.doi.org/10.1155/2009/283147

    Darus M. , '', in J. Inequal. Appl. , (2009 ) -.

  • Bai, R.-F., Qi, F. and Xi, B.-Y., Hermite-Hadamard type inequalities for the m- and (α, m)-logarithmically convex functions, Filomat, 27 (2013), no. 1, 1–7; Available online at http://dx.doi.org/10.2298/FIL1301001B

    Xi B.-Y. , 'Hermite-Hadamard type inequalities for the m- and (α, m)-logarithmically convex functions ' (2013 ) 27 Filomat : 1 -7.

    • Search Google Scholar
  • Bai, S.-P. and Qi, F., Some inequalities for (s 1, m 1)-(s 2, m 2)-convex functions on the co-ordinates, Glob. J. Math. Anal., 1 (2013), no. 1, 22–28; Available online at http://dx.doi.org/10.14419/gjma.v1i1.776

    Qi F. , 'Some inequalities for (s1, m1)-(s2, m2)-convex functions on the co-ordinates ' (2013 ) 1 Glob. J. Math. Anal. : 22 -28.

    • Search Google Scholar
  • Chun, L. and Qi, F., Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, J. Inequal. Appl., 2013, 2013:451, 10 pages; Available online at http://dx.doi.org/10.1186/1029-242X-2013-451

    Qi F. , 'Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex ' (2013 ) 2013 J. Inequal. Appl. : 451 -.

    • Search Google Scholar
  • Dragomir, S. S., Refinements of the Hermite-Hadamard integral inequality for log-convex functions, Austral. Math. Soc. Gaz., 28 (2001), no. 3, 129–134.

    Dragomir S. S. , 'Refinements of the Hermite-Hadamard integral inequality for log-convex functions ' (2001 ) 28 Austral. Math. Soc. Gaz. : 129 -134.

    • Search Google Scholar
  • Dragomir, S. S. and Agarwal, R. P., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), no. 5, 91–95; Available online at http://dx.doi.org/10.1016/S0893-9659(98)00086-X

    Agarwal R. P. , 'Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula ' (1998 ) 11 Appl. Math. Lett. : 91 -95.

    • Search Google Scholar
  • Dragomir, S. S. and Pearce, C. E. M., Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000; Available online at http://rgmia.org/monographs/hermite_hadamard.html

    Pearce C. E. M. , '', in Selected Topics on Hermite-Hadamard Type Inequalities and Applications , (2000 ) -.

    • Search Google Scholar
  • Gill, P. M., Pearce, C. E. M. and Pečarić, J., Hadamard’s inequality for r-convex functions, J. Math. Anal. Appl., 215 (1997), no. 2, 461–470; Available online at http://dx.doi.org/10.1006/jmaa.1997.5645

    Pečarić J. , 'Hadamard’s inequality for r-convex functions ' (1997 ) 215 J. Math. Anal. Appl. : 461 -470.

    • Search Google Scholar
  • Guo, B.-N. and Qi, F., A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms, 52 (2009), no. 1, 89–92; Available online at http://dx.doi.org/10.1007/s11075-008-9259-7

    Qi F. , 'A simple proof of logarithmic convexity of extended mean values ' (2009 ) 52 Numer. Algorithms : 89 -92.

    • Search Google Scholar
  • Guo, B.-N. and Qi, F., The function (b xa x)/x: Logarithmic convexity and applications to extended mean values, Filomat, 25 (2011), no. 4, 63–73; Available online at http://dx.doi.org/10.2298/FIL1104063G

    Qi F. , 'The function (bx − ax)/x: Logarithmic convexity and applications to extended mean values ' (2011 ) 25 Filomat : 63 -73.

    • Search Google Scholar
  • Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1934.

    Pólya G. , '', in Inequalities , (1934 ) -.

  • Kirmaci, U. S., Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147 (2004), no. 1, 137–146; Available online at http://dx.doi.org/10.1016/S0096-3003(02)00657-4

    Kirmaci U. S. , 'Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula ' (2004 ) 147 Appl. Math. Comput. : 137 -146.

    • Search Google Scholar
  • Li, W.-H. and Qi, F., Some Hermite-Hadamard type inequalities for functions whose n-th derivatives are (α, m)-convex, Filomat, 27 (2013), no. 8, 1575–1582; Available online at http://dx.doi.org/10.2298/FIL1308575L

    Qi F. , 'Some Hermite-Hadamard type inequalities for functions whose n-th derivatives are (α, m)-convex ' (2013 ) 27 Filomat : 1575 -1582.

    • Search Google Scholar
  • Ngoc, N. P. N., Vinh, N. V. and Hien, P. T. T., Integral inequalities of Hadamard type for r-convex functions, Int. Math. Forum, 4 (2009), no. 33–36, 1723–1728.

    Hien P. T. T. , 'Integral inequalities of Hadamard type for r-convex functions ' (2009 ) 4 Int. Math. Forum : 1723 -1728.

    • Search Google Scholar
  • Niculescu, C. P. and Persson, L.-E., Convex Functions and their Applications, CMS Books in Mathematics, Springer-Verlag, 2005.

    Persson L.-E. , '', in Convex Functions and their Applications , (2005 ) -.

  • Noor, M. A., Qi, F. and Awan, M. U., Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis (Berlin), 33 (2013), no. 4, 367–375; Available online at http://dx.doi.org/10.1524/anly.2013.1223

    Awan M. U. , 'Some Hermite-Hadamard type inequalities for log-h-convex functions ' (2013 ) 33 Analysis (Berlin) : 367 -375.

    • Search Google Scholar
  • Pearce, C. E. M. and Pečarić, J. Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13 (2000), no. 2, 51–55; Available online at http://dx.doi.org/10.1016/S0893-9659(99)00164-0

    Pečarić J. , 'Inequalities for differentiable mappings with application to special means and quadrature formulae ' (2000 ) 13 Appl. Math. Lett. : 51 -55.

    • Search Google Scholar
  • Pearce, C. E. M., Pečarić, J. and Šimić, V., Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl., 220 (1998), no. 1, 99–109; Available online at http://dx.doi.org/10.1006/jmaa.1997.5822

    Šimić V. , 'Stolarsky means and Hadamard’s inequality ' (1998 ) 220 J. Math. Anal. Appl. : 99 -109.

    • Search Google Scholar
  • Qi, F. and Xi, B.-Y., Some integral inequalities of Simpson type for GA-ε-convex functions, Georgian Math. J., 20 (2013), no. 4, 775–788; Available online at http://dx.doi.org/10.1515/gmj-2013-0043

    Xi B.-Y. , 'Some integral inequalities of Simpson type for GA-ε-convex functions ' (2013 ) 20 Georgian Math. J. : 775 -788.

    • Search Google Scholar
  • Shuang, Y., Wang, Y. and Qi, F., Some inequalities of Hermite-Hadamard type for functions whose third derivatives are (α, m)-convex, J. Comput. Anal. Appl., 17 (2014), no. 2, 272–279.

    Qi F. , 'Some inequalities of Hermite-Hadamard type for functions whose third derivatives are (α, m)-convex ' (2014 ) 17 J. Comput. Anal. Appl. : 272 -279.

    • Search Google Scholar
  • Shuang, Y., Yin, H.-P. and Qi, F., Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions, Analysis (Munich), 33 (2013), no. 2, 197–208; Available online at http://dx.doi.org/10.1524/anly.2013.1192

    Qi F. , 'Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions ' (2013 ) 33 Analysis (Munich) : 197 -208.

    • Search Google Scholar
  • Stolarsky, K. B., Generalizations of the logarithmic mean, Math. Mag., 48 (1975), 87–92.

    Stolarsky K. B. , 'Generalizations of the logarithmic mean ' (1975 ) 48 Math. Mag. : 87 -92.

  • Sulaiman, W. T., Refinements to Hadamard’s inequality for log-convex functions, Appl. Math., 2 (2011), no. 7, 899–903; Available online at http://dx.doi.org/10.4236/am.2011.27120

    Sulaiman W. T. , 'Refinements to Hadamard’s inequality for log-convex functions ' (2011 ) 2 Appl. Math. : 899 -903.

    • Search Google Scholar
  • Sun, M.-B. and Yang, X.-P., Inequalities for the weighted mean of r-convex functions, Proc. Amer. Math. Soc., 133 (2005), no. 6, 1639–1646; Available online at http://dx.doi.org/10.1090/S0002-9939-05-07835-4

    Yang X.-P. , 'Inequalities for the weighted mean of r-convex functions ' (2005 ) 133 Proc. Amer. Math. Soc. : 1639 -1646.

    • Search Google Scholar
  • Wang, S.-H. and Qi, F., Inequalities of Hermite-Hadamard type for convex functions which are n-times differentiable, Math. Inequal. Appl., 16 (2013), no. 4, 1269–1278; Available online at http://dx.doi.org/10.7153/mia-16-97

    Qi F. , 'Inequalities of Hermite-Hadamard type for convex functions which are n-times differentiable ' (2013 ) 16 Math. Inequal. Appl. : 1269 -1278.

    • Search Google Scholar
  • Xi, B.-Y. and Bao, T.-Y., Some properties of r-mean convex functions, Math. Practice Theory, 38 (2008), no. 12, 113–119. (Chinese).

    Bao T.-Y. , 'Some properties of r-mean convex functions ' (2008 ) 38 Math. Practice Theory : 113 -119.

    • Search Google Scholar
  • Xi, B.-Y. and Qi, F., Hermite-Hadamard type inequalities for functions whose derivatives are of convexities, Nonlinear Funct. Anal. Appl., 18 (2013), no. 2, 163–176.

    Qi F. , 'Hermite-Hadamard type inequalities for functions whose derivatives are of convexities ' (2013 ) 18 Nonlinear Funct. Anal. Appl. : 163 -176.

    • Search Google Scholar
  • Xi, B.-Y. and Qi, F., Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacet. J. Math. Stat., 42 (2013), no. 3, 243–257.

    Qi F. , 'Some Hermite-Hadamard type inequalities for differentiable convex functions and applications ' (2013 ) 42 Hacet. J. Math. Stat. : 243 -257.

    • Search Google Scholar
  • Xi, B.-Y. and Qi, F., Some inequalities of Hermite-Hadamard type for h-convex functions, Adv. Inequal. Appl., 2 (2013), no. 1, 1–15.

    Qi F. , 'Some inequalities of Hermite-Hadamard type for h-convex functions ' (2013 ) 2 Adv. Inequal. Appl. : 1 -15.

    • Search Google Scholar
  • Xi, B.-Y., Wang, Y. and Qi, F., Some integral inequalities of Hermite-Hadamard type for (s, m)-convex functions, Transylv. J. Math. Mechanics, 5 (2013), no. 1, 69–84.

    Qi F. , 'Some integral inequalities of Hermite-Hadamard type for (s, m)-convex functions ' (2013 ) 5 Transylv. J. Math. Mechanics : 69 -84.

    • Search Google Scholar
  • Yang, G.-S. and Hwang, D.-Y., Refinements of Hadamard’s inequality for r-convex functions, Indian J. Pure Appl. Math., 32 (2001), no. 10, 1571–1579.

    Hwang D.-Y. , 'Refinements of Hadamard’s inequality for r-convex functions ' (2001 ) 32 Indian J. Pure Appl. Math. : 1571 -1579.

    • Search Google Scholar
  • Zhang, X.-M., Chu, Y.-M. and Zhang, X.-H., The Hermite-Hadamard type inequality of GA-convex functions and its application, J. Inequal. Appl., 2010 (2010), Article ID 507560, 11 pages; Available online at http://dx.doi.org/10.1155/2010/507560

    Zhang X.-H. , '', in J. Inequal. Appl. , (2010 ) -.

  • Zhang, T.-Y., Ji, A.-P. and Qi, F., Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means, Matematiche (Catania), 68 (2013), no. 1, 229–239; Available online at http://dx.doi.org/10.4418/2013.68.1.17

    Qi F. , 'Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means ' (2013 ) 68 Matematiche (Catania) : 229 -239.

    • Search Google Scholar