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Some new integral inequalities for n-times differentiable log-convex functions

Year 2017, Volume: 5 Issue: 2, 10 - 15, 30.03.2017

İmdat Iscan Huriye Kadakal Mahir Kadakal

Abstract

 In this work, by using an integral identity together with both the Ho¨lder and the Power-Mean integral inequality we establish several new inequalities for n-time differentiable log-convex functions.

Keywords

Convex function Log-Convex function

References

  • [1] M. Alomari and M. Darus, “On The Hadamard’s Inequality for Log-Convex Functions on the Coordinates”, Hindawi Publishing Corporation Journal of Inequalities and Applications, Volume 2009, Article ID 283147, 13 pages, doi:10.1155/2009/283147.
  • [2] M. A. Ardıc¸ and M. Emin ¨Ozdemir, “Inequalities for log-convex functions vıa three times differentiability”, arXiv:1405.7480v1 [math.CA] 29 May 2014.
  • [3] S. P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable ( α ,m)-convex functions”, Jour. of Ineq. and Appl., 2012, 2012:267.
  • [4] P. Cerone, S.S. Dragomir and J. Roumeliotis, “Some Ostrowski type inequalities for n-time differentiable mappings and applications”, Demonstratio Math., 32 (4) (1999), 697–712.
  • [5] P. Cerone, S.S. Dragomir, J. Roumeliotis and J. Sunde, “A new generalization of the trapezoid formula for n-time differentiable mappings and applications”, Demonstratio Math., 33 (4) (2000), 719–736.
  • [6] S. S. Dragomir and C.E.M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000, online: http://www.staxo.vu.edu.au/RGMIA/monographs/hermite hadamard.html.
  • [7] S.S. Dragomir, “New jensen’s type inequalities for differentiable log-convex functions of selfadjoint operators in Hilbert spaces”, Sarajevo Journal of Mathematics, Vol.7 (19) (2011), 67-80.
  • [8] D. Y. Hwang, “Some Inequalities for n-time Differentiable Mappings and Applications”, Kyung. Math. Jour., 43 (2003), 335–343.
  • [9] I. Is¸can, “Ostrowski type inequalities for p-convex functions”, New Trends in Mathematical Sciences, 4 (3) (2016), 140-150.
  • [10] I. Is¸can and S. Turhan, “Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral”, Moroccan J. Pure and Appl. Anal.(MJPAA), Volume 2(1) (2016), 34-46. [11] I. Is¸can, “Hermite-Hadamard type inequalities for harmonically convex functions”, Hacettepe Journal of Mathematics and Statistics, Volume 43 (6) (2014), 935–942.
  • [12] W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard inequality to n-time differentiable function which are s -convex in the second sense”, Analysis (Munich), 32 (2012), 209–220.
  • [13] S. Maden, H. Kadakal, M. Kadakal and ˙ I. ˙Is¸can, “Some new integral inequalities for n-times differentiable convex and concave functions”, https://www.researchgate.net/publication/312529563, (Submitted).
  • [14] M. Mansour, M. A. Obaid, “A Generalization of Some Inequalities for the log-Convex Functions”, International Mathematical Forum, 5, 2010, no. 65, 3243 – 3249. [15] C. P. Niculescu, “The Hermite–Hadamard inequality for log-convex functions”, Nonlinear Analysis 75 (2012) 662–669.
  • [16] J. Park, “Some Hermite-Hadamard-like Type Inequalities for Logarithmically Convex Functions”, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 45, 2217-2233.
  • [17] 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), 32 (2012), 247–262.
  • [18] G. S. Yang, K. L. Tseng and H. T. Wang, “A note on ıntegral inequalıtıes of Hadamard type for log-convex and log-concave functıons”, Taıwanese Journal of Mathematıcs, Vol. 16, No. 2, pp. 479-496, April 2012.
  • [19] C. Yıldız, “New inequalities of the Hermite-Hadamard type for n-time differentiable functions which are quasiconvex”, Journal of Mathematical Inequalities, 10, 3(2016), 703-711.
  • [20] X. Zhang, W. Jiang, “Some properties of log-convex function and applications for the exponential function”, Computers and Mathematics with Applications 63 (2012) 1111–1116.

Year 2017, Volume: 5 Issue: 2, 10 - 15, 30.03.2017

İmdat Iscan Huriye Kadakal Mahir Kadakal

Abstract

References

  • [1] M. Alomari and M. Darus, “On The Hadamard’s Inequality for Log-Convex Functions on the Coordinates”, Hindawi Publishing Corporation Journal of Inequalities and Applications, Volume 2009, Article ID 283147, 13 pages, doi:10.1155/2009/283147.
  • [2] M. A. Ardıc¸ and M. Emin ¨Ozdemir, “Inequalities for log-convex functions vıa three times differentiability”, arXiv:1405.7480v1 [math.CA] 29 May 2014.
  • [3] S. P. Bai, S.-H. Wang and F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable ( α ,m)-convex functions”, Jour. of Ineq. and Appl., 2012, 2012:267.
  • [4] P. Cerone, S.S. Dragomir and J. Roumeliotis, “Some Ostrowski type inequalities for n-time differentiable mappings and applications”, Demonstratio Math., 32 (4) (1999), 697–712.
  • [5] P. Cerone, S.S. Dragomir, J. Roumeliotis and J. Sunde, “A new generalization of the trapezoid formula for n-time differentiable mappings and applications”, Demonstratio Math., 33 (4) (2000), 719–736.
  • [6] S. S. Dragomir and C.E.M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000, online: http://www.staxo.vu.edu.au/RGMIA/monographs/hermite hadamard.html.
  • [7] S.S. Dragomir, “New jensen’s type inequalities for differentiable log-convex functions of selfadjoint operators in Hilbert spaces”, Sarajevo Journal of Mathematics, Vol.7 (19) (2011), 67-80.
  • [8] D. Y. Hwang, “Some Inequalities for n-time Differentiable Mappings and Applications”, Kyung. Math. Jour., 43 (2003), 335–343.
  • [9] I. Is¸can, “Ostrowski type inequalities for p-convex functions”, New Trends in Mathematical Sciences, 4 (3) (2016), 140-150.
  • [10] I. Is¸can and S. Turhan, “Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral”, Moroccan J. Pure and Appl. Anal.(MJPAA), Volume 2(1) (2016), 34-46. [11] I. Is¸can, “Hermite-Hadamard type inequalities for harmonically convex functions”, Hacettepe Journal of Mathematics and Statistics, Volume 43 (6) (2014), 935–942.
  • [12] W.-D. Jiang, D.-W. Niu, Y. Hua and F. Qi, “Generalizations of Hermite-Hadamard inequality to n-time differentiable function which are s -convex in the second sense”, Analysis (Munich), 32 (2012), 209–220.
  • [13] S. Maden, H. Kadakal, M. Kadakal and ˙ I. ˙Is¸can, “Some new integral inequalities for n-times differentiable convex and concave functions”, https://www.researchgate.net/publication/312529563, (Submitted).
  • [14] M. Mansour, M. A. Obaid, “A Generalization of Some Inequalities for the log-Convex Functions”, International Mathematical Forum, 5, 2010, no. 65, 3243 – 3249. [15] C. P. Niculescu, “The Hermite–Hadamard inequality for log-convex functions”, Nonlinear Analysis 75 (2012) 662–669.
  • [16] J. Park, “Some Hermite-Hadamard-like Type Inequalities for Logarithmically Convex Functions”, Int. Journal of Math. Analysis, Vol. 7, 2013, no. 45, 2217-2233.
  • [17] 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), 32 (2012), 247–262.
  • [18] G. S. Yang, K. L. Tseng and H. T. Wang, “A note on ıntegral inequalıtıes of Hadamard type for log-convex and log-concave functıons”, Taıwanese Journal of Mathematıcs, Vol. 16, No. 2, pp. 479-496, April 2012.
  • [19] C. Yıldız, “New inequalities of the Hermite-Hadamard type for n-time differentiable functions which are quasiconvex”, Journal of Mathematical Inequalities, 10, 3(2016), 703-711.
  • [20] X. Zhang, W. Jiang, “Some properties of log-convex function and applications for the exponential function”, Computers and Mathematics with Applications 63 (2012) 1111–1116.
There are 18 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

İmdat Iscan This is me

Huriye Kadakal This is me

Mahir Kadakal

Publication Date March 30, 2017
Published in Issue Year 2017 Volume: 5 Issue: 2

Cite

APA Iscan, İ., Kadakal, H., & Kadakal, M. (2017). Some new integral inequalities for n-times differentiable log-convex functions. New Trends in Mathematical Sciences, 5(2), 10-15.
AMA Iscan İ, Kadakal H, Kadakal M. Some new integral inequalities for n-times differentiable log-convex functions. New Trends in Mathematical Sciences. March 2017;5(2):10-15.
Chicago Iscan, İmdat, Huriye Kadakal, and Mahir Kadakal. “Some New Integral Inequalities for N-Times Differentiable Log-Convex Functions”. New Trends in Mathematical Sciences 5, no. 2 (March 2017): 10-15.
EndNote Iscan İ, Kadakal H, Kadakal M (March 1, 2017) Some new integral inequalities for n-times differentiable log-convex functions. New Trends in Mathematical Sciences 5 2 10–15.
IEEE İ. Iscan, H. Kadakal, and M. Kadakal, “Some new integral inequalities for n-times differentiable log-convex functions”, New Trends in Mathematical Sciences, vol. 5, no. 2, pp. 10–15, 2017.
ISNAD Iscan, İmdat et al. “Some New Integral Inequalities for N-Times Differentiable Log-Convex Functions”. New Trends in Mathematical Sciences 5/2 (March 2017), 10-15.
JAMA Iscan İ, Kadakal H, Kadakal M. Some new integral inequalities for n-times differentiable log-convex functions. New Trends in Mathematical Sciences. 2017;5:10–15.
MLA Iscan, İmdat et al. “Some New Integral Inequalities for N-Times Differentiable Log-Convex Functions”. New Trends in Mathematical Sciences, vol. 5, no. 2, 2017, pp. 10-15.
Vancouver Iscan İ, Kadakal H, Kadakal M. Some new integral inequalities for n-times differentiable log-convex functions. New Trends in Mathematical Sciences. 2017;5(2):10-5.
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