On Giaccardi's Inequality and Associated Functional in the Plane

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Atiq Ur Rehman
M. Hassaan Akbar
G. Farid

Abstract

In this paper the authors extend Giaccardi's inequality to coordinates in the plane. The authors consider the nonnegative associated functional due to Giaccardi's inequality in plane and discuss its properties for certain class of parametrized functions. Also the authors proved related mean value theorems.

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References

  1. S. Butt, J. Peˇ cari ´ c and Atiq ur Rehman, Exponential Convexity of Petrovi ´ c and Related Functional, J. Inequal. Appl. 2011 (2011), Art. ID 89.
  2. S. S Dragomir, On Hadamards Inequality for Convex Functions on the Co-ordinates in a Rectangle from the Plane, Taiwanese J Mat. 4 (2001), 775-788
  3. G. Farid, M. Marwan, and A. U. Rehman, Fejer-Hadamard Inequality for Convex Functions on the Coordinates in a Rectangle from the Plane, Int. J. Analysis Appl. 10(1) (2016), 40-47.
  4. G. Farid, M. Marwan and A. U. Rehman, New Mean Value Theorems and Generalization of Hadamard Inequality via Coordinated m-Convex Functions, J. Inequal. Appl. 2015 (2015), Art. ID 283.
  5. D. S. Mitrinovic, J. Peˇ cari ´ c and A.M Fink, Classical and New Inequalities in Analysis, Vol. 61, Springer Science & Business Media, 2013.
  6. M. A. Noor, F. Qi and M. U. Awan , Some Hermite-Hadamard Type Inequalities for Log-h-Convex Functions, Analysis, 33(4) (2013), 367-375.
  7. C.P. Niculescu, The Hermite-Hadamard Inequality for Log-convex Functions, Nonlinear Analysis, 75 (2012) 662-669.
  8. J. Peˇ cari ´ c, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, New York, 1992.
  9. J. Peˇ cari ´ c and Atiq Ur Rehman, On Logarithmic Convexity for Power Sums and Related Results, J. Inequal. Appl. 2008 (2008), Art. ID 389410.
  10. J. Peˇ cari ´ c and Atiq Ur Rehman, On Logarithmic Convexity for Giaccardi's Difference, Rad HAZU. 515 (2013), 01-10.
  11. A. U. Rehman, Muhammad Mudessir, Hafiza Tahira Fazal and Ghulam Farid, Petrovi ´ c's Inequality on Coordinates and Related Results, Cogent Math. 3(1) (2016), Art. ID 1227298.
  12. Xiaoming Zhang and Weidong Jiang, Some Properties of Log-convex Function and Applications for the Exponential Function, Comput. Math. Appl. 63(6) (2012), 1111-1116.