Inequalities of Hermite–Hadamard type for GA-convex functions
Abstract
Some inequalities of Hermite–Hadamard type for GA-convex functions defined on positive intervals are given.
Keywords
Hermite–Hadamard type inequalities; convex functions; integral inequalities; GA-convex functions
References
1. Anderson G.D., Vamanamurthy M.K., Vuorinen M., Generalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007), no. 2, 1294–1308.
2. Beckenbach E.F., Convex functions, Bull. Amer. Math. Soc. 54 (1948), no. 5, 439–460.
3. Bombardelli M., Varošanec S., Properties of h-convex functions related to the Hermite–Hadamard–Fejér inequalities, Comput. Math. Appl. 58 (2009), no. 9, 1869–1877.
4. Cristescu G., Hadamard type inequalities for convolution of h-convex functions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3–11.
5. Dragomir S.S., An inequality improving the first Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 31, 8 pp.
6. Dragomir S.S., An inequality improving the second Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 3, Article 35, 8 pp.
7. Dragomir S.S., An Ostrowski like inequality for convex functions and applications, Rev. Math. Complut. 16 (2003), no. 2, 373–382.
8. Dragomir S.S., Fitzpatrick S., The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32 (1999), no. 4, 687–696.
9. Dragomir S.S., Fitzpatrick S., The Jensen inequality for s-Breckner convex functions in linear spaces, Demonstratio Math. 33 (2000), no. 1, 43–49.
10. Dragomir S.S., Mond B., On Hadamard’s inequality for a class of functions of Godunova and Levin, Indian J. Math. 39 (1997), no. 1, 1–9.
11. Dragomir S.S., Pearce C.E.M., On Jensen’s inequality for a class of functions of Godunova and Levin, Period. Math. Hungar. 33 (1996), no. 2, 93–100.
12. Dragomir S.S., Pearce C.E.M., Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc. 57 (1998), no. 3, 377–385.
13. Dragomir S.S., Pečarić J., Persson L.E., Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335–341.
14. Dragomir S.S., Rassias Th.M. (eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, Dordrecht, 2002.
15. Godunova E.K., Levin V.I., Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numerical mathematics and mathematical physics, 138–142, 166, Moskov. Gos. Ped. Inst., Moscow, 1985 (in Russian).
16. Noor M.A., Noor K.I., Awan M.U., Some inequalities for geometrically-arithmetically h-convex functions, Creat. Math. Inform. 23 (2014), no. 1, 91–98.
17. Zhang X.-M., Chu Y.-M., Zhang X.-H., The Hermite–Hadamard type inequality of GAconvex functions and its application, J. Inequal. Appl. 2010, Art. ID 507560, 11 pp.
2. Beckenbach E.F., Convex functions, Bull. Amer. Math. Soc. 54 (1948), no. 5, 439–460.
3. Bombardelli M., Varošanec S., Properties of h-convex functions related to the Hermite–Hadamard–Fejér inequalities, Comput. Math. Appl. 58 (2009), no. 9, 1869–1877.
4. Cristescu G., Hadamard type inequalities for convolution of h-convex functions, Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 8 (2010), 3–11.
5. Dragomir S.S., An inequality improving the first Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 31, 8 pp.
6. Dragomir S.S., An inequality improving the second Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 3, Article 35, 8 pp.
7. Dragomir S.S., An Ostrowski like inequality for convex functions and applications, Rev. Math. Complut. 16 (2003), no. 2, 373–382.
8. Dragomir S.S., Fitzpatrick S., The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32 (1999), no. 4, 687–696.
9. Dragomir S.S., Fitzpatrick S., The Jensen inequality for s-Breckner convex functions in linear spaces, Demonstratio Math. 33 (2000), no. 1, 43–49.
10. Dragomir S.S., Mond B., On Hadamard’s inequality for a class of functions of Godunova and Levin, Indian J. Math. 39 (1997), no. 1, 1–9.
11. Dragomir S.S., Pearce C.E.M., On Jensen’s inequality for a class of functions of Godunova and Levin, Period. Math. Hungar. 33 (1996), no. 2, 93–100.
12. Dragomir S.S., Pearce C.E.M., Quasi-convex functions and Hadamard’s inequality, Bull. Austral. Math. Soc. 57 (1998), no. 3, 377–385.
13. Dragomir S.S., Pečarić J., Persson L.E., Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335–341.
14. Dragomir S.S., Rassias Th.M. (eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, Dordrecht, 2002.
15. Godunova E.K., Levin V.I., Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions, Numerical mathematics and mathematical physics, 138–142, 166, Moskov. Gos. Ped. Inst., Moscow, 1985 (in Russian).
16. Noor M.A., Noor K.I., Awan M.U., Some inequalities for geometrically-arithmetically h-convex functions, Creat. Math. Inform. 23 (2014), no. 1, 91–98.
17. Zhang X.-M., Chu Y.-M., Zhang X.-H., The Hermite–Hadamard type inequality of GAconvex functions and its application, J. Inequal. Appl. 2010, Art. ID 507560, 11 pp.
DragomirS. S. (2018). Inequalities of Hermite–Hadamard type for GA-convex functions. Annales Mathematicae Silesianae, 32, 145-168. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13918
Sever S. Dragomir
sever.dragomir@vu.edu.au
Mathematics, College of Engineering & Science, Victoria University, Australia Australia
Mathematics, College of Engineering & Science, Victoria University, Australia Australia
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