Some New Refined Hardy Type Inequalities with Breaking Points p = 2 or p = 3
For usual Hardy type inequalities the natural “breaking point” (the parameter value where the inequality reverses) is p = 1. Recently, J. Oguntuase and L.-E. Persson proved a refined Hardy type inequality with breaking point at p = 2. In this paper we show that this refinement is not unique and can be replaced by another refined Hardy type inequality with breaking point at p = 2. Moreover, a new refined Hardy type inequality with breaking point at p = 3 is obtained. One key idea is to prove some new Jensen type inequalities related to convex or superquadratic funcions, which are also of independent interest.
KeywordsInequalities refined Hardy type inequalities convex functions superquadratic functions Jensen type inequalities.
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- 1.S. Abramovich, G. Jameson and G. Sinnamon, Refining Jensen’s inequality, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (95), (2004), 3–14.Google Scholar
- 3.G.H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math. 54 (1925), 150–156.Google Scholar
- 4.G.H. Hardy, Notes on some points in the integral calculus, LXIV. Further inequalities between integrals, Messenger of Math. 57 (1928), 12–16.Google Scholar
- 5.C.O. Imoru, On some integral inequalities related to Hardy’s, Canad. Math. Bull 20(3) (1977), 307 312.Google Scholar
- 6.V. Kokitashivili, A Meshki and L.-E. Persson,Weighted norm inequalities for integral transform with product kernels, Mathematics Research Development Series, Nova Science Publishers Inc., New York, 2010.Google Scholar
- 10.J.A. Oguntuase and L.-E. Persson, Refinement of Hardy’s inequality for “all” p. Banach and function spaces II, Yokohama Publ., Yokohama, 2008, 129–144.Google Scholar
- 11.J.A. Oguntuase and L.-E. Persson, Hardy type inequalities via convexity – the journey so far, Aust. J. Math. Anal. Appl. 7 (2010), no 2, Art. 18, 19pp.Google Scholar
- 12.L.-E. Persson and N. Samko, What should have happened if Hardy had discovered this?, J. Inequal. Appl., 2012:29.Google Scholar