Niaz Tasadduq, Khan Khuram Ali, Pečarić Josip
Department of Mathematics, University of Sargodha, Sargodha, 40100 Pakistan.
RUDN University, Miklukho-Maklaya str. 6, 117198, Moscow, Russia.
J Inequal Appl. 2017;2017(1):254. doi: 10.1186/s13660-017-1521-x. Epub 2017 Oct 10.
In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the 'two-point right focal' problem. We use Fink's identity and a new Abel-Gontscharoff-type Green's function for a 'two-point right focal' to generalize the refinement of Jensen's inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for -convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen's inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed.
在本文中，我们针对“两点右焦点”问题，利用新的格林函数建立了新的阿贝尔 - 贡恰罗夫型恒等式。我们使用芬克恒等式以及“两点右焦点”问题的一个新的阿贝尔 - 贡恰罗夫型格林函数，将（霍尔瓦特和佩查里奇，《数学不等式及其应用》14: 777 - 791，2011）中给出的詹森不等式的精细化从凸函数推广到高阶凸函数。此外，我们利用关于点处 - 凸函数不等式的最新理论，阐述了由这些恒等式得到的线性泛函的单调性。进一步，我们利用切比雪夫泛函不等式给出了与詹森不等式精细化推广相关恒等式的界。构建了一些与格鲁斯和奥斯特罗夫斯基型不等式相关的结果。
