Jensen Inequalities Of Convex Function And Their Applications

Inequalities related to convex function play a very important role in the basic theory and application of mathematics. Jensen inequality of convex function, which has a wide range of application, is one of the important classical contents of convex function. In 1905, J.L.W.V.Jensen first defined convex function with inequality, so Jensen inequality was named. Jensen inequality was generated and developed along with convex function, but then the researches on the basic theory and its application of convex function were all realized through it. The fact that the convexity of convex function and its definition is based on inequality makes convex function become a very important tool to prove inequality, so that a lot of inequalities were established, reformed and extended. And the reform, expansion, extension and application of Jensen inequality of convex function have become a hot research issue for mathematicians.Firstly, on the base of [22, 25, 30], this paper, using the combined method, proposes a Jensen inequality between the weighted average with repetitive samples and the weighted average with non-repetitive samples, and this Jensen inequality is an extension of Jensen inequality in Theorem 2 of [22]. Then two new mappings, whose properties are investigated using the convexity of convex function, are established on this new inequality and some new discrete variables inequalities are obtained.Secondly, on the base of [41], this paper studies the new properties of two mappings in [41] with the integral property and the monotonicity of the derivative on the left and the derivative on the right of convex function. As a result, two new continuous variable inequalities, which are used to prove the two known properties of paper [41], are obtained (this method is much simpler).Finally, three theorems associated with the Chebyshev inequality are given as the preparatory knowledge for the application part of this paper. Then the above discrete variable inequalities and the two properties proved by the continuous variable inequalities are applied to Chebyshev inequality and these three related theorems, so a lot of results are obtained and the inequality theory has been enriched.