Abstract
In this study, we present some new refinements of the Jensen inequality with the help of majorization results. We use the concept of convexity along with the theory of majorization and obtain refinements of the Jensen inequality. Moreover, as consequences of the refined Jensen inequality, we derive some bounds for power means and quasiarithmetic means. Furthermore, as applications of the refined Jensen inequality, we give some bounds for divergences, Shannon entropy, and various distances associated with probability distributions.
Highlights
Introduction eJensen inequality is one of the most significant and fundamental inequalities in the current literature of mathematical inequalities. e Jensen inequality got more importance due to the fact that many inequalities such as Holder inequality, Minkowski’s inequality, Ky Fan’s inequality, Levinson’s inequality, and Hermite–Hadamard inequality can be obtained from this inequality
We begin this section with the following result, in which we obtain a refinement of the Jensen inequality with the help of eorem 2
Conclusion e Jensen inequality is one of the most powerful and attractive inequalities among the mathematical inequalities. is inequality generalized the definition of convex function and generated many new inequalities
Summary
Introduction eJensen inequality is one of the most significant and fundamental inequalities in the current literature of mathematical inequalities. e Jensen inequality got more importance due to the fact that many inequalities such as Holder inequality, Minkowski’s inequality, Ky Fan’s inequality, Levinson’s inequality, and Hermite–Hadamard inequality can be obtained from this inequality. One of the most attractive features of the Jensen inequality is that it generalized the definition of the convex function. In 2006, Dragomir [10] presented new refinements of the Jensen inequality by using a convex function defined on the linear space. Pecaricand Peric [11] established some improvements of the converse of the Jensen inequality for linear functional, and some applications of the obtained results were presented therein.
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