Abstract
Jensen’s and its related inequalities have attracted the attention of several mathematicians due to the fact that Jensen’s inequality has numerous applications in almost all disciplines of mathematics and in other fields of science. In this article, we propose new bounds for the difference of two sides of Jensen’s inequality in terms of power means. An example has been presented for the importance and support of the main results. Related results have been given in quantum calculus. As consequences, improvements of quantum integral version of Hermite-Hadamard inequality have been derived. The obtained inequalities have been applied for some well-known inequalities such as Hermite-Hadamrd, Hölder, and power mean inequalities. Finally, some applications are given in information theory. The tools performed for obtaining the main results may be applied to obtain more results for other inequalities.
Highlights
There is no doubt that one of the most important classes of functions is the class of convex functions
The beauty of convex functions is due to its unique graphical representation, geometrical interpretation, and developments in the theory of inequalities
The integral version of the Jensen inequality is presented in the following theorem [3]
Summary
There is no doubt that one of the most important classes of functions is the class of convex functions. In 2018, Pecarić et al [12] focused to find the bounds for the difference of two sides of Jensen’s inequality They considered some Green convex functions and their related identities and derived bounds for the Jensen gap for the class of C2 functions without using convexity condition. In 2021, Khan et al [14] further modified the method given in [3, 13] and derived several results for Jensen and related inequalities In this method, they have used real weights and found the integrals of some functions which pertaining Green functions, in a very simple way with the help of the obtained identity for the Jensen gap. We focused to give applications for Shannon-entropy, Csiszár, and Zipf-Mandelbrot entropy etc
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