Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.

Bounds for the Jensen Gap in terms of Power Means with Applications

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.

Trumpism as an Alternative Look at the Great America

The article analyzes the phenomenon of “Trumpism,” which brought hopes for its renewal and expectation of changes into traditional American politics. “Trumpism” swung at the power of the Deep State and showed America that it is historically, mentally, culturally, and economically fragmented. The paternalism of American democrats toward a certain part of the country’s population, which Trump positions as “communism,” opposes the true America, which is personified by the states of the “rusty belt” and its working class. “Trumpism” sees the true rebirth of America in a policy of sound national egoism and in America’s focus on itself. For opponents of President Donald Trump, his return to power means another immersion of the ruling elite in the fight against many unknown things, when the political “behind the scenes” has already come out of the shadows and has shown its actual goals.

Refinements of Jensen’s Inequality via Majorization Results with Applications in the Information Theory

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.

One-Kind Hybrid Power Means of the Two-Term Exponential Sums and Quartic Gauss Sums

The main purpose of this article is using the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the hybrid power mean of the two-term exponential sums and quartic Gauss sums and then prove two interesting linear recurrence formulas. As applications, some asymptotic formulas are obtained.

On New Means with Interesting Practical Applications: Generalized Power Means

Means of positive numbers appear in many applications and have been a traditional matter of study. In this work, we focus on defining a new mean of two positive values with some properties which are essential in applications, ranging from subdivision and multiresolution schemes to the numerical solution of conservation laws. In particular, three main properties are crucial—in essence, the ideas of these properties are roughly the following: to stay close to the minimum of the two values when the two arguments are far away from each other, to be quite similar to the arithmetic mean of the two values when they are similar and to satisfy a Lipchitz condition. We present new means with these properties and improve upon the results obtained with other means, in the sense that they give sharper theoretical constants that are closer to the results obtained in practical examples. This has an immediate correspondence in several applications, as can be observed in the section devoted to a particular example.