Probability inequalities for multiplicative sequences of random variables

涉及乘法混沌和的尖锐不等式

Sharp inequalities involving multiplicative chaos sums

本文通过引理和方法获得通用极值不等式，并证明由乘法系统生成的函数系统收敛。

Midpoint and trapezoid type inequalities for multiplicatively convex functions

In this paper, we first prove two new identities for multiplicative differentiable functions. Based on this identity, we establish a midpoint and trapezoid type inequalities for multiplicatively convex functions. Applications to special means are also given.

Sharp multiplicative inequalities with $\mathrm{BMO}$ $\mathrm{II}$

We find the best possible constant $C$ in the inequality $$\|φ\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|φ\|_{L^p}^{\frac{p}{r}}\|φ\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. We employ the Bellman function technique to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. First, we solve the problem on an interval and then transfer our results to the circle and the line. We also …

Some New Inequalities for (h-s)_{1,2}-convex Functions via Further Properties

In this paper, we establish some new inequalities of the Hermite-Hadamard like for class of (h-s)_{1,2}-convex functions which are ordinary, super-multiplicative or similarly ordered and nonnegative.

Lyapunov-Type Inequalities for Third Order Nonlinear Equations

We derive Lyapunov-type inequalities for general third order nonlinear equations involving multiple $ψ$-Laplacian operators of the form \begin{equation*} (ψ_{2}((ψ_{1}(u'))'))' + q(x)f(u) = 0, \end{equation*} where $ψ_{2}$ and $ψ_{1}$ are odd, increasing functions, $ψ_{2}$ is super-multiplicative, $ψ_{1}$ is sub-multiplicative, and $\frac{1}{ψ_{1}}$ is convex, and $f$ is a continuous function which satisfies a sign condition. Our results utilize $q_{+}$ and $q_{-}$, as opposed to $|q|$ which appears in most results in the literature. Additionally, these new inequalities generalize previously o…

Some families of generalized Mathieu-type power series, associated probability distributions and related functional inequalities involving complete monotonicity and log-convexity

By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with some generalizations of the Mathieu series. In particular, the mathematical expectation, variance and the characteristic functions, related to the probability density functions of the considered probability distributions are derived. As a consequence, some interesting inequalities involving complete monotonicity and log-convexity are derived.