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In this paper, we investigate the complete monotonicity of some functions involving gamma function. Using the monotonic properties of these functions, we derived some inequalities involving gamma and beta functions. Such inequalities…

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued…

First, we give the definition for quasi-nearly subharmonic functions, now for general, not necessarily nonnegative functions, unlike previously. We point out that our function class incudes, among others, quasisubharmonic functions, nearly…

Score matching is an estimation procedure that has been developed for statistical models whose probability density function is known up to proportionality but whose normalizing constant is intractable, so that maximum likelihood is…

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic…

Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial…

Weighted mean value identities over balls are considered for harmonic functions and their derivatives. Logarithmic and other weights are involved in these identities for functions. Some applications of weighted identities are presented.…

The mean flux theorems are proved for solutions of the Helmholtz equation and its modified version. Also, their converses are considered along with some other properties which generalise those that guarantee harmonicity.

We recall the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic functions and essentially almost subharmonic…

Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate stable distributions under various conditions. A esult potentially applicable in a…

We provides some useful estimates for solving martingale representation problem under G-expectations. We also study the corresponding conditions for the existence and uniqueness.

This work is about a new class of martingales: the vertical martingales. We construct the vertical martingale for smooth submersions and we develop a stochastic calculus for one. Furthermore, we gives a stochastic characterization for…

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we…

We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of…

We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties…

There has been much recent attention on $h$-functions, so named since they describe the distribution of harmonic measure for a given multiply connected domain with respect to some basepoint. In this paper, we focus on a closely related…

Let $\Delta_k$ be the Dunkl Laplacian on $\mathbb{R}^d$. The main goal of this paper is to characterize $\Delta_k$-harmonic functions by means of a mean value property

In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $\mathbb{R}^m$. Being defined as the…

The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown…