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The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as…
In this paper, we present the definitions and some properties of the general fractional integrals (GFIs) and general fractional derivatives (GFDs) of a function f(x) with respect to another function g(x). Examples of special cases of…
We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…
In the paper, the authors show that the weighted geometric mean and the logarithmic mean are Bernstein functions and establish integral representations of these means by Cauchy's integral theorem in the theory of complex functions.
The Mittag-Leffler function plays an important role in Geometric Function Theory, particularly in the study of analytic and meromorphic function classes. Among its various generalizations, the Barnes-Mittag-Leffler function stands out due…
In the present paper, we first prove the logarithmic convexity of the elementary function $\frac{b^x-a^x}x$, where $x\ne0$ and $b>a>0$. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values,…
In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact…
We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm…
This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
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…
We present inequalities related to generalized matrix function for positive semidefinite block matrices. We introduce partial generalized matrix functions corresponding to partial traces, and then provide a unified extension of the recent…
The two-parametric Mittag-Leffler function (MLF), $E_{\alpha,\beta}$, is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical…
In its additive version, Bohr-Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution $f(x)$ to the equation $\Delta f(x)=\ln x$ on the open half-line $(0,\infty)$ is the log-gamma function…
Objectives: This paper develops two algorithms to achieve federated generalized linear mixed effect models (GLMM), and compares the developed model's outcomes with each other, as well as that from the standard R package (`lme4'). Methods:…
We introduce the beta generalized exponential distribution that includes the beta exponential and generalized exponential distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. We derive the…
We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.
Generalization of the integral representation of the gamma function has been obtained, which shows that the Hankel contour assumes rotation in the complex plane. The range of admissible values for the contour rotation angle is set. Using…
In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and…
In this paper, we introduce new classes of functions that extend the known classes of functions of complex variable, such as entire functions, meromorphic functions, rational functions and polynomial functions and take values in the set of…