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Our aim in this paper is to derive several new integral representations of the Fox-Wright functions. In particular, we give new Laplace and Stieltjes transform for this special functions under a special restriction on parameters. From the…

Due to their flexibility, Fox-$H$ functions are widely studied and applied to many research topics, such as astrophysics, mechanical statistic, probability, etc. Well-known special cases of Fox-$H$ functions, such as Mittag-Leffler and…

We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of…

In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient…

The Fox $H$-function is a special function which is defined via the Mellin-Barnes integrals and produces, as particular cases, Wright generalized hypergeometric functions, MacRobert's $E$-functions and Meijer $G$-functions, to name but few.…

We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has different scaling factor. We give necessary and sufficient conditions in terms…

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, 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 give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite . We present some properties and relationships involving logarithmically completely monotonic…

Let $ f:(0,\infty)\rightarrow \Bbb{R} $ be a completely monotonic function. In this paper, we present some properties of this functions and several new classes of completely monotonic functions. We also give some special functions such that…

In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $\Gamma(x)$, the $n$-th harmonic…

The purpose of this paper is to provide a set of sufficient conditions so that the normalized form of the Fox-Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit…

In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\infty)$. As applications of the necessary and…

We state some sufficient or equivalent conditions to GRH of general L-functions in terms of monotonicity of certain weighted summatory functions.

In the article, a notion "logarithmically absolutely monotonic function" is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity…

In this note we prove a condition of monotonicity for the integral functional $ F(g) = \int_a^b h(x)\, d[-g(x)] $ with respect to $g$, a function of bounded variation. This condition is applied to analyze the behavior of a generalized…

In this article, logarithmically complete monotonicity properties of some functions such as $\frac1{[\Gamma(x+1)]^{1/x}}$, $\frac{[{\Gamma(x+\alpha+1)}]^{1/(x+\alpha)}}{[{\Gamma(x+1)}]^{1/x}}$, $\frac{[\Gamma(x+1)]^{1/x}}{(x+1)^\alpha}$ and…

In this paper we find several new properties of a class of Fox's H functions which we call delta neutral. In particular, we find an expansion in the neighborhood of finite nonzero singularity and give new Mellin transform formulas under a…

We point out the connection of the so-called H\^opital-style rules for monotonicity and oscillation to some well-known properties of concave/convex functions. From this standpoint, we are able to generalize the rules under no…

We consider the following question: if a function of the form $\int_0^{\infty}\varphi(t)\, e^{-xt}dt$ is completely monotonic, is it then $\varphi\ge0$? It turns out that the question is related to a moment problem. In the end we apply…