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In the present paper, we prove the monotonicity property of the ratios of the generalized Volterra function. As consequences, new and interesting monotonicity concerning ratios of the exponential integral function, as well as it yields some new functional inequalities including Turán-type inequalities. Moreover, two-side bounding inequalities are then obtained for the generalized Volterra function. The main mathematical tools are some integral inequalities. As applications, a few of upper and lower bound inequalities for the exponential integral function are derived. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples accompanied by graphical representations to substantiate the accuracy of the obtained results. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section. [ABSTRACT FROM AUTHOR]

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox–Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these representations, we found sufficient conditions on parameters of the generalized Volterra function to prove its complete monotonicity. As applications, a Turán type inequality for generalized Volterra functions is derived, infinite integral of some special functions are expressed in terms of the generalized Volterra functions and closed-form integral representations for a family of convergent Mathieu-type series defined in terms of generalized Volterra functions are established. [ABSTRACT FROM AUTHOR]

In this paper we prove some monotonicity, log-convexity and log-concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Turán type inequalities) as well as we determined sharp upper and lower bounds for the normalized incomplete Volterra functions in terms of weighted power means. [ABSTRACT FROM AUTHOR]

In this article, we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler-Poisson-Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter k, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations. [ABSTRACT FROM AUTHOR]

In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, generalized Taylor formula with Bessel operators, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag-Leffler functions. At the end, some topics are indicated for further study and possible generalizations. Also the aim of the paper is to attract attention and give references to not widely known results on fractional powers of the Bessel differential operator. [ABSTRACT FROM AUTHOR]

This short survey paper contains brief historical information, main known facts and original author's results on the theory of the Buschman- Erdélyi transmutations and some of their applications. The operators of Buschman-Erdélyi type were first studied by E.T. Copson, R.G. Buschman and A. Erdélyi as integral operators. In 1990's the author was first to prove the transmutational nature of these operators and published papers with detailed study of their properties. This class include as special cases such famous objects as the Sonine-Poisson-Delsarte transmutations and the fractional Riemann-Lioville integrals. In this paper, the Buschman-Erdelyi transmutations are logically classified as operators of the first kind with special case of zero order smoothness operators, second kind and third kind with special case of unitary Sonine-Katrakhov and Poisson-Katrakhov transmutations. We study such properties as transmutational conditions, factorizations, norm estimates, connections with classical integral transforms. Applications are considered to singular partial differential equations, embedding theorems with sharp constants in Kipriyanov spaces, Euler-Poisson-Darboux equation including Copson lemma, generalized translations, Dunkl operators, Radon transform, generalized harmonics theory, Hardy operators, V. Katrakhov's results on pseudodifferential operators and boundary-value problems of new kind for equations with solutions of arbitrary growth at the isolated singularity for elliptic partial differential equations. [ABSTRACT FROM AUTHOR]

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. These lead to a more general conjecture on monotonicity of ratios of Kummer hypergeometric functions, which remained open ever since. In this paper we prove some conjectures for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turán-type inequalities. [ABSTRACT FROM AUTHOR]

In this paper we prove monotonicity of some ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions. The results are also closely connected with Turán type inequalities. In order to obtain main results we apply methods developed for the case of classical Kummer and Gauss hypergeometric functions in [S.M. Sitnik, Inequalities for the exponential remainder, preprint, Institute of Automation and Control Process, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok 1993 (in Russian)] and [S.M. Sitnik, Conjectures on Monotonicity of Ratios of Kummer and Gauss Hypergeometric Functions, RGMIA Research Report Collection 17 (2014), Article 107]. [ABSTRACT FROM AUTHOR]

In this article we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singular ones. We consider well-known classes of singular differential equations with Bessel operators, such as classical and generalized Euler-Poisson-Darboux equation and the generalized radiation problem of A. Weinstein. Methods of this paper are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel-type equations, GASPT (Generalized Axially Symmetric Potential Theory) equations of Weinstein, Bessel-type generalized wave equations with variable coefficients, ultra B-hyperbolic equations and others. So with many results and examples the main conclusion of this paper is illustrated: the integral transforms composition method (ITCM) of constructing transmutations is very important and effective tool also for obtaining connection formulas and explicit representations of solutions to a wide class of singular differential equations, including ones with Bessel operators. [ABSTRACT FROM AUTHOR]