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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 we prove some Tur\'an type inequalities for generalized Volterra functions and derive closed-form integral representations for a family of convergent Mathieu-type series defined in terms of generalized Volterra functions., Comment: 10 pages. Submitted to Applied Mathematics and Computation

In the paper 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 A.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., Comment: 23 pages, submitted to Electronic Journal of Differential Equations (EJDE)

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 Tur\'an type inequalities) as well as we determined sharp upper and lower bounds for the normalized incomplete Volterra functions in terms of weighted power means., Comment: Submitted to Integral Transforms and Special Functions. In this version acknowledgement is added and some references are corrected

In this paper, our aim is to show some mean value inequalities for the Fox-Wright functions, such as Tur\'an--type inequalities, Lazarevi\'c and Wilker--type inequalities. As applications we derive some new type inequalities for hypergeometric functions and the four--parametric Mittag--Leffler functions. Furthermore, we prove monotonicity of ratios for sections of series of Fox-Wright functions, the results is also closely connected with Tur\'an--type inequalities. Moreover, some other type inequalities are also presented. At the end of the paper, some problems stated, which may be of interest for further research.

In the paper 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., Comment: Electronic Journal of Differential Equations, 2017

In this paper some Tur\'an type inequalities for classical and generalized Mittag-Leffler functions are considered. The method is based on proving monotonicity for special ratio of sections for series of Mittag-Leffler functions. Some applications are considered to Lazarevi\'c and Wilker-type inequalities for classical and generalized Mittag-Leffler functions. At the end some open problems are formulated., Comment: 15 pages

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 Tur\'an type inequalities. In order to obtain main results we apply methods developed earlier by the authors for the case of classical Kummer and Gauss hypergeometric functions., Comment: 9 pages. arXiv admin note: substantial text overlap with arXiv:1410.6120

In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from 1993. In this paper we prove these conjectures in the more general setting for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turan--type inequalities., Comment: 8 pages, minor correction of formulas and references

This survey paper contains brief historical information, main known facts and original author's results on the theory of transmutations and some applications. Operators of Buschman-Erdelyi type were first studied by E.T.Copson, R.G.Buschman and A.Erdelyi as integral operators. In 1990's the author was first to prove transmutational nature of these operators and published papers with detailed study of their properties. This class include as special cases such famous objects as Sonine-Poisson-Delsarte transmutations and fractional Riemann-Lioville integrals. In this paper Buschman-Erdelyi transmutations are fully 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.V.Katrakhov's results on pseudodifferential operators and problems of new kind for equations with solutions of arbitrary growth at singularity. The Russian original paper version is also included, it is slightly different from english one., Comment: This paper is dedicated to Professor Sergei V. Rogosin on the occasion of his 60th birthday. English version - 31 pages, Russian version - 35 pages

In this survey we consider generalizations for Young and Cauchy--Bunyakowsky inequalities with applications., Comment: 51 pages, english introduction and russian text

This is an extended version of originally published survey in the book: "Advances in Modern Analysis and Mathematical Modeling". Editors: Yu.F.Korobeinik, A.G.Kusraev. Vladikavkaz: Vladikavkaz Scientific Center, of the Russian Academy of Sciences and Republic of North Ossetia--Alania, 2008. P. 226-293. (In Russian). In this survey we consider main transmutation theory topics with many applications, including author's own results. The topics covered are: transmutations for Sturm--Liouville operators, Vekua-Erdelyi-Lowndes transmutations, transmutations for general differential operators with variable coefficients, Sonine and Poisson transmutations, transmutations and fractional integrals, Buschman--Erdelyi transmutations, in the search for Volterra unitary transmutations, transmutations for singular differential operators with variable coefficients, composition method for transmutations, some applications and open problems., Comment: 141 pages, 393 references, english contents and russian text