Projects

This page contains information on the projects PN-II-RU-PD-388 (Special functions: inequalities and applications) and PN-II-RU-TE-2012-3-0190 (Study of special functions with methods of real and complex analysis), supported by Romanian National Authority for Scientific Research, CNCS-UEFISCDI.

The main goal through of the postdoctoral research project PN-II-RU-PD-388 it was to study in details the Turán type inequalities concerning modified Bessel functions of the first and second kinds, Gauss hypergeometric functions and the generalized Marcum function in order to apply the results in information theory. Principal investigator: Árpád Baricz.
Through the young research team project PN-II-RU-TE-2012-3-0190 our aim it was to study in details the properties of special functions, like Bessel, modified Bessel and hypergeometric functions, by using methods of real and complex analysis. The members of the research group were Szilárd András (researcher) and Árpád Baricz (principal investigator).

Postdoctoral Research Project PN-II-RU-PD-388

Title of project: Special Functions: Inequalities and Applications

Project's brief summary

Since the publication of G. Szegő in 1948 of the famous Turán inequality for Legendre polynomials many researchers have extended this inequality for [orthogonal] polynomials [like Gegenbauer, Hermite, Jacobi, Lommel, Bernstein, Pollaczek, Appell] and special functions [like Bessel, modified Bessel, q-Bessel, Riemann zeta]. The Turán type inequalities now have an extensive literature and some of the results have been applied successfully in problems that arise in economics and information theory. The main goal of this project on the one hand is to study in details the Turán type inequalities concerning Bessel and modified Bessel functions of the first and second kinds, and on the other hand to deduce some tight lower and upper bounds for the generalized Marcum Q-function in order to apply these results in information theory.

Objective 1 (2010): Determination of some tight lower and upper bounds for the generalized Marcum Q-function

Bounds for the generalized Marcum Q-function [with Y. Sun]. Applied Mathematics and Computation 217(5) (2010) 2238-2250.
Corrections to "Unified Laguerre polynomial-series-based distribution of small-scale fading envelopes'' [with Y. Sun, S. Zhou]. IEEE Transactions on Vehicular Technology 60(1) (2011) 347-349.
Talk Turán type inequalities for Bessel functions on 2010.11.18. Seminar of the Society of Mathematicians and Physicists of Rijeka, Rijeka, Croatia.

Objective 2 (2011): Determination of some Turán type inequalities for modified Bessel functions of the first and second kinds

On Turán type inequalities for modified Bessel functions [with S. Ponnusamy]. Proceedings of the American Mathematical Society 141(2) (2013) 523-532.
Functional inequalities for modified Bessel functions [with S. Ponnusamy, M. Vuorinen]. Expositiones Mathematicae 29(4) (2011) 399-414.
Talk Turán type inequalities for some special functions on 2011.05.30. International Conference on Asymptotics and Special Functions, City University of Hong Kong, Hong Kong, China.
Talk Turán type inequalities for Tricomi confluent hypergeometric functions on 2011.10.10. Analysis Seminar at Department of Mathematics, University of Helsinki, Helsinki, Finland.
Talk Turán type inequalities concerning modified Bessel functions on 2011.11.05. Scientific Conference, Transylvanian Museum Society, Sovata [Szováta], Romania.

Objective 3 (2011): The study of the generalized Marcum Q-function

The generalized Marcum Q-function: an orthogonal polynomial approach [with S. András, Y. Sun]. Acta Universitatis Sapientiae Mathematica 3(1) (2011) 60-76.
Functional inequalities for the incomplete gamma function [with H. Alzer]. Journal of Mathematical Analysis and Applications 385(1) (2012) 167-178.
Talk An orthogonal polynomial approach to generalized Marcum Q-function on 2011.10.14. Analysis Seminar at Department of Mathematics, University of Turku, Turku, Finland.

Objective 4 (2012): Application of the results on the generalized Marcum Q-function in information theory

New approximations for DQPSK transmission bit error rate [with S. András, J. Fodor]. Proceedings of IEEE 8th International Symposium on Applied Computational Intelligence and Informatics, May 23-25, Timişoara (2013) 73-77.

Young Research Team Project PN-II-RU-TE-2012-3-0190

supported by Romanian National Authority for Scientific Research, CNCS-UEFISCDI, with contract no. 43/30.04.2013

Title of project: Study of special functions with methods of real and complex analysis

Researcher: Szilárd András. Principal investigator: Árpád Baricz.

Project's brief summary

Because of their remarkable properties, special functions have been used frequently by scientists. For example, a wide range of problems concerning the most important areas of mathematical physics and various engineering problems are linked into application of Bessel and hypergeometric functions. These functions are often used in the solution of problems of hydrodynamics, acoustics, radio physics, atomic and nuclear physics, information theory, wave mechanics and elasticity theory. Special functions play also an important role in geometric function theory. Maybe the most known application is the solution of the famous Bieberbach conjecture by de Branges. The surprising use of generalized hypergeometric functions by de Branges has generated considerable interest, and the geometric properties of these functions have been investigated by many authors. The main goal through this project is to study in details the properties of special functions, such as Bessel and hypergeometric functions, by using methods of real and complex analysis. Topics which we would like to study are the followings: starlikeness, convexity and close-to-convexity of Bessel and hypergeometric functions, close-to-convexity of the derivatives of Bessel and hypergeometric functions, Ulam-Hyers stability of Bessel and hypergeometric functions, properties of series of Bessel functions, properties of modified Bessel functions and of the product of modified Bessel functions of the first and second kind.

Objective 1 (2013): Study of Bessel and modified Bessel functions with applications

Integral representations and summations of modified Struve function [with T.K. Pogány]. Acta Mathematica Hungarica 141(3) (2013) 254-281.

Objective 2 (2013): The radius of starlikeness of normalized Bessel functions of the first kind

The radius of starlikeness of normalized Bessel functions of the first kind [with P.A. Kupán, R. Szász]. Proceedings of the American Mathematical Society 142(6) (2014) 2019-2025.
Talk The radii of starlikeness and convexity of normalized Bessel functions on 2013.06.11. International Conference on Computational Methods and Function Theory, Shantou University, Shantou, China.
Talk The radius of convexity of three kind of normalized Bessel functions of the first kind on 2013.10.09. Mathematics and Statistics Seminar of Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus.

Objective 3 (2013): Discrete Chebyshev inequalities for series of product of modified Bessel functions of the first and second kind

Properties of the product of modified Bessel functions [with T.K. Pogány]. Springer Volume: Analytic Number Theory, Approximation Theory and Special Functions (2013) 809-820.

Objective 4 (2014): Geometrical concavity of the product of modified Bessel functions of the first and second kind

Bounds for Turánians of modified Bessel functions. Expositiones Mathematicae 33(2) (2015) 223-251.
Talk Infinitely divisible distributions, Stieltjes transforms and Turán type inequalities on 2014.06.04. Seminar of Department of Mathematics, Kafkas University, Kars, Turkey.

Objective 5 (2014): Geometric properties of Bessel and hypergeometric functions and their derivatives

Close-to-convexity of some special functions and their derivatives [with R. Szász]. Bulletin of the Malaysian Mathematical Sciences Society 39(1) (2016) 427-437.
Talk Close-to-convexity of some special functions and their derivatives on 2014.03.07. International Conference Natura-Econ, Babeş-Bolyai University, Faculty of Economics and Business Administration, Department of Business Administration, Extension Sfântu-Gheorghe, Romania.
Differential subordinations involving generalized Bessel functions [with E. Deniz, M. Çağlar, H. Orhan]. Bulletin of the Malaysian Mathematical Sciences Society 38(3) (2015) 1255-1280.

Objective 6 (2015): Study of Bessel and modified Bessel functions with applications

Starlikeness of Bessel functions and their derivatives [with M. Çağlar, E. Deniz]. Mathematical Inequalities and Applications 19(2) (2016) 439-449.
Talk Starlikeness of normalized Bessel functions and their derivatives on 2015.03.05. Seminar of the Indian Statistical Institute, Chennai, India.
The radius of alpha convexity of normalized Bessel functions of the first kind [with H. Orhan, R. Szász]. Computational Methods and Function Theory 16(1) (2016) 93-103.
Workshop on special functions at Faculty of Mathematics and Informatics, Babeş-Bolyai University on 2015.09.08.
The topics include classical special functions going back to Bessel, inequalities, asymptotic analysis, the role of special functions in complex analysis and sampling theory. Speakers: Árpád Baricz, Dragana Jankov Masirevic, Tibor Pogány, Saminathan Ponnusamy and Sanjeev Singh.

Objective 7 (2015): The radius of convexity of normalized Bessel functions of the first kind

The radius of convexity of normalized Bessel functions of the first kind [with R. Szász]. Analysis and Applications 12(5) (2014) 485-509.
Talk Radii of starlikeness and convexity of three kind of normalized Bessel functions of the first kind on 2014.06.05. Seminar of Department of Mathematics, Kafkas University, Kars, Turkey.
Geometric properties of some Lommel and Struve functions [with N. Yağmur]. Ramanujan Journal 42(2) (2017) 325-346.
Talk Radii of convexity of some Lommel and Struve functions on 2015.08.26. International Symposium The Real World is Complex, University of Copenhagen, Copenhagen, Denmark.
The radius of convexity of normalized Bessel functions [with R. Szász]. Analysis Mathematica 41(3) (2015) 141-151.
Radii of starlikeness and convexity of some q-Bessel functions [with D.K. Dimitrov, I. Mező]. Journal of Mathematical Analysis and Applications 435(1) (2016) 968-985.
Talk Radii of starlikeness and convexity of some q-Bessel functions on 2015.05.28. Seminar of Department of Mathematics, Erzincan University, Erzincan, Turkey.

Objective 8 (2015): Ulam-Hyers stability of Bessel functions of the first and second kind

Talk Ulam-Hyers stability of integral equations with weak singularities on 2014.06.06 [by Sz. András]. Conference on Ulam's type stability, Rytro, Poland.
Ulam-Hyers stability of singular integral equations, via weakly Picard operators [with Sz. András, T.K. Pogány]. Fixed Point Theory 17(1) (2016) 21-36.

Objective 9 (2016): Study of Bessel and modified Bessel functions with applications

Close-to-convexity of normalized Dini functions [with E. Deniz, N. Yağmur]. Mathematische Nachrichten 14-15(289) (2016) 1721-1726.
Talk Radii of starlikeness and convexity of normalized Lommel and Struve functions of the first kind on 2016.02.25. Seminar of Department of Mathematics, Pukyong National University, Busan, South Korea.
Talk Laguerre-Pólya class of real entire functions, Fourier critical points and geometric properties of some special functions on 2016.02.26. Seminar of Department of Mathematics, Pusan National University, Busan, South Korea.
Bounds for the radii of univalence of some special functions [with I. Aktaş, N. Yağmur]. Mathematical Inequalities and Applications 20(3) (2017) 825-843.
Talk Bounds for the radii of univalence of some special functions on 2016.06.14. International Conference on Constructive Theory of Functions, Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, Sozopol, Bulgaria.

Objective 10 (2016): Ulam-Hyers stability of hypergeometric functions

Talk Ulam-Hyers stability of Bessel and Legendre type integral equations on 2015.06.14 [by Sz. András]. International Conference on Nonlinear Operators, Differential Equations and Applications, Cluj-Napoca, Romania.
Ulam-Hyers stability of singular integral equations, via weakly Picard operators [with Sz. András, T.K. Pogány]. Fixed Point Theory 17(1) (2016) 21-36.

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