Grigor Barsegian

    Some (more than 20) new principles concerning the mentioned concepts were established in my works. All these results are novel, i.e. have no predecessors. The results have led to some new trends (for which I got an "International Marie Curie Award" in 2012).

    I think similar information deserves to be reflected somehow in CV. This is why I present below two complements which describe shortly the principles and trends and give corresponding references.

 

 

    Here we deal with different basic concepts in mathematics such as arbitrary enough smooth real functions of one and two variables, plane curves and surfaces in R³, arbitrary meromorphic (particularly analytic) functions in a given domain arise very rare. Results of similar generality were established mostly in 19th century and quite few in 20th century).

    In this section we present shortly several novel results (among them five trends) related to the mentioned above basic concepts in mathematics which were obtained since 1970s. Two of the mentioned trends (which are quite recent) relate respectively to real and complex differential equations; three of them (which are comparatively old) relate to general theory of meromorphic functions, particularly to classical theories of Nevanlinna (value distribution) and Ahlfors (covering surfaces). It is interesting that the mentioned results transfer ideas of Nevanlinna-Ahlfors theories into real analysis and geometry.

    All these results and trends are novel, i.e. have no predecessors. 

    We list here some of them concerning respectively to the real analysis, geometry and complex analysis.

 

    A result (principle of zeros) giving bounds for the number of zeros of arbitrary smooth real functions of one variable. (This is the only result of similar nature). It permits to initiate the trend in real differential equations.

    A result (principle of angles) concerning arbitrary smooth plane curves. It connects rotations around a given center and tangential rotations of the curves. (This is the only result of similar nature and the second, after Crofton formula, result related to the curves of similar generality). The principle permits to initiate one of the mentioned trends in complex analysis, leads to a principle (of logarithmic derivatives) in complex analysis and also have applications on real differential equations.

    More than 15 new results on arbitrary meromorphic (analytic) functions in arbitrary domains. (Phenomena considered in most of these results are novel: not considered in Cauchy or Ahlfors theories).

 

These new results permit to initiate studies of two trends in complex analysis and the trend in complex differential equations. 

 

    Below we give a more complete list of results of general nature and add also corresponding references.

    Among them we mention:

    1. two generalities related to arbitrary smooth real functions of one variable, see [BarsBook2002], chapter.4, [Bars2009], item 2;

    2. a generality related to arbitrary smooth real functions of two variables, see [Bars2010b];

    3. a generality related to arbitrary smooth curve in the plane - principle of angles, see [Bars2009] (plenary lecture at the 6th ISAAC Congress;

    4. a generality related to arbitrary smooth surface in R³, see [Bars2017];

    5. more than 15 generalities related to arbitrary analytic or meromorphic functions in a given domain and even more generalities related to functions in the complex plane, see [Bars1978],  [Bars1981], [Bars1981a], [Bars1983], [Bars1986], [Bars1999], [Bars2010], [Bars2013, [Bars2017], [Bars2018], especially see my book [BarsBook2002] and survey [Bars2019]. Among them principle of logarithmic derivatives and principle of simple a-points, see the first section in [Bars2019].

    We can add here also some results related to arbitrary complex polynomials, see, for instant, [Bars2018b]. 

 

 

    The above mentioned generalities arose in my attempts to create some new trends in real and complex analysis, however they concerns also other areas such as, for instant, differential geometry, as well as real and complex differential equations.

    In this subsections we present shortly five trends and describe shortly their novelty and applicability. The old three trends (3, 4 and 5) relate to analytic meromorphic functions and published during 1978-2007. Two trends (1 and 2) established quite recently. They relate respectively to real and complex differential equations. 

 

    Trend 1: relates to studies of oscillations, i.e. zeros, of solutions of systems of real differential equations.

    Novelty and applicability. The zeros of similar solutions weren't considered before. This is an essential gap in the field since the zeros admit a lot of interpretations in different applied fields.

 

We start studies of zeros of similar solutions. 

 

    Trend 2: relates to studies of analytic or meromorphic solutions in a given domain of complex differential equations.

    Novelty and applicability. Complex differential equations admitting solutions in the complex plane were widely studied. The main attention was paid to the zeros of similar solutions, more widely to value distribution of the solutions. As to the zeros of solutions in a given domain, they weren't considered before. This is an essential gap in the field since many applied scientists are, obviously, interested namely in the solution in a given domain.

 

We start studies of similar solutions in a given domain.

For several well known equations we consider as classical problems (for instance zeros, more generally a-points of solutions), as well as some new problems. 

 

    The key results of this trend are in process of publication; issued only one paper [BarsYuan]. 

    The next three trends relate to some new developments closely connected with the classical Nevanlinna value distribution theory [Nev] and Ahlfors theory of covering surfaces [Ahlfors]. The trends discover several new phenomena related to meromorphic functions. It is interesting that they give also wide generalizations of the first and the second fundamental theorems in the mentioned theories, see [Bars1985].

 

    Trend 3: relates to Gamma-lines of meromorphic functions w(z) particularly level sets of functions Rew(z).

    Solutions of equations u(x,y)=0 (more generally u(x,y)=const) arise very often in pure and applied mathematics. Remember that the level sets u-A admit a lot of interpretations (streaming line, potential line, isobar, isoterm) in different applied fields of engineering, physics, environmental problems etc. Notice that in "non degenerating cases" these solutions are some curves. In the case u(x,y) is a polynomial in two variables P(x,y) the solutions P(x,y)=const were studied very largely in the frame of Hilbert's problem 16, part 1; namely problem was to give bounds for the number of connected components of the solutions  As to other geometric aspects of the solutions P(x,y)=const they were almost non touched. Also, till end of 1970s, we have no any study related to level sets of any large class of real functions u(x,y), even harmonic functions or more generally functions Rew or |w|. Definitely this was a big gap since solutions of Rew=A∈(-∞,∞), or of |w|=R∈(0,∞), have important interpretations in different applied sciences.

 

Thus studies of these (very important in applications) solutions Rew=A may affect largely on the complex analysis

(where just a-points of functions w(z), i.e. solutions of w(z)=a, were studied before).

 

    We consider now much larger concept. Let w(z) be a meromorphic function in a domain D and Γ be a curve in w-plane; for simplicity we assume that Γ is an analytic curves. Denoting by w⁻¹ the inverse functions we consider the set w⁻¹(Γ) lying in D (that is preimages of Γ under mapping by w). We call this set w⁻¹(Γ) as Gamma-lines of the curve Γ of the function w. Now we see that the solutions of Rew=A and of |w|=R are particular cases of Gamma-lines which we obtain by taking respectively Γ=γ(A):={w:Imw=A} and Γ=Γ(R):={w:|w|=R}. It is easy to see that in this case the γ(A)-lines are curves for any A and Γ(R)-lines are curves for any R≠0.

    The concept of Gamma-lines (which is the set w⁻¹(Γ)) is quite similar to the concept of set a-points (which is the set w⁻¹(a), for a given point a∈C). This similarity raises a question: can theory of Gamma-lines developed in analogy with Nevanlinna-Ahlfors theories describing a-points?

    At the end of 1970s we initiated studies in this directions and obtained an analog of the second fundamental theorems in Nevanlinna-Ahlfors theories. The topic was started in [Bars1978]; the results, later on, were enlarged in [Bars1981], [Bars1992] and summarized in the book [BarsBook2002]. 

    Notice that the Gamma-lines are the sets of curves. Respectively the key aspects in their study should be their length or cardinality (like in the mentioned above Hilbert problem).

    In this trend we present various estimates for the length L(D,Γ) of Γ-lines of w lying in D and give some integrated formulas for the length. This permits to get bounds for cardinalities of Γ-lines for w in D. This means particularly that we consider the Hilbert problem for harmonic functions (instead of polynomials in Hilbert's case).

    At present this topic counted usually as theory of Gamma-lines, see, for instants, «MathSciNet» , review MR2002433 (2005f:30001) related to my book [BarsBook2002] published in 2002. 

 

    Trend 4: relates to studies of the geometry of a-points of meromorphic functions w(z).

    Fundamental results of the classical Nevanlinna-Ahlfors theories deal with the numbers of a-points of meromorphic functions w(z).

 

A new "proximity phenomenon" for meromorphic functions 

(as in the complex plane as well as in a given domain)

deals with the geometric locations of the a-points

and strengthen the mentioned fundamental results of Nevanlinna-Ahlfors theories related to the number of a-points.

 

    To describe the "proximity phenomenon' or "proximity property" of a-points of meromorphic functions w(z) in the complex plane we consider two generic values a and b and for the set of a-points and the set of b-points define a distance between these sets and give upper bounds for this distance. Thus we estimate the closeness between these two sets and come to the proximity (or closeness) phenomenon, which, speaking qualitatively, says that for the generic a and b the sets of a-points and b-points should be close to each other to some extend. Different versions of this property were established in 1978-85, see [Bars1978], [Bars1981], [Bars1983], [Bars1985].

    A similar phenomenon can be traced also in the case of an arbitrary function in a given domain, [Bars1997]

    It is interesting that for the basic functions the proximity phenomenon implies also the main conclusions of preceding theories related to the numbers of a-points; see [Bars1983], [Bars1985].

    This topic also counted as a theory in some reviews: see «Zentralblatt» , review 060230037 and «MathSciNet» , review MR1680525, (2000d:30041). 

 

    Trend 5: Universal version of value distribution for meromorphic functions in a given domain.

    The Nevanlinna-Ahlfors theories reveal value distribution type phenomena of meromorphic functions w which describe interrelations between the numbers n(r,a_{ν}) of a-points for different values a₁,...,a_{q}, q≥2. Typical examples are the second fundamental theorems in these theories. The theories work well for the functions w in the complex plane while for the functions in a given disk they work only when w has "sufficiently large" growth.

    Meantime, there are numerous, well known classes of functions defined in the unit disk (H^{p}, Dirichlet, Blaschke product etc.) for which the value distribution was not studied at all.

    Can there be a general value distribution type result which is valid for an arbitrary meromorphic function in an arbitrary domain D? Since the main result in Nevanlinna-Ahlfors theories are the second fundamental theorems in both these theories the most ambitious task is as follows: can there be some analogs of these classical results are valid also for function in an arbitrary domain D? In other words the question is is there a generality for any analytic and meromorphic function analogous to these fundamental theorems?

    It is easy to show that the mentioned classical theories do not contain enough characteristics to describe the numbers of zeros (or a-points) in general case.

 

We introduce corresponding new characteristic 

and obtain a universal version of value distribution 

which is valid for any meromorphic function in a given domain.

 

    This leads to brand new type phenomena for functions in a given domain and in addition, for functions in the complex plane we get conclusions quite similar to those established earlier in Nevanlinna-Ahlfors theories.

    The results were published in [Bars2010]; see also [Bars2019], Section 5.

    

 

    1. Barsegian G.A. (quest editor), Value Distribution Theory and Geometric complex functions theory, In the journal "Izvestia Acad. Nauk Armenii", vol. 3, 1997; Translation in Journal of Contemporary Math. Analysis, Allerton Press.

    2. Barsegian G.A. (quest editor), Value Distribution Theory and Geometric complex functions theory, In the journal "Izvestia Acad. Nauk Armenii", vol. 4, 1997; Translation in Journal of Contemporary Math. Analysis, Allerton Press.

    3. Barsegian G.A., Gamma-lines: on the geometry of real and complex functions, Taylor and Francis, London, New York, 2002.

    4. Barsegian, I. Laine and C.C. Yang (editors), Value distribution and related topics, Kluwer, series "Advances in complex analysis and related topics", 2004.

    5. G. Barsegian, H. Begehr (editors), Topics in Analysis and its Applications, (the NATO Advanced Research Workshop, September 2002, Yerevan), Kluwer, series NATO Science Publications, 2004.

    6. Barsegian G., Begehr H., Nersessyan A. and Ghazaryan H. (editors), Complex Analysis, Differential Equations and Related Topics, (ISAAC Conference, September 2002, Yerevan), Nat. Acad. Sci. Armenia, 2004.

    

 

    1. Barsegian G., Deficient values and the structure of covering surfaces, Izvestia Acad. Nauk Armenii, v.12, 1977, p.46-53.

    2. Barsegian G., Distribution of sums of a-points of meromorphic functions, Dokl. Acad. Nauk SSSR, v. 234, n. 4, 1977, p. 761-763.

    3. Barsegian G., A Geometric approach to the problem of ramifications of Riemann surfaces, Dokl. Acad. Nauk SSSR, v.237, n. 4, 1977, p.

    4. Barsegian G., To the distribution of distortions under mapping by meromorphic functions, Dokl. Acad. Nauk SSSR, v.237, n. 5, 1977, p. 1009-1011. (in Russian, translated in Soviet Math. Dokl.)

    5. Barsegian G., A Geometric approach to the problem of ramifications of Riemann surfaces, Dokl. Acad. Nauk Armenii, v.65, n. 4, 1977, p.216-220.

    6. Barsegian G., On distribution of zeros of imaginary parts of meromorphic functions, Math. Zametki, v.24, n. 2, 1978, p.183-194.

    7. Barsegian G. New results in the theory of meromorphic functions, Dokl. Acad. Nauk SSSR, v.238, n.4, 1978, p.777-780.

    8. Barsegian G. On geometric structure of image of disks under mappings by meromorphic functions, Math. Sbornic, v. 106(148), n. 1, 1978, p.35-43.

    9. Barsegian G. On deficient values and growth of meromorphic functions with finite lower orders, Dokl. Acad. Nauk Armenii, v.67, 1978, p.291-294.

    10. Barsegian G., Distribution of sums of a-points and Riemann surfaces of the class F-g, Math. Zametki, v.25, n. 1, 1979, p.51-59.

    11. Barsegian G., On the geometry of meromorphic functions, Math. Sbornic, v.114(156), n.2, 1981, p.179-226. (in Russian); Trans. in "Math. USSR Sbornic.

    12. Barsegian G., Exceptional values associated with logarithmic derivatives of meromorphic functions, Izvestia Acad. Nauk Armenii, v. 16, n. 5, 1981, p.408-423.

    13. Barsegian G., Proximity property of a-points of meromorphic functions, Math. Sbornic, Vol. 120(162), n. 1, 1983, p.42-63, (in Russian); Transl. In "Math. USSR Sbornik ".

    14. Barsegian G., On a necessary condition of existence of solutions of the general interpolation problem of R. Nevanlinna, Izvestia Acad. Nauk Armenii, v. 18, n.2, 1983, p.442-447.

    15. Barsegian G. United approach to the study of deficient values, Borel's rays and cercle de remplicages, Dokl. Acad. Nauk SSSR, v.271, n. 1, 1983, p.11-14.

    16. Barsegian G., On mutual arrangement of asymptotic paths and a-points of meromorphic functions, Izvestia Acad. Nauk Armenii, v. 18, n. 2, 1983, p.124-133.

    17. Barsegian G., Proximity property of meromorphic functions and the structure of the univalent domains on Riemann surfaces, Izvestia Acad. Nauk Armenii, Matematika, v. 20, n. 5, (1985), 375--400; v. 20, n. 6, (1985), 407--425, (in Russian); Transl. in "Journal of Contemporary Math. Analysis ", Armenian Acad. of Sci., Allerton Press.

    18. Barsegian G., On a method for estimations of orders of meromorphic solutions of algebraic differential equations based on proximity property of a-points, Dokl. Acad. Nauk Armenii, v. 82, n. 5, 1986, p.495-598.

    19. Barsegian G., Estimates of derivative of meromorphic functions oh sets of a-points, Journal of London Math. Soc., (2), 34, 1986, p.534-540.

    20. Barsegian G., On locations of a-points of meromorphic functions, Review Romaine de mathematiques Pures at Appliquess, v.32, n. 10, p. 865-873.

    21. Barsegian G., Applications of L.Ahlfors theory of covering surfaces to the study of algebraic differential equations of the second orders, Differentsialnye Uravnenija, v.23, n. 2, 1987, p. 341-344.

    22. Barsegian G. and Masrtirosian K., On the values accepted by analytical functions on the set of a-points of the meromorphic function in given domain, Izvestia Acad. Nauk Armenii, v.24, n.3, 1989, p.248-258.

    23. Barsegian G. and Sukiasian G., Proximity property of a-points for meromorphic functions with simple zeros and poles on rays, ARMNIINTI, n.16, 1989, p.1-46.

    24. Barsegian G. and Sukiasian G., Proximity property of a-points for meromorphic functions with close zeros and poles, Izvestia Acad. Nauk Armenii, v. 25, n.1, 1990, p.21-33.

    25. Barsegian G. and Sukiasian G., Distributions of values and proximity of a-points for quotients of Blaschke products with nearly zeros. Studio Sci. Math. Hungarica, v. 25, 1990, p. 419-428.

    26. Barsegian G. and Petrosian V., Cartan's identity for some magnitudes associated with logatithmic derivatives of meromorphic function, Izvestia Acad. Nauk Armenii, v 25, n.6, 1990.

    27. Barsegian G., Property of comparability of derivatives, and property of comparability of distances between a-points of meromorphic functions, Izvestia Acad. Nauk Armenii, v. 26, n.1, 1991, p. 52-70.

    28. Barsegian G. and Sukiasian G., Meromorphic functions with nearly zeros and poles lying on rays, Izvestia Acad. Nauk Armenii, v. 26, n.2, 1991, p.

    29. Barsegian G. and Belyi V., On polynomial approximations of function on Riemann surface which are inverse to the given meromorphic functions, Izvestia Acad. Nauk Armenii, v. 26, n.5, 1991, p.

    30. Barsegian G., Two new principles concerning presenting derivatives and distances between the a-points of meromorphic functions, Review Romaine de mathematiques Pures at Appliquess, v.34, n. 7-8, 1991, p. 420-424.

    31. Barsegian G., Tangent variation principle in complex analysis, Izvestia Acad. Nauk Armenii, v.27, n.3, 1992, p.37-60. (in Russian); Transl. In "Journal of contemporary math. Analysis" (Armenian Acad. of Sci.), Allerton Press.

    32. Barsegian G. and Avakian V. and Karachanian M., Space generalized quasiconformal mappings, ARMNIINTI, n.5, 1993, p. 1-35.

    33. Barsegian G.A., Principle of partitioning of meromorphic functions, Math. Montisnigri, v. 5, 1995, p. 18-26.

    34. Barsegian G., Some remarks on links between mutual arrangements of a-points of meromorphic functions and ramifications, Math. Montisnigri, v. 4, 1996, p.1-7.

    35. Barsegian G. and Petrosian V., Exeptional values defined by means of logarithmic derivatives of higher orders , v.31, 1996,

    36. Barsegian G., Principle of closeness of sufficiently large sets of a-points of meromorphic functions, Hokkaido Math. Journal, v.26, n.2, 1997, p. 451-456.

    37. Barsegian G.A., Some applications of principle of partitioning of meromorphic functions, Part 1. Comparability property. Izvestia Acad. Nauk Armenii, v. 31, n. 3-4, 1997, p. 5-50.

    38. Barsegian G.A. and Sikiasian G.A., Some applications of principle of particioning of meromorphic functions, Part 2. Accumulations of a-points and Littlewood property. Izvestia Acad. Nauk Armenii, v. 32, n. 3-4, 1997, p. 5-10.

    39. Barsegian G.A. (quest editor), Value Distribution Theory and Geometric complex functions theory, In the journal "Izvestia Acad. Nauk Armenii", vol. 3, 1997; Translation in Journal of Contemporary Math. Analysis, Allerton Press.

    40. Barsegian G.A. (quest editor), Value Distribution Theory and Geometric complex functions theory, In the journal "Izvestia Acad. Nauk Armenii", vol. 4, 1997; Translation in Journal of Contemporary Math. Analysis, Allerton Press.

    41. Barsegian G., On a method of the study of algebraic differential equations of higher orders, Bulletin of Honk Kong Math. Soc., vol. 2, n. 1, (1998), p. 159-164.

    42. Barsegian G., Estimates of higher derivatives of meromorphic functions on sets of its a-points, Bulletin of Honk Kong Math. Soc., vol. 2, n. 2, (1999), p. 341-346

    43. Barsegian G., Additions to Ahlfors theory of covering surfaces, Review Romaine de mathematiques Pures at Appliquess, v. 44, n.2, 1999. p. 503-519.

    44. Barsegian G.A. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications. Part 1. On the magnitudes of functions on the sets of a-points of derivatives. Complex Variables, v. 41, 2000, p. 293-313.

    45. Barsegian G.A., Laine I. and Yang C.C., Stability phenomenon and problems for complex differential equations with relations to sharing values, Mathematichni Studii, v. 13, 2000, p. 224-228.

    46. Barsegian G.A. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications. Part 2. On the derivatives of meromorphic and inverse functions, Complex Variables, v. 44, 2001, p.13-27.

    47. G. A. Barsegian, I. Laine and C. C. Yang, Locations of values of meromorphic solutions of algebraic differential equations, Complex Variables, v. 46, 2001, p. 323 - 336.

    48. Barsegian G.A. and Walker P., On the mutual arrangement of a-points of a meromorphic functions and its derivatives, 2000,Southeast Asian Bulletin of Mathematics, v. 26, 2002, p.1--6.

    49. Barsegian G., Laine I. and Yang C. C., On a method of estimating derivatives in complex differential equations, Journal Math. Soc. Japan, v. 54, n. 4, 2002, p. 923-935.

    50. Barsegian G., Begehr H. and Laine I., Stability phenomenon for generalizations of algebraic differential equations, Zeitshirft fur Analysis und ihre Anwendungen, 21, n. 2, 2002, pp. 495 -503.

    51. Barsegian G., Gamma-lines: on the geometry of real and complex functions, Taylor and Francis, London, New York, 2002.

    52. Barsegian G. and Yang C. C., A new property of meromorphic functions and its applications. Analysis and applications---ISAAC 2001 (Berlin), 109--120, Int. Soc. Anal. Appl. Comput., 10, Kluwer Acad. Publ., Dordrecht, 2003.

    53. Barsegian, I. Laine and C.C. Yang (editors), Value distribution and related topics, Kluwer, series "Advances in complex analysis and its Applications", 2004.

    54. G. Barsegian, H. Begehr (editors), Topics in Analysis and its Applications, (the NATO Advanced Research Workshop, September 2002, Yerevan), Kluwer, series NATO Science Publications, 2004.

    55. Barsegian G., Begehr H., Nersessyan A. and Ghazaryan H. (editors), Complex Analysis, Differential Equations and Related Topics, (ISAAC Conference, September 2002, Yerevan), Nat. Acad. Sci. Armenia, 2004.

    56. Barsegian G. and Yang C.C., On some new concept of exceptional values of meromorphic functions and its applications. In book: Value distribution and related topics, editors G. Barsegian, I. Laine and C.C. Yang; Kluwer, Series: Advances in Complex Analysis and its Applications, 2004.

    57. Barsegian G. and Sukiasyan G., Methods for study level sets of enough smooth functions, Analysis and applications. In book "Topics in Analysis and Applications": the NATO Advanced Research Workshop, August 2002, Yerevan", editors G. Barsegian, H. Begehr, Kluwer, Series: NATO Science Publications, 2004.

    58. Barsegian G., A new program of investigations in Analysis: Gamma-lines approaches, pp. 1-73; in book: Value distribution and related topics, editors G. Barsegian, I. Laine and C.C. Yang; Kluwer, Series: Advances in Complex Analysis and its Applications, 2004.

    59. Barsegian G., Sarkisian A., and Yang C.C., On quasimeromorphic solutions of complex differential equations. In book: Value distribution and related topics, editors G. Barsegian, I. Laine and C.C. Yang, Kluwer, Series: Advances in Complex Analysis and its Applications, 2004.

    60. Barsegian G. and Begehr H., Lines of catastrophe: problems, examples of solutions, connections with Nevanlinna theory and Gamma-lines, Third ISAAC Congress, Freie Universitat Berlin, August 20-25, 2001, Plenary and selected lectures, World Scientific, 2003.

    61. Barseghyan K. and Barsegian G., A method for study oscillations of ordinary differential equations with an application to population dynamics, In book "Topics in Analysis and Applications": the NATO Advanced Research Workshop, August 2002, Yerevan", editors G. Barsegian, H. Begehr, Kluwer, Series: NATO Science Publications, 2004.

    62. Barsegian G., Gamma-lines of polynomials and a problem by Erdosh-Herzog-Piranian, In book "Topics in Analysis and Applications": the NATO Advanced Research Workshop, August 2002, Yerevan", editors G. Barsegian, H. Begehr, Kluwer, Series: NATO Science Publications, 2004.

    63. Barseghyan K. G. and Barsegian G., On a Nevanlinna type result for solutions of some nonlinear first order differential equations, in book: Barsegian G., Begehr H., Ghazaryan H. and Nersessyan A. (editors), Complex Analysis, Differential Equations and Related Topics, ISAAC Conference, September 17 -21, 2002, Yerevan, Armenia, Nat. Acad. Sci. Armenia, 2004.

    64. Barsegian G. and Lê D. T., On a topological property of multi-valued solutions of some general classes of complex differential equations, Complex Variables Theory Appl., v. 50, n. 5, 2005, p. 307-318.

    65. Barseghyan K. and Barsegian G., Oscillations of Solutions of ODE and Nevanlinna Theory, Global Journal of Mathematics & Mathematical Science (GJMMS), 2005.

    66. Barseghyan K. and Barsegian G., On a Nevanlinna type result for solutions of nonautonomous equations y′=F₁(x,y), x′=F₂(x,y), Complex Variables Theory Appl., v. 50, n. 7-11, 2005, p. 469-477.

    67. Barsegian G., An analog of Nevanlinna fundamental theorem for real functions. Complex Variables Theory Appl., v. 50, n. 7-11, 2005, p. 535-541.

    68. Barsegian G., Some new inequalities in geometry and analysis, "Izvestia Acad. Nauk Armenii", v. 42, n. 2, 2007, (in Russian); Translation in Journal of Contemporary Math. Analysis, Allerton Press.

    69. Barsegian G., Fernandez A. and and Lê D. T., On closeness of a- and b-points of arbitrary polynomials, J. Appl. Funct. Anal. 2 (2007), no. 1, 13-20.

    70. Barsegian G., Laine I. and Lê D. T., On topological behavior of solutions of some algebraic differential equations, Complex Variables and Elliptic Equations, v. 53 (2008), p. 411 - 421.

    71. Barsegian G., Turbulence of real functions, Georgian Mathematical Journal, 15, n.2, 2008, p. 225--240.

    72. Barsegian G, Some interrelated results in different branches of geometry and analysis, p.3-33, in book "Further progress in analysis. Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, 13 -- 18 August 2007", editors Begehr H., Celebi O. and Gilbert R., World Scientific, 2009.

    73. Barsegian G. and Lê D. T., On the length of level sets of real functions, Analysis, 30, 2010, p. 135-146.

    74. Barsegian G., A purely geometric inequality for analytic functions admitting no zeros, Complex Variables and Elliptic Equations, v. 55, n. 1, 2010, p. 249 - 251.

    75. Barsegian G., Universal value distribution for arbitrary meromorphic functions in a given domain, p. 123-128, in book "Progress in Analysis and its Applications, Proceedings of the 7th International ISAAC Congress, Imperial College London, UK, 13-18 July 2009", editors Ruzhansky M.and Wirth J.,World Scientific, 2010.

    76. Barsegian G. and Sargsyan A., A method of counting the zeros of the Riccati equation and its application to biological and economical prognoses, p. 129-135, in book "Progress in Analysis and its Applications, Proceedings of the 7th International ISAAC Congress, Imperial College London, UK, 13-18 July 2009", editors Ruzhansky M.and Wirth J.,World Scientific, 2010.

    77. Barsegian G., Miscellanea in integral geometry, algebraic geometry and real analysis, Reports of the National Academy of Sciences of Armenia, v. 103, n. 3, 2010, p. 210-219.

    78. Alonso A., Barsegian G. and Fernandez A., Gamma-lines approach in the study of global behavior of solutions of complex differential equations, Complex Variables and Elliptic Equations, v 56, Issue 1-4, 2011, p. 119-135.

    79. Barsegian G., On a principle in the theory of complex polynomials implying Gauss-Lucas Theorem, p. 14-20, in "Recent Developments in Generalized Analytic Functions and Their Applications", edited by G.Giorgadze, Tbilisi State University, 2011.

    80. Barsegian G., On the proximity property of meromorphic functions initiating some novel studies in the complex analysisp, p. 20-25, in "Recent Developments in Generalized Analytic Functions and Their Applications", edited by G.Giorgadze, Tbilisi State University, 2011.

    81. Barsegian G., Revisiting a general property of meromorphic functions, Contemporary Mathematics (USA), v. 591, 2013, p. 45-49.

    82. Barsegian G., Idea of complex conjugacy and counterparts of some complex analytic principles in differential geometry, Reports and Studies in Forestry and Natural Sciences (Finland), n. 14, p. 27-44.

    83. Barsegian G., A triple principle (for curves, surfaces and complex functions) and universal version of value distribution theory, 699, 2017, pp.67-86, In book: Complex Analysis and Dynamical Systems VII, Contemporary Mathematics (USA), 2017.

    84. Barsegian G., Sergeev A. and Montes-Rodrigues A., A new sharp inequality for arbitrary complex polynomials related to Gamma-lines, Complex variables and elliptic equations, 2018.

    85. Barsegian G., A new principle for arbitrary meromorphic functions in a given domain, Georgian Math. Journal, 25, 2, 2018, p. 181-186.

    86. Barsegian G. and Yuan W., On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain, Georgian Math. Journal, 25, 2, 2018, p. 187-194.

    87. Barsegian G., Survey of Some General Properties of Meromorphic Functions in a Given Domain, p. 47-67, in book "Analysis as a Life. Dedicated to Heinrich Begehr on the Occasion of his 80th Birthday, " Editors: Rogosin S., Çelebi, O., Birkhauser, 2019.