This article about mathematical results of F.G. Avkhadiev

Aksentev Leonid Aleksandrovich

Aptekarev A. I.

Bikchentaev  A. M.

Dubinin V. N.

Elizarov A. M.

Goryainov V. V.

Kayumov I. R.

Laptev A.

Nasyrov S. R.

Kazan Federal University Digital Repository

Russian Math. Surveys 73:1 181–185 Uspekhi Mat. Nauk 73:1 187–190DOI: https://doi.org/10.1070/RM9809MATHEMATICAL LIFEFarit Gabidinovich Avkhadiev(on his 70th birthday)On 20 August 2017 Professor Farit Gabidi-novich Avkhadiev, doctor of the physicaland mathematical sciences, observed his 70thbirthday.He was born in a family of teachers in thevillage of Chembulat in the Kzyl-Yul district ofthe Tatar Autonomous Soviet Socialist Repub-lic (now the Atnya district of the Republic ofTatarstan). In 1964 he enrolled in the Fac-ulty of Mechanics and Mathematics of Kazan’State University, where he graduated in 1969with distinction. Already during his studentyears Avkhadiev showed exceptional mathe-matical abilities, started doing research underthe supervision of Professor L.A. Aksent’ev,and published his first papers. He recalls thatafter defending his Ph.D. thesis in 1972, it wasnot in his plans to become a professional scientist. He went to Algeria and taughtmathematics there for several years. However, after returning to his country hisinterest in research grew stronger, and he proceeded to work in the ChebotarevInstitute of Mathematics and Mechanics at Kazan’ University, where he worked hisway up from a junior researcher to the head of the Department of MathematicalAnalysis and the Division of Mathematics of the institute. In 2008 he became thehead of the Department of the Theory of Functions and Approximations in the Fac-ulty of Mechanics and Mathematics at Kazan’ State University.Avkhadiev is a prominent expert in complex analysis and mathematical physics.He obtained brilliant results on sufficient conditions for univalence, inverse bound-ary value problems for analytic functions, and integral inequalities in Sobolev spacesfor plane and space domains. His acute intelligence, rich imagination, scrupulosityin investigating problems, his ability to correctly formulate a problem and proposean optimal approach to it— all these qualities soon made him a highly respectedauthority in complex analysis.AMS 2010 Mathematics Subject Classification. Primary 01A70.c⃝ 2018 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.182 Farit Gabidinovich AvkhadievHis achievements and results of his colleagues on sufficient conditions for univa-lence and related questions were presented in the surveys [1], [2], [5]–[7], [11] andthe book [3]. Here we give a brief description of the most important and interestingparts of his research work.He obtained many beautiful results related to the univalence of analytic func-tions. He found new sufficient conditions for univalence in terms of the Schwarzianderivative (f ′′/f ′)′− (1/2)(f ′′/f ′)2 and pre-Schwarzian f ′′/f ′ of a function f . Sev-eral of his conditions for univalence and p-valence of analytic functions are statedin terms of their boundary behaviour. Results by M. Morse, F.D. Gakhov, andYu.M. Krikunov on topological methods in the theory of functions of a complexvariable influenced him to extend the scope of application of these conditions byconsidering more general classes of maps such as, for example, quasi-conformalmappings and then interior mappings in the sense of Stoilow, both in the plane andin the spaces Rn.The problem of constructing a Riemann surface over a sphere given the projectionof its boundary is closely connected with these topics. It was stated by manyfamous mathematicians, such as Picard, Lo¨wner, Hopf, and others. Avkhadiev (inconjunction with S.R. Nasyrov) obtained necessary and sufficient conditions forthis problem to be solvable for boundary curves in a a fairly broad class. Togetherwith Aksent’ev and G.G. Bil’chenko he found algorithms for constructing uni- andmultivalent polygons with prescribed interior angles and applied these algorithmsto the univalent solvability of inverse boundary-value problems and the analysis ofthe geometric properties of Hilbert boundary problems.In the 1980s Avkhadiev developed a general approach to sufficient conditionsfor the uni- and p-valence of interior mappings in Rn in the sense of Stoilow, based onthe concept of p-admissible functional. This approach was one of the main advanceshe made in his D.Sc. thesis (defended in 1990, at the Institute of Mathematics ofthe Siberian Branch of the USSR Academy of Sciences). In the early 1990s heand his student I. R. Kayumov turned to an investigation of Bloch spaces, and asa result, they solved the well known Anderson–Clunie–Pommerenke problem onfinding estimates for the coefficients of the logarithms of derivatives in the class Sof univalent functions on the disk. The many years of Avkhadiev’s collaborationwith his German colleague K.-J. Wirths were very fruitful: they obtained a numberof new sharp estimates in Banach spaces of analytic functions, and in particular,they found sharp estimates that extend the classical principle of the hyperbolicmetric to higher derivatives. They presented their results in the monograph [12].Avkhadiev’s research topics changed significantly from the mid-1990s: most ofhis results are now connected with various integral inequalities for functions inSobolev spaces. The turning point was the paper [4], where he solved the generalizedSaint-Venant problem of finding a geometric characteristic that would be equivalentto the torsional stiffness of elastic beams with simply connected cross-section. Ananalogue of Hardy’s integral inequality can be derived from the inequalities in [4].Avkhadiev’s interests have been connected with this inequality and its generaliza-tions in recent years. Here we present some of these striking results.Let Ω be a proper open subset of Rn, and let δ = dist(x, ∂Ω), p > 1, ands > 1. Denote by c(p, s) = cΩ(p, s) the minimum possible constant in the integralFarit Gabidinovich Avkhadiev 183inequality ∫Ω|f |pδsdx 6 c(p, s)∫Ω|∇f |2δs−pdx, f ∈ C∞0 (Ω).For n = 2, Avkhadiev [8] found a 2-sided estimate for c(p, 2) in terms of a geometriccharacteristic of Ω which is easy to determine. For n > 2 he proved that if s > n,then the following sharp inequality holds for each p with 1 6 p <∞:cΩ(p, s) 6ps− p .This is a brilliant solution of a problem stated by J. L. Lewis (1988) and A. Wan-nebo (1990).Another result [10] computes the sharp constant λν in the inequality(14− ν2)∫Ω|f |2δ2dx+λ2νδ20∫Ω|f |2 dx 6∫Ω|∇f |2 dx, f ∈ C∞0 (Ω),for a convex domain Ω in Rn, where δ0 = supΩ δ and ν ∈ [0, 1/2] is a constant.The authors showed that λν is the first positive root of Lamb’s equation for Besselfunctions. In particular, λ0 = 0.940, which provides a sharp bound in a Hardy-typeinequality proposed by H. Brezis and M. Marcus, and λ1/2 = π/2, which general-izes and refines the well-known isoperimetric inequalities due to Poincare´ (n = 1),Hersch (n = 2), and Payne and Stakgold (n > 3).It his work Avkhadiev has always devoted considerable attention to applied prob-lems, and his analysis of them has provided him with new formulations of purelymathematical problems. In this connection we can point out his statement andsolution (with D.V. Maklakov) of the inverse problem of recovering a hydrofoilsection from a given cavitation diagram or the problem of maximizing the criticalMach number for aerofoil sections (together with A.M. Elizarov and D.A. Fokin).Avkhadiev is an excellent lecturer and teacher. He presented results of hisresearch in a clear and accessible form in his textbooks [9] and [14]. His stu-dents have defended six Ph.D. theses, and he has been the scientific advisor for twoD.Sc. theses.Mention should be made of the considerable time and effort he devotes to hisposition as the editor-in-chief of the journal Izvestiya Vysshikh Uchebnykh Zave-denii.Matematika.1Avkhadiev’s interests outside mathematics are also very broad. He is interestedin history, politics, and literature, and collects notable sayings of prominent scholars(some of these sayings decorate his office). Despite his numerous posts and duties,he finds time for outdoor activities. For many years he pursued whitewater sportstogether with friends and colleagues on catamarans, down swiftly flowing rivers inthe Volga Region, the Urals, and Siberia. In some extreme situations during thesetrips Avkhadiev’s best qualities became apparent: his calmness, self-confidence,openness, charm, and his philosophical attitude to life.Farit Avkhadiev is a family man, a wonderful husband and father, selflesslydevoted to his wife Nina Vasil’evna and daughter Kamila. He enjoys telling storiesof his little grandson in Moscow, and is proud of his successes.1Translated as Russian Mathematics (Izvestiya VUZ. Matematika).184 Farit Gabidinovich AvkhadievFrom the bottom of our hearts we wish Farit Gabidinovich and his family everysuccess, happiness, and good health.L.A. Aksent’ev, A.I. Aptekarev, A.M. Bikchentaev, V.V. Goryainov,V.N. Dubinin, A.M. Elizarov, I.R. Kayumov, A. Laptev,S.R. Nasyrov, D.V. Prokhorov, A.G. Sergeev, and V.D. StepanovSelected publications of F.G. Avkhadiev[1] “Основные результаты в достаточных условиях однолистности аналитическихфункций”, УМН 30:4(184) (1975), 3–60 (совм. с Л.А. Аксентьевым); Englishtransl., “The main results on sufficient conditions for an analytic function to beschlicht”, Russian Math. Surveys 30:4 (1975), 1–63 (with L.A. Aksent’ev)[2] “Достаточные условия конечнолистности аналитических функций и ихприложения”, Итоги науки и техн. Сер. Матем. анал., 25, ВИНИТИ, М.1987, с. 3–121 (совм. с Л.А. Аксентьевым, А.М. Елизаровым); Englishtransl., “Sufficient conditions for the finite-valence of analytic functions and theirapplications”, J. Soviet Math. 49:1 (1990), 715–799 (with L.A. Aksent’ev andA.M. Elizarov)[3] Конформные отображения и краевые задачи, Изд-во Казанский фонд“Математика”, Казань 1996, 216 с. [Conformal mappings and boundary-valueproblems, Kazanskii Fond “Matematika”, Kazan’ 1996, 216 pp.][4] “Решение обобщенной задачи Сен-Венана”, Матем. сб. 189:12 (1998), 3–12;English transl., “Solution of the generalized Saint Venant problem”, Sb. Math.189:12 (1998), 1739–1748[5] “Научный семинар по геометрической теории функций: основные результатыдвух последних десятилетий”, Тр. Матем. центра им. Н.И. Лобачевского,14, Казан. матем. о-во, Казань 2002, с. 7–38 (совм. с Л.А. Аксентьевым,А.М. Елизаровым, С. Р. Насыровым) [“A research seminar on the geometrictheory of functions: basic results of the last two decades”, Tr. Mat. Tsentra im.N. I. Lobachevskogo, vol. 14, Kazan’ Math. Society, Kazan’ 2002, pp. 7–38 (withL.A. Aksent’ev, A.M. Elizarov, and S.R. Nasyrov)][6] “Исследования по теории функций и изопериметрическим задачам”, Нарубеже веков. НИИММ Казанского университета, Изд-во Казан. матем.о-ва, Казань 2003, с. 37–50 (совм. с И.Р. Каюмовым, Р. Г. Салахудиновым)[“Investigations in the theory of functions and isoperimetric problems”, At theturn of the 20th-21st centuries. Research Institute of Mathematics and Mechanicsat Kazan’ University, Kazan’ Math. Society, Kazan’ 2003, pp. 37–50 (withI. R. Kayumov and R.G. Salakhudinov)][7] “Геометрическая теория функций”, История науки в Казанскомуниверситете, 1980–2003, Изд-во КГУ, Казань 2005, с. 15–20 [“Geometricfunction theory”, History of science at Kazan’ University, 1980–2003, Publishinghouse of Kazan’ State University, Kazan’ 2005, pp. 15–20][8] “Hardy type inequalities in higher dimensions with explicit estimate of constants”,Lobachevskii J. Math. 21 (2006), 3–31[9] Неравенства для интегральных характеристик областей, Учеб. пособие,Казан. гос. ун-т, Казань 2006, 140 с. [Inequalities for integral characteristics ofdomains, Textbook, Kazan’ State University, Kazan’ 2006, 140 pp.][10] “Unified Poincare´ and Hardy inequalities with sharp constants for convex domains”,ZAMM Z. Angew. Math. Mech. 87:8-9 (2007), 632–642 (with K.-J. Wirths)Farit Gabidinovich Avkhadiev 185[11] “Экстремальные проблемы, связанные с интегральными характеристиками”,НИИММ им. Н.Г. Чеботарева КГУ. 2003–2007 годы, Изд-во КГУ, Казань 2008,с. 36–53 [“Extremal problems connected with integral characteristics”, ChebotarevResearch Institute of Mathematics and Mechanics at Kazan’ University. Years2003–2007, Publishing house of Kazan’ State University, Kazan’ 2008, pp. 36–53][12] Schwarz–Pick type inequalities, Front. Math., Birkha¨user, Basel 2009, viii+156 pp.(with K.-J. Wirths)[13] С. Р. Насыров (ред.), “Кафедра теории функций и приближений”,Механико-математический факультет Казанского университета.Очерки истории, изд. 3-е, перераб. и доп., Казан. ун-т, Казань 2011,с. 104–113 [S.R. Nasyrov (ed.), “Department of the Theory of Functions andApproximations”, Faculty of Mechanics and Mathematics of Kazan’ University.Essays on history, 3d rev. aug. ed., Kazan’ University, Kazan’ 2011, pp. 104–113][14] Введение в геометрическую теорию функций, Учеб. пособие, Казан. ун-т,Казань 2012, 127 с. [Introduction to geometric function theory, Textbook, Kazan’University, Kazan’ 2012, 127 pp.][15] “Геометрическое описание областей, для которых константа Харди равна 1/4”,Изв. РАН. Сер. матем. 78:5 (2014), 3–26; English transl., “A geometricdescription of domains whose Hardy constant is equal to 1/4”, Izv. Math. 78:5(2014), 855–876[16] “Интегральные неравенства в областях гиперболического типа и ихприменения”, Матем. сб. 206:12 (2015), 3–28; English transl., “Integralinequalities in domains of hyperbolic type and their applications”, Sb. Math. 206:12(2015), 1657–1681Translated by N. KRUZHILIN

Farit Gabidinovich Avkhadiev (on his 70th birthday)

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Farit Gabidinovich Avkhadiev (on his 70th birthday)

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