TOPICS IN SPECIAL FUNCTIONS. II

In geometric function theory, conformally invariant extremal problems often have expressions in terms of special functions. Such problems occur, for instance, in the study of change of euclidean and noneuclidean distances under quasiconformal mappings. This fact has led to many new results on special functions. Our goal is to provide a survey of such results.

[1]  M. Vuorinen.  GEOMETRIC PROPERTIES OF QUASICONFORMAL MAPS AND SPECIAL FUNCTIONS , 2007 .

[2]  G. V. Kuzʹmina.  Moduli of families of curves and quadratic differentials , 1982 .

[3]  Topics in Special Functions , 2007, 0712.3856.

[4]  M. Vuorinen,et al.  Some inequalities for the Poincaré metric of plane domains , 2005 .

[5]  M. Vuorinen,et al.  Teichmüller's extremal ring problem , 2006 .

[6]  Matti Vuorinen,et al.  Generalized elliptic integrals and modular equations , 2000 .

[7]  Árpád Baricz.  Landen-Type Inequality for Bessel Functions , 2006 .

[8]  Extremal functions for plane quasiconformal mappings , 2003 .

[9]  Árpád Baricz,et al.  Functional inequalities involving special functions , 2006 .

[10]  J. A. Hempel.  Precise Bounds in the Theorems of Schottky and Picard , 1980 .

[11]  Distortion functions for plane quasiconformal mappings , 1988 .

[12]  S. Qiu.  Agard’s η-distortion function and Schottky’s theorem , 1997 .

[13]  Matti Vuorinen,et al.  Inequalities for quasiconformal mappings in space , 1993 .

[14]  A. Eremenko,et al.  SHARP ESTIMATES FOR HYPERBOLIC METRICS AND COVERING THEOREMS OF LANDAU TYPE , 2005 .

[15]  Iosif Pinelis,et al.  L'Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality , 2004, Am. Math. Mon..

[16]  M. Vamanamurthy,et al.  Some inequalities for the Hersch–Pfluger distortion function , 1999 .

[17]  V. N. Dubinin,et al.  Capacities of certain plane condensers and sets under simple geometric transformations , 2008 .

[18]  S. Yamashita.  Inverse functions of Grötzsch’s and Teichmüller’s modulus functions , 2003 .

[19]  Árpád Baricz,et al.  Turán type inequalities for generalized complete elliptic integrals , 2007 .

[20]  M. Taniguchi,et al.  Equivalence problem for annuli and Bell representations in the plane , 2007 .

[21]  M. Vamanamurthy,et al.  Generalized elliptic integrals and the Legendre M-function , 2007, math/0701438.

[22]  A. Vasilʹev,et al.  Moduli of Families of Curves for Conformal and Quasiconformal Mappings , 2002 .

[23]  The Distortion Theorem for quasiconformal mappings, Schottky’s Theorem and holomorphic motions , 1997 .

[24]  A. Lyzzaik.  The Modulus of the Image Annuli Under Univalent Harmonic Mappings and a Conjecture of Nitsche , 2001 .

[25]  Lokenath Debnath,et al.  A new generalized and sharp version of Jordan's inequality and its applications to the improvement of the Yang Le inequality, II , 2007, Appl. Math. Lett..

[26]  Minimal Harmonic Measure on Complementary Regions , 2003 .

[27]  The bounds of the solutions to generalized modular equations , 2006 .

[28]  M. Vuorinen,et al.  EXTREMAL PROBLEMS AND SYMMETRIZATION FOR PLANE RING DOMAINS , 1996 .

[29]  L. Ahlfors,et al.  The boundary correspondence under quasiconformal mappings , 1956 .

[30]  T. Sugawa,et al.  ON CONFORMAL REPRESENTATIONS OF THE INTERIOR OF AN ELLIPSE , 2006 .

[31]  Submultiplicative properties of the $φ_{K}$-distortion function , 1996 .

[32]  Q. Songliang.  On Mori's theorem in quasiconformal theory , 1997 .

[33]  S. Agard.  Distortion theorems for quasiconformal mappings , 1968 .

[34]  L. Ahlfors,et al.  Conformal invariants and function-theoretic null-sets , 1950 .

[35]  Q. I. Rahman,et al.  On the Monotonicity of Certain Functionals in the Theory of Analytic Functions , 1967, Canadian Mathematical Bulletin.

[36]  M. Vamanamurthy,et al.  Bounds for Quasiconformal Distortion Functions , 1997 .

[37]  Frederick W. Gehring,et al.  The coefficients of quasiconformality of domains in space , 1965 .

[38]  Quasimultiplicative properties for the η-distortion function , 1996 .

[39]  R. Barnard,et al.  The Poincaré metric and isoperimetric inequalities for hyperbolic polygons , 2005 .

[40]  M. Vuorinen,et al.  Estimates for the hyperbolic metric of the punctured plane and applications , 2001 .

[41]  Matti Vuorinen,et al.  Generalized Elliptic Integrals , 2007, math/0701436.